Singularly Perturbed Boundary Value Problems: A Functional Analytic Approach 3030762580, 9783030762582

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Matteo Dalla Riva Massimo Lanza de Cristoforis Paolo Musolino

Singularly Perturbed Boundary Value Problems A Functional Analytic Approach

Singularly Perturbed Boundary Value Problems

Matteo Dalla Riva • Massimo Lanza de Cristoforis Paolo Musolino

Singularly Perturbed Boundary Value Problems A Functional Analytic Approach

Matteo Dalla Riva College of Engineering and Natural Science The University of Tulsa Tulsa, OK, USA

Massimo Lanza de Cristoforis Dipartimento di Matematica Universit` a degli Studi di Padova Padova, Italy

Paolo Musolino Dipartimento di Scienze Molecolari e Nanosistemi Universit`a Ca’ Foscari Venezia Venezia, Italy

ISBN 978-3-030-76258-2 ISBN 978-3-030-76259-9 (eBook) https://doi.org/10.1007/978-3-030-76259-9 Mathematics Subject Classification: 31B10, 35B10, 35B25, 35B30, 35C15, 35C20, 35J25, 35J66, 35P15, 45P05, 46N20, 47H30, 47G40, 42B20 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To our families

Preface

This book is the first introductory and self-contained presentation of a method for the analysis of singularly perturbed boundary value problems that we have called the Functional Analytic Approach. The method was proposed some 20 years ago by the second author and, since then, it has developed in many directions and been implemented in linear and nonlinear problems, including problems that arise from Continuum Mechanics and Material Sciences. Before the publication of this book, the presentation of the theoretical aspect and of the applications of the Functional Analytic Approach was disseminated in a number of different journal articles. This book is the first comprehensive introduction to the topic, and it covers both the theoretical material that stands at the basis of the Functional Analytic Approach and its applications to a series of problems that ranges from simple illustrative examples to more involved research results. The Functional Analytic Approach makes constant use of Potential Theory, of the integral representation method for the solutions of boundary value problems, of the Theory of Analytic Functions both in finite and infinite dimension, and of Nonlinear Functional Analysis. All theoretical results used in the Functional Analytic Approach are collected in the first seven chapters of the book and in an appendix at the end of the book. Despite the fact that these initial chapters and the appendix are mostly self-contained, we did not include proofs that can be found in widely available textbooks and for which we cannot claim any paternity. Some classical results have, however, been revisited in a way that fits our purposes, and the corresponding proofs have been presented in full detail. We begin describing the Functional Analytic Approach in Chapter 8, where we show its application to a Dirichlet problem for the Laplace equation in a domain with a hole that shrinks to a point. In the chapters that follow Chapter 8, we study different kinds of boundary conditions, including nonlinear boundary conditions, and different geometric settings, including unbounded domains with periodic sets of holes or inclusions. To keep the length of the book within reasonable limits, we excluded some of the directions along which the Functional Analytic Approach has developed. In particular, we do not treat differential equations other than the Laplace and Poisson equations and we stick to domains with shrinking holes. For the analysis involving vii

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different equations and, in particular, elliptic systems, and for the study of different domain perturbations, we refer the reader to the works of the authors and their collaborators in the bibliography. Throughout the book we have tried to keep the presentation at a level which we think adequate for the graduate student. As for prerequisites, we expect the reader to be familiar with basic concepts of Lebesgue integration for functions of several variables and of Lp spaces, with some elementary Functional Analysis, and with Differential Calculus in normed spaces. We have tried to keep the use of Distribution Theory to a minimum. In addition, the book provides references to all accessory results. The book is designed to serve various purposes. • The large introductory part can be used as a reference textbook for graduate courses on classical Potential Theory and its applications to boundary value problems for the Laplace and Poisson equations. Students and instructors can find, here, material on Lipschitz and Schauder functions and sets (Chapter 2), Green formulas and layer potentials (Chapter 4), Potential Theory for the Laplace equation in multiconnected domains (Chapter 6), and volume potentials and their applications to the Poisson equation (Chapter 7). The periodic counterpart of these results of Potential Theory is presented later in the book, in Chapter 12. • The initial chapters also contain results that are rarely presented in the literature and may also, therefore, attract the interest of more expert readers. For example, in Section 4.3 we present a theorem on the H¨older continuity of singular integrals of convolution type. This result was originally proven by Carlo Miranda [212], but the corresponding paper was published in Italian and, to the best of our knowledge, has never appeared in English and in the “modern” form included in the book. • In Chapter 8 we start introducing the Functional Analytic Approach. A reader looking for a quick introduction to the method can find simple illustrative examples specifically designed for this purpose in Section 8.3, where we study the Dirichlet problem in dimension three or bigger; Section 9.2, where we consider a mixed problem for the Laplace equation; Section 9.3, about a mixed problem for the Poisson equation; Section 10.3, on a problem with two “moderately close” holes; and Section 11.2, for the analysis of a nonlinear problem. • More expert readers will find a comprehensive presentation of the Functional Analytic Approach, which allows a comparison between our approach and the more classical expansion methods of Asymptotic Analysis and offers insights on the specific features of the approach and its applications to linear boundary value problems (Chapters 8, 9, and 10), nonlinear boundary value problems (Chapter 11), and problems in unbounded periodic domains (Chapter 13). • Finally, applications of the method to problems that arise in the study of composite or porous materials, and more specifically in the study of the average properties of such materials, can be found in Section 13.5, on the longitudinal flow along a periodic array of thin cylinders, and in Sections 13.6 and 13.7, on the effective conductivity of a composite with a thermal resistance at the materials interface.

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We would like to thank the students and colleagues that helped us in writing this book. In particular, we acknowledge the help received from Prof. Alberto Cialdea, Prof. David Natroshvili, Prof. Sergei V. Rogosin, and Prof. Promarz M. Tamrazov on technical issues related to the use of integral equations and Potential Theory for the solution of boundary value problems. We are indebted to Prof. Joan Verdera for several recent references on issues related to the theory of singular integrals in H¨older spaces. We also mention the help received from Prof. Mark L. Agranovsky on technical issues related to the theory of Analytic Functions, and the help of Prof. G´erard Bourdaud, Prof. Victor I. Burenkov, and Prof. Winfried Sickel on issues related to Function Space Theory. Finally, we wish to thank Dr. Tu˘gba Akyel, Dr. Riccardo Falconi, Dr. Paolo Luzzini, Prof. Pier Domenico Lamberti, Dr. Riccardo Molinarolo, Mr. Jonathan Pinkey, and Dr. Roman Pukhtaievych for the generous gift of their time in proofreading preliminary versions of the book and making precious suggestions and remarks. Dr. Paolo Luzzini also prepared the three pictures in the book, for which we owe him additional thanks. Tulsa, OK, USA

Matteo Dalla Riva

Padova, Italy

Massimo Lanza de Cristoforis

Venezia, Italy March 2021

Paolo Musolino

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 A Selection of Problems . . . . . . . . . . . . . . . . . . 1.2.1 Perturbation Problems for the Riemann Map 1.2.2 Linear Elliptic Boundary Value Problems . . 1.2.3 Eigenvalues Problems . . . . . . . . . . . . . . 1.2.4 Nonlinear Boundary Value Problems . . . . . 1.2.5 Problems in Periodic Domains . . . . . . . . . 1.2.6 Different Boundary Perturbations . . . . . . . 1.2.7 Perturbation Results for Integral Operators . 1.3 Structure of the Book . . . . . . . . . . . . . . . . . . .

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1 4 6 7 7 7 8 8 8 9 9

2

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Preliminaries of Linear Functional Analysis . . . . . . . . . . . . . . 2.3 Spaces of Classically Differentiable Functions . . . . . . . . . . . . . 2.4 Distributions and Weak Derivatives . . . . . . . . . . . . . . . . . . . . 2.5 Real Analytic Functions and Spaces of Real Analytic Functions . . 2.6 Spaces of H¨older and Lipschitz Continuous Functions . . . . . . . . 2.7 Coordinate Cylinders and Local Strict Hypographs . . . . . . . . . . 2.8 Tangent Space to a Local Strict Hypograph . . . . . . . . . . . . . . . 2.9 Lipschitz Subsets of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Elementary Inequalities on the Boundary of a Lipschitz Subset of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Schauder Spaces in Open Subsets of Rn . . . . . . . . . . . . . . . . . 2.12 Composition of Functions in Schauder Spaces . . . . . . . . . . . . . 2.13 Local Strict Hypographs of a Schauder Class . . . . . . . . . . . . . . 2.14 Extendibility of Functions of Schauder Spaces on an Open Subset of Class C m,α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15 On the Extendibility of Continuous Functions to the Closure of Open Sets of Class C 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.16 A Consequence of the Rule of Change of Variables for Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.17 A Fundamental Inequality of the Unit Normal on the Boundary of a Set of Class C 1,α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.18 Existence of Tubular Neighborhoods of the Boundary of Bounded Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 2.19 A Sufficient Condition for the H¨older Continuity of Continuously Differentiable Functions, in the Wake of the Work of Carlo Miranda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.20 Schauder Spaces on a Compact Manifold and on the Boundary of a Bounded Open Subset of Rn . . . . . . . . . . . . . . . . . . . . . . . 99 2.21 Tangential Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2.22 Schauder Spaces in Open Subsets of Rn , a Case of a Negative Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3

Preliminaries on Harmonic Functions . . . . . . . . . . 3.1 Basic Properties of Harmonic Functions . . . . . . . 3.2 A Fundamental Solution for the Laplace Operator . 3.3 Isolated Singularities of Harmonic Functions . . . . 3.4 Behavior at Infinity of Harmonic Functions . . . . .

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Green Identities and Layer Potentials . . . . . . . . . . . . . . . . . . 4.1 Green Identities for Bounded Domains . . . . . . . . . . . . . . . . 4.2 Green Identities for Harmonic Functions on Exterior Domains . 4.3 Preliminaries on Singular Integrals and Layer Potentials . . . . . 4.4 The Single Layer Potential . . . . . . . . . . . . . . . . . . . . . . . 4.5 The Double Layer Potential . . . . . . . . . . . . . . . . . . . . . . 4.6 A Regularizing Property of the Double Layer Potential on the Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Preliminaries on the Fredholm Alternative Principle . . . . . . . . . . 175 5.1 Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.2 Fredholm Alternative in a Dual System . . . . . . . . . . . . . . . . . 176

6

Boundary Value Problems and Boundary Integral Operators . . . . 6.1 The Geometric Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Dirichlet and Neumann Boundary Value Problems . . . . . . . 6.3 Uniqueness for the Interior and Exterior Dirichlet and Neumann Boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Boundary Integral Operators Associated to the Single and Double Layer Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The Null Spaces of 12 I + WΩ and 12 I + WΩt . . . . . . . . . . . . . 6.6 The Null Spaces of − 12 I + WΩ and − 12 I + WΩt . . . . . . . . . . 6.7 The Dirichlet Problem in Ω . . . . . . . . . . . . . . . . . . . . . . . 6.8 The Dirichlet Problem in Ω − . . . . . . . . . . . . . . . . . . . . . . 6.9 The Neumann Problem in Ω and Ω − . . . . . . . . . . . . . . . . . .

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183 186 191 195 201 206

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6.10 6.11 6.12 6.13 6.14 7

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Further Mapping Properties of VΩ . . . A Mixed Boundary Value Problem . . . The Operators I + λWΩ and I + λWΩt A Linear Transmission Problem . . . . . A Robin Problem . . . . . . . . . . . . . .

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Poisson Equation and Volume Potentials . . . . . . . . . . . . . . . . . . 7.1 Preliminary Remarks on the Poisson Equation . . . . . . . . . . . . . 7.2 Volume Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Volume Potentials with Weakly Singular Kernels . . . . . . 7.2.2 Volume Potentials with Kernels Which are Weakly Singular Together with Their First Order Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Volume Potentials with Singular Kernels and with a Constant Density . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Volume Potentials with Kernels Which are Weakly Singular and Which Have a Strong Singularity in the First Order Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . 7.2.5 The Newtonian Potential in Schauder Spaces . . . . . . . . . 7.2.6 Volume Potentials in Roumieu Classes . . . . . . . . . . . . . 7.3 Boundary Value Problems for the Poisson Equation in Schauder Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 The Interior Dirichlet Problem for the Poisson Equation in Schauder Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 The Interior Neumann Problem for the Poisson Equation in Schauder Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 The Interior Robin Problem for the Poisson Equation in Schauder Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A Dirichlet Problem in a Domain with a Small Hole . . . . . . . 8.1 The Geometric Setting . . . . . . . . . . . . . . . . . . . . . . . . 8.2 A Dirichlet Problem for the Laplace Equation . . . . . . . . . 8.3 Analysis for n ≥ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Analysis for n = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Analysis of System (8.32) . . . . . . . . . . . . . . . . . 8.4.2 Analysis of System (8.33) . . . . . . . . . . . . . . . . . 8.4.3 Real Analytic Representation of the Map  → u . . . 8.4.4 Some Remarks on the Logarithmic Behavior . . . . . 8.5 How to Compute the Coefficients (in Dimension 2) . . . . . . 8.5.1 Series Expansions of (Φi [], Φo []) and (Ψ i [], Ψ o []) 8.5.2 Series Expansion of u . . . . . . . . . . . . . . . . . . . 8.5.3 Principal Terms in the Series Expansion of u . . . . 8.5.4 Series Expansion for the Energy of u . . . . . . . . . 8.5.5 Series Expansions in a Circular Annulus . . . . . . . .

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Contents

Other Problems with Linear Boundary Conditions in a Domain with a Small Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 The Geometric Setting . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 A Mixed Boundary Value Problem for the Laplace Equation . . 9.3 A Mixed Boundary Value Problem for the Poisson Equation . . 9.4 A Steklov Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . 9.4.1 Some Basic Facts on Steklov Eigenvalues and Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Formulation of the Steklov Problem (9.31) in Terms of Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Real Analytic Representations for the Simple Steklov Eigenvalues and Eigenfunctions . . . . . . . . . . . . . . .

10

A Dirichlet Problem in a Domain with Two Small Holes . . . . 10.1 The Geometric Setting . . . . . . . . . . . . . . . . . . . . . . . . 10.2 A Dirichlet Problem in Ω(1 , 2 ) . . . . . . . . . . . . . . . . . 10.3 Close and Moderately Close Holes in Dimension n ≥ 3 . . . 10.3.1 Moderately Close Holes in Dimension n ≥ 3 . . . . . 10.3.2 Close Holes in Dimension n ≥ 3 . . . . . . . . . . . . 10.4 Moderately Close Holes in Dimension n = 2 . . . . . . . . . . 10.4.1 Integral Representation of the Solution . . . . . . . . . 10.4.2 Analysis of System (10.39) . . . . . . . . . . . . . . . . 10.4.3 Analysis of System (10.40) . . . . . . . . . . . . . . . . o ξ ξ , HΩ , and HxΩ . . . . 10.4.4 The Auxiliary Functions HΩ 1 2 10.4.5 Representation of u1 ,2 in Terms of Analytic Maps . 10.4.6 Asymptotic Behavior of u1 ,2 as (1 , 2 ) → (0, 0) .

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11

Nonlinear Boundary Value Problems in Domains with a Small Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Geometric Setting . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 A Nonlinear Robin Problem . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Formulation of a Nonlinear Robin Problem in Terms of Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Formulation of Problems (11.1) and (11.2) in Terms of Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Analytic Representation for the Family {u(, ·)}∈]0, [ . 11.2.4 Local Uniqueness of the Family {u(, ·)}∈]0,0 [ . . . . . 11.2.5 Analytic Representation for the Energy Integral of the Family {u(, ·)}∈]0, [ . . . . . . . . . . . . . . . . . . . . . 11.3 A Nonlinear Transmission Problem . . . . . . . . . . . . . . . . . . 11.3.1 Formulation of the Nonlinear Transmission Problem in Terms of Integral Equations . . . . . . . . . . . . . . . . . . 11.3.2 Analytic Representation for the Family of Solutions {(ui (, ·), uo (, ·))}∈]0, [ . . . . . . . . . . . . . . . . . . .

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11.3.3 A Property of Local Uniqueness for the Family of Solutions {(ui (, ·), uo (, ·))}∈]0, [ . . . . . . . . . . . . . . 474 11.3.4 Analytic Representation for the Energy Integrals of the Family of Solutions {(ui (, ·), uo (, ·))}∈]0, [ . . . . . . . . 479 12

13

Boundary Value Problems in Periodic Domains, a Potential Theoretic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 A Periodic Analog of the Fundamental Solution . . . . . . . . . . . 12.2 Periodic Layer Potentials for the Laplace Equation . . . . . . . . . 12.2.1 Geometric Setting . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Definition and Properties of the Periodic Layer Potentials 12.3 Uniqueness Results for Periodic Boundary Value Problems . . . . t ..... 12.4 Mapping Properties of ± 12 I + Wq,ΩQ and ± 12 I + Wq,Ω Q 12.5 Existence Results for Periodic Boundary Value Problems . . . . . Singular Perturbation Problems in Periodic Domains . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 The Geometric Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Perturbed Problems in Periodic Domains . . . . . . . . . . . . . . . 13.4 Preliminaries and Notation . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Asymptotic Behavior of the Longitudinal Flow . . . . . . . . . . . 13.5.1 Asymptotic Behavior of ΣII [] . . . . . . . . . . . . . . . . . 13.6 A Singularly Perturbed Non-ideal Transmission Problem . . . . . 13.6.1 Transmission Problems with Non-ideal Contact Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6.2 Formulation of the Singularly Perturbed Transmission Problem in Terms of Integral Equations . . . . . . . . . . . 13.6.3 A Functional Analytic Representation Theorem for the Solutions of the Singularly Perturbed Transmission Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6.4 A Functional Analytic Representation Theorem for the Effective Conductivity . . . . . . . . . . . . . . . . . . . . . . 13.7 Series Expansion for the Effective Conductivity . . . . . . . . . . . 13.7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7.2 Power Series Expansion for ρ() ≡ 1/r# . . . . . . . . . . 13.7.3 Power Series Expansions for ρ() ≡ /r# . . . . . . . . . . 13.8 A Quasilinear Heat Transmission Problem . . . . . . . . . . . . . . 13.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.8.2 An Equivalent Formulation of Problem (13.132) . . . . . . 13.8.3 Formulation of Problem (13.135) in Terms of Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.8.4 A Representation Theorem for the Family of Solutions of Problem (13.132) . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 The Homomorphism Theorem . . . . . . . . . . . . . . . . . . . . . A.2 The Inductive Topology . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Lebesgue Number of an Open Cover . . . . . . . . . . . . . . . . . A.4 Perforated Connected Domains Are Connected . . . . . . . . . . A.5 Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Calculus in Banach Spaces and the Implicit Function Theorem . A.7 Composition Operators . . . . . . . . . . . . . . . . . . . . . . . . . A.8 Integral Operators with Real Analytic Kernel . . . . . . . . . . . . A.9 Sard’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.10 Theorem of Invariance of Domain . . . . . . . . . . . . . . . . . . A.11 Mollifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.12 The Partition of Unity . . . . . . . . . . . . . . . . . . . . . . . . . .

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615 615 615 617 617 619 620 633 635 645 646 646 647

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667

Chapter 1

Introduction

This book is devoted to the analysis of the basic boundary value problems for the Laplace equation in singularly perturbed domains. More precisely, we will focus on domains with small holes and develop an alternative approach to those of Asymptotic Analysis. While one could develop most of the book’s material for Lipschitz domains and a large class of elliptic operators, we have decided to write a primer of sorts, one easily accessible to beginners. Therefore, we have tried to explain our point of view under the most elementary conditions possible: we treat domains of class C 1,α for some α ∈]0, 1[ (the classical Lyapunov sets) and the model of elliptic operators, i.e., the Laplace operator. In a typical problem found in this book, we have a bounded open subset Ω o of class C 1,α of Rn and we assume that Ω o contains the origin. Then we consider a certain boundary value problem in Ω o , which we assume has a unique solution uo . Here the superscript ‘o’ stands for ‘outer.’ (Fittingly, the superscript ‘i’ will stand for ‘inner’.) Next we make in Ω o a hole of size  > 0 around the origin. For example, we can remove from Ω o a ball centered at the origin and with radius . We denote by Ω() the perforated domain that we obtain and we consider in Ω() a boundary value problem which is, in a sense, of the same nature of the problem on Ω o . For example, we can have a Dirichlet problem for the same equation both in Ω() and in Ω o . We assume that the problem in Ω() also has a unique solution u . At this point, we can say that the problem in Ω o is ‘unperturbed’, and that the problem in Ω() has been perturbed by the presence of the hole. It is then natural to ask how the solution u , or its energy integral, or other functionals associated with u behave when  approaches 0. The difficulty is that the limiting set Ω o \ {0} has a nature which is different from that of Ω() and, in particular, Ω o \ {0} is no longer a set of class C 1,α . As can be seen in a large body of mathematical literature, problems of this type are not only interesting in themselves, but also have importance in Continuum Mechanics and in applied sciences and engineering. For example, they find several applications in shape and topological optimization problems and in the inverse prob© Springer Nature Switzerland AG 2021 M. Dalla Riva et al., Singularly Perturbed Boundary Value Problems, https://doi.org/10.1007/978-3-030-76259-9 1

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2

1 Introduction

lems related to nondestructive testing. A description of such applications can be found in the monographs of Novotny and Sokołowski [240] and of Ammari and Kang [8], which contain many further references on the topic. In an effort to approximate u or related functionals, a commonly accepted way to tackle the problem is to figure out an asymptotic expansion of u for  > 0 small. So for example, we fix a point p of the unperturbed set, with that point being away from the hole, and we write an equality of the type u (p) = a(γ1 (), . . . , γs ()) + R()

as  tends to 0+ ,

(1.1)

where s is a natural number, a(·, . . . , ·) is an ‘approximating’ function of s real variables defined in a neighborhood of the origin in Rs , γ1 , . . . , γs are s explicitly known functions which tend to zero as  tends to 0+ , such as  ,  log  , log−1  , and so on, and R() is a reminder term. Here p is kept fixed and thus we did not include it as a variable in the right-hand side of (1.1). Now the task is to show that the expansion we have written is actually of some use. For example, to show that (1.1) gives an approximation of the unknown u (p) by means of a(γ1 (), . . . , γs ()), we have to prove that the reminder term R() tends to zero more rapidly than certain functions of γ1 (),. . . , γs () as  tends to zero. The faster R() tends to zero, the better is our approximation. As is well known, guessing the explicit form of the functions γ1 ,. . . , γs and of the function a is a part of the problem that may be utterly nontrivial, especially if the problem we are looking at is nonlinear. In some cases, one can expect that a is a polynomial and that (1.1) takes the form  u (p) = aβ γ1 ()β1 . . . γs ()βs + Rr () as  tends to 0+ . (1.2) |β|≤r

If that is the case, then the coefficients aβ can usually be computed with means at our disposal, after which we wish to show that the reminder Rr () converges to zero faster than order r with respect to γ1 , . . . , γs . There has been an enormous effort in the mathematical literature to accomplish the plan described above, not only for problems with a shrinking hole, but also for singular perturbations of Ω o generated by the formation of peaks, wedges, slits, thin bridges, and so on. The most common approach adopted in the literature is that of the expansion methods of the so-called Asymptotic Analysis. The large number of authors contributing to the development of such methods prevents us from attempting any complete body of references. Nevertheless, we want to mention the early results in the monographs of Cherepanov [48] and [49] on the formation of cracks and in the books of Nayfeh [234], Van Dyke [270], and Cole [57], which present an extensive review of the expansion methods known at the time. For the rigorous description of the method of matching outer and inner asymptotic expansions we refer to the book

1 Introduction

3

of Il’in [127], and for the Compound Expansion Method (also known as MultiScale Expansion Method) we mention the two volumes of Mazya et al. [203, 204] where, among other results, the authors introduce a systematic approach for analyzing general Douglis and Nirenberg elliptic boundary value problems in domains with perforations and corners. Different domain perturbations can be found in Kozlov et al. [146], where the authors consider boundary value problems in domains depending on a small parameter  in such a way that the limit regions as  tends to zero consist of subsets of different space dimensions. For the analysis of problems in domains with two holes that collide into one another while shrinking in size we mention the works of Bonnaillie-No¨el et al. [32] and Bonnaillie-No¨el et al. [31] and for the analysis of problems in a domain containing ‘clouds’ of small holes we refer to the book of Maz’ya et al. [200], where the authors exploit Meso-Scale Asymptotic Approximations to compute the asymptotic expansion of Green kernels. We also mention Dauge et al. [85] where the authors consider self-similar perturbations of a corner domain and make a comparison between the Multi-Scale and the Matched Asymptotic Expansion Methods. Further references on expansion methods and their applications can be found in the books and papers mentioned above. Expansion methods are not, however, the only means available for dealing with domain perturbations. For example, in the two dimensional case one can resort to complex analytic techniques, as in the works of Mityushev et al. [221] and [224] and Rogosin and Vaitekhovich [251]. The Functional Analytic Approach of this book can be considered as a complement to expansion methods, though we take, in a sense, a different standpoint. We no longer look for asymptotic expansions such as (1.1) but, in the example above, we look for representation formulas of the form u (p) = A(γ1 (), . . . , γs ())

for  > 0 small enough ,

(1.3)

where A(·, . . . , ·) is a real analytic function of s real variables defined in a neighborhood of the origin in Rs and γ1 ,. . . , γs are explicitly known functions which tend to zero as  tends to 0+ (just as in (1.1)). In particular, formula (1.3) implies that u (p) =

 Dβ A(0, . . . , 0) γ1 ()β1 . . . γs ()βs β! s

β∈N

for  > 0 sufficiently small, and that accordingly u (p) has an asymptotic expansion in terms of the powers of γ1 (), . . . , γs (). Namely, for each natural r, we have  cβ γ1 ()β1 . . . γs ()βs + Rr () (1.4) u (p) = |β|≤r

for  > 0 sufficiently small, where Rr () converges to zero faster than order r with respect to γ1 ,. . . , γs , and where the coefficients cβ are delivered by the equality cβ = Dβ A(0, . . . , 0)/β!.

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1 Introduction

We emphasize that the main point of this book is demonstrating the existence of the function A for a suitable choice of γ1 ,. . . , γs . The matter of computing A or the coefficients cβ is secondary from our point of view, although we regard such a problem as being of great importance in various applications, and we do also consider it in some cases. We also note that a formula such as (1.3) conveys the message that if we look for an asymptotic expansion of the form (1.2) and if we wish to make the approximation actually converge to u (p) when r tends to +∞ and  is fixed, then an effective choice of the functions γ1 , . . . , γs is that given by formula (1.3). Especially in the case of nonlinear problems, such information may be of interest for people who look for asymptotic expansions by means of Asymptotic Analysis techniques.

1.1 An Example To better illustrate our point of view, we now introduce a concrete example where our ‘unperturbed’ problem is a Dirichlet boundary value problem on Ω o with a boundary datum g o ∈ C 1,α (∂Ω o ) (cf. [163], [166]). Namely,  Δu = 0 in Ω o , (1.5) u = g o on ∂Ω o . As is well known, problem (1.5) has one and only one classical solution uo ∈ C 1,α (Ω o ). In addition to the assumptions introduced above, we assume that the domain Ω o has a connected exterior. To put it simply, Ω o has no holes. Then we perturb problem (1.5) by introducing a hole in Ω o . To do so, we fix another domain Ω i which satisfies the same assumptions as Ω o . The set Ω i will serve as a reference domain for the hole, the letter ‘i’ standing for ‘inner.’ Indeed, for 0 > 0 small enough we have Ω i ⊆ Ω o

∀ ∈ [−0 , 0 ]

(we recall that Ω i and Ω o contain the origin). Accordingly, for  ∈ [−0 , 0 ] we can define the perforated domain Ω() ≡ Ω o \ Ω i . We observe that the boundary ∂Ω() consists of two connected components, ∂Ω i

and

∂Ω o .

Also, if  shrinks to 0, then Ω() degenerates to the punctured domain Ω(0) = Ω o \ {0}.

1.1 An Example

5

Our next step is to introduce a Dirichlet problem in Ω(). To assign the Dirichlet datum on the component ∂Ω i of the boundary, we fix a function g i ∈ C 1,α (∂Ω i ) and we rescale it. Then, for each  ∈]0, 0 [ the ‘perturbed’ problem is given by ⎧ in Ω() , ⎨ Δu = 0 u(x) = g i (x/) ∀x ∈ ∂Ω i , (1.6) ⎩ on ∂Ω o . u = go Like problem (1.5), problem (1.6) also has a unique classical solution in C 1,α (Ω()). We denote it by u and we wish to study its dependence upon . To do so, we fix a point p in Ω o \ {0}. Then we observe that, possibly shrinking 0 , we can assume that p ∈ Ω() ∀ ∈]0, 0 [ . Accordingly, u (p) is defined for all  ∈]0, 0 [ and we can ask the question: What can we say about u (p) as  > 0 approaches 0? The answer given in the book accords with our introductory explanation and turns out to be dependent on the dimension n. In Chapter 8, we will prove that for n ≥ 3 the map which takes  to u (p) has a real analytic continuation in a neighborhood of zero. Namely, there exist p > 0 and a real analytic function Up from ] − p , p [ to R such that ∀ ∈]0, p [. (1.7) u (p) = Up () Then, with reference to formula (1.3), we have the case where s = 1 and γ1 () = . If instead the dimension n is two, then we will show that there are three real analytic functions Up , Wp , and R from ] − p , p [ to R such that u (p) = Up () +

Wp ()  R() + log 2π

∀ ∈]0, p [ .

(1.8)

The presence of the logarithmic term implies that the right-hand side cannot be continued analytically in a neighborhood of zero, unless we consider some special cases where Wp () = 0. However, if we divide both the numerator and the denominator of the quotient by log , we obtain u (p) = Up () +

Wp () log1  R() log1  +

1 2π

∀ ∈]0, p [ .

Then we observe that for tp > 0 small enough we have |R()t| < and accordingly

1 2π

∀(, t) ∈] − p , p [×] − tp , tp [

6

1 Introduction

R()t + 1/(2π) = 0

∀(, t) ∈] − p , p [×] − tp , tp [ .

It follows that the function A from ] − p , p [×] − tp , tp [ to R which takes a pair (, t) to Wp ()t A(, t) ≡ Up () + 1 R()t + 2π is real analytic. Also, possibly shrinking p , we can assume that 1 ∈] − tp , tp [ log  Then we conclude that



1 u (p) = A , log 

∀ ∈]0, p [ .

 ∀ ∈]0, p [ .

With reference to formula (1.3), we now have the case where s = 2, γ1 () = , and γ2 () = 1/log . As a consequence, the function u (p) can be written as an analytic function of  and 1/log . So, the message conveyed to us here is that, if we want to write an asymptotic expansion associated with a convergent series, we should try using an expansion of powers of  and 1/log . In Chapter 8, we explain how to compute the Taylor coefficients of the analytic functions which appear in the above formulas for u (p) by following the ideas of the paper [82] with Rogosin. In particular, we shall see that rather than using the expansion in  and 1/log , it is convenient to consider a rearrangement of it. Indeed, it turns out that an expansion in powers of  and 1  r0 + log 2π is more efficient. Here r0 is a real constant that depends only on Ω o and Ω i . Then the results that we obtain can be compared with those of Il’in [127] and Maz’ya et al. [203, 204].

1.2 A Selection of Problems We now indicate some of the problems that have been investigated by means of the Functional Analytic Approach.

1.2 A Selection of Problems

7

1.2.1 Perturbation Problems for the Riemann Map The first problem to which the Functional Analytic Approach was applied concerns the Riemann map in planar perforated domains. Indeed, when n = 2, the set Ω o and the inner hole Ω i that we introduced above are Jordan domains. Then the Riemann Mapping Theorem implies that there exist a unique r ∈]0, 1[ and a unique (suitably normalized) holomorphic diffeomorphism from an annulus with inner radius equal to r and outer radius equal to 1 to Ω() (the so-called Riemann map). By the Functional Analytic Approach one can analyze the behavior of both r and of the Riemann map as  tends to zero, even in the case where the Jordan curves which bind the domains Ω o and Ω i are considered as variables and we have a problem of nonlocal nature. In particular, one can prove that r admits a real analytic continuation for small and negative values of  (see [160], [161]).

1.2.2 Linear Elliptic Boundary Value Problems The analysis of the Dirichlet problem for the Poisson equation, which develops what we have illustrated above for the Laplace operator, has been presented in [162] and [167]. In these papers we have also analyzed the dependence of the solution upon suitable diffeomorphisms that parametrize the boundaries of Ω o and Ω i . Then in [74] we have investigated the meaning of equality (1.7) when  is negative. Indeed, one can observe that the solution u is defined only for positive values of , whereas the map Up is defined in an open neighborhood of  = 0. Thus, it makes sense to ask what happens for  < 0. In [79] we have considered an analogous question for the case of dimension two and we have investigated the logarithmic behavior that appears in (1.8). The Functional Analytic Approach has also been applied to linear elliptic systems of differential equations. In particular, it has been used to deal with problems for the Lam´e equations of linearized elasticity [66], [67] and for the Stokes system, which describes the steady-state flow of an incompressible viscous fluid [62].

1.2.3 Eigenvalues Problems By the Functional Analytic Approach one can also study the asymptotic behavior of eigenvalues and eigenfunctions in domains with a small hole. For the analysis of a Neumann eigenvalue problem for the Laplace operator, we refer to [169], and for the analysis of the Steklov eigenvalue problem, we refer to the papers [113] and [114] with Gryshchuk and to paper [171].

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1 Introduction

1.2.4 Nonlinear Boundary Value Problems Further applications of the Functional Analytic Approach concern nonlinear boundary value problems in domains with small holes and inclusions. Typically, in the applications of the Functional Analytic Approach to such problems, one first shows the existence of an  dependent family of solutions u of the perturbed problem in Ω() (which may not be the only one). Then one looks for representation formulas such as (1.3), again for u or related functionals. This being accomplished, one can investigate (local) uniqueness properties of the family of solutions. This approach has been used to study nonlinear boundary value problems for linear differential operators with nonlinear boundary conditions, and certain nonlinear boundary value problems for quasi-linear equations that can be reduced to linear differential equations with nonlinear boundary conditions. For example, a problem for the Laplace equation with a nonlinear Robin boundary condition has been analyzed in [164] and nonlinear transmission conditions are studied in [168], in the paper [72] with Molinarolo, and in the paper [225] of Molinarolo. Nonlinear traction problems for the equations of the linearized elastostatics have been considered in [66] and nonlinear traction problems in which the boundary data are allowed to depend singularly on the parameter  have been studied in [63], [64], and [67].

1.2.5 Problems in Periodic Domains Along with problems in bounded perforated domains, we have considered problems in periodic domains with an infinite number of holes or inclusions. As a first step, we have considered problems where the periodicity cell is fixed and each of the holes is shrinking to a point. We mention, for example, [228], which concerns a Dirichlet problem for the Laplace equation, and [179], where we consider a Neumann problem for the Poisson equation. Nonlinear Robin boundary conditions have been studied in [176] and [68] and quasilinear heat transmission problems have been investigated in [177] and [70]. Then we have turned to problems in which the periodicity cell is also shrinking: for the analysis of a linear problem, we refer to [178] and [181], and for a nonlinear problem to [182]. Then we have turned to problems in which the periodicity cell is also shrinking: for the analysis of a linear problem, we refer to [178] and [181], and for a nonlinear problem to [182, 183].

1.2.6 Different Boundary Perturbations Some papers concern applications of the Functional Analytic Approach to boundary perturbations other than holes or inclusions that shrink to interior points. For example, the case of two shrinking holes colliding into one another has been considered in [78], [79], and [80]. The case of a small hole that approaches the outer boundary

1.3 Structure of the Book

9

of a domain has been studied in the paper [29] with Bonnaillie-No¨el and Dambrine, and self-similar perturbations of a plane sector domain have been analyzed in [60] with Costabel and Dauge.

1.2.7 Perturbation Results for Integral Operators To carry out the analysis of singularly perturbed elliptic boundary value problems using the approach introduced in this book, we need to understand the dependence of certain integral operators upon perturbations both of the density and of the support. In this sense, we mention the paper [184] with Preciso on the Cauchy integral; the papers [185] and [186] with Rossi on the layer potentials associated with the fundamental solution of the Laplace and of the Helmholtz operator; the papers [62], [65], and the paper [89] with Dondi on layer potentials associated with a fundamental solution of general elliptic operators with constant coefficients; the paper [174] on periodic layer potentials associated with a fundamental solution of general elliptic operators with constant coefficients; the paper [69] on volume potentials corresponding to parametric families of fundamental solutions; and the papers [172], [173] with Luzzini on the layer potentials associated with the fundamental solution of the heat equation. In addition, special families of parameter-dependent fundamental solutions are introduced and studied in [61] and in paper [73] with Morais, which deal with the case of general elliptic operators with constant real and quaternion coefficients, respectively, and in [180], where we consider the periodic case and we investigate the dependence upon the periodic structure. We also mention the work on nonlinear integral operators with analytic kernels in [175].

1.3 Structure of the Book In Chapter 2 we present some preliminary material, mainly on spaces of H¨older continuous functions and Schauder spaces, as well as some basics about regular subsets of Rn . Although this part is mostly self-contained, we sometimes refer to other textbooks for basic results of Calculus and of Real and Functional Analysis. In Chapter 3 we summarize some basic properties of harmonic functions and for most of the proofs we refer to major monographs, in particular to Evans [95], Folland [102], and Gilbarg and Trudinger [107]. In Chapter 4 we present the Green Identities both in a bounded domain and in the corresponding unbounded exterior domain. Then we introduce the notion of single and double layer potentials and show certain mapping properties of these potentials. Although these properties can be found in the classical monographs of G¨unter [115] and of Kupradze et al. [151], here we prove the corresponding statements with optimal H¨older exponents. To do so, we exploit a result of Miranda [212] on the H¨older

10

1 Introduction

continuity of singular integrals of convolution type. We also provide a complete proof of Miranda’s result (that was originally published in Italian). In Chapter 5 we introduce some basic properties of Fredholm operators and the Fredholm Alternative Theorem both in its classical setting and in the version of the duality pairs developed by Wendland [273], [274]. In Chapter 6, we obtain existence and uniqueness results for the classical boundary value problems for the Laplace equation in Schauder spaces using a Potential Theoretic Method and the Fredholm Alternative Theorem. For recent contributions on integral equation methods for the solution of boundary value problems we refer to the monographs of Constanda [58], Constanda et al. [59], Gewinner and Stephan [116], Hsiao and Wendland [126], Maz’ya and Soloviev [205], McLean [206], Medkov´a [208], Mitrea and Mitrea [215], Kohr and Pop [143], Sauter and Schwab [254]. In Chapter 7, we introduce some basic properties of the volume potentials in Schauder spaces and prove existence results for the classical boundary value problems of the Poisson equation. Then we also consider the case of Roumieu spaces. Although the properties of volume potentials in Schauder spaces can be found in the classical monographs of G¨unter [115] and of Kupradze et al. [151], here we prove the corresponding statements with optimal H¨older exponents by following a proof of Miranda [212] and by exloiting a known lemma (cf. Majda and Bertozzi [196, Prop. 8.12, pp. 348–350]) for which we provide a proof of Mateu et al. [198]. In Chapter 8, we begin to explain the Functional Analytic method of the present book and we do so in detail for the Dirichlet problem for the Laplace operator in a domain with a single hole which shrinks to a point, as we have outlined above. Then in Chapter 9 we turn to a mixed boundary value problem for the Laplace and Poisson equations and the Steklov eigenvalue problem. In Chapter 10 we illustrate in detail the case of two holes which shrink to a common point. We focus on the Dirichlet problem for the Laplace equation. In Chapter 11 we consider the case of nonlinear boundary conditions in domains with a small hole. We do so for nonlinear Robin and nonlinear transmission conditions for the Laplace equation. In Chapter 12 we consider boundary value problems for the Laplace equation in periodic domains obtained by removing from Rn a periodic set of holes. To do so, we develop a periodic version of Potential Theory based on layer potentials which are defined by replacing the fundamental solution of the Laplace equation with a periodic analogue. In Chapter 13, we consider both linear and nonlinear boundary value problems in the entire space with a periodic set of perforations which shrink to points. In the appendix, we have included a number of known results that are necessary for our purposes throughout the book.

Chapter 2

Preliminaries

Abstract In this chapter we review some preliminary material from Calculus and Functional Analysis and also recall the basic properties of real analytic functions. We introduce the notion of Roumieu classes, then summarize the classical properties of H¨older continuous functions. Next, we introduce the notion of coordinate cylinders and of sets that are hypographs of Cartesian functions locally around the boundary points, including the case of Lipschitz sets. Finally, we turn to Schauder spaces and the corresponding embedding theorems and to the subsets of Rn of class C m,α . Although this part of the book is mostly self-contained, we sometimes refer to other textbooks for basic results of Calculus and of Real and Functional Analysis.

2.1 Basic Notation In this section we introduce some symbols that we use all throughout the book. We begin with the symbol = that we use for equalities and equations and the symbol ≡ that we use for definitions (instead of := that other authors use). We denote by N, Z, R, C, the set of natural numbers including zero, the set of integer numbers, the set of real numbers, and the set of complex numbers, respectively. We denote by K either the field R or C. Let X be a set. Then we denote by DX the diagonal of X × X, i.e., we set DX ≡ {(x1 , x2 ) ∈ X × X : x1 = x2 } . If Y is also a set then we denote by Y X the set of maps from X to Y . Let (X,  · X ) and (Y,  · Y ) be normed spaces on K. Let U and V be subsets of X and Y , respectively. Then U denotes the closure of U , ∂U denotes the boundary of U , and diam (U ) denotes the diameter of U , i.e., diam (U ) ≡ sup {x − yX : x, y ∈ U } .

© Springer Nature Switzerland AG 2021 M. Dalla Riva et al., Singularly Perturbed Boundary Value Problems, https://doi.org/10.1007/978-3-030-76259-9 2

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2 Preliminaries

˚ the interior of U and by U − the exterior of U , i.e., we We denote by either U + or U set ˚, U− ≡ X \ U . (2.1) U+ ≡ U In particular, U + = U if U is open. We say that a function f from U to Y is locally constant provided that for each u0 ∈ U there exists a neighborhood W0 of u0 in U such that f is constant in W0 . We denote by C 0 (U, V ) the set of continuous functions from U to V . We denote by B(U, V ) the set of bounded functions from U to V . As it is well known, if Y is complete then the space B(U, Y ) with the sup-norm f B(U,Y ) ≡ sup f (x)Y x∈U

is complete and the subspace Cb0 (U, Y ) ≡ C 0 (U, Y ) ∩ B(U, Y ) of B(U, Y ) is also complete. When no ambiguity can arise, we simply write B(U ) instead of B(U, K) and similarly for other function spaces defined on U . If U is a subset of R and if f is a function from U to R, we say that f is increasing provided that f (ρ1 ) ≤ f (ρ2 ) whenever ρ1 , ρ2 ∈ U and ρ1 < ρ2 . Then we say that f is strictly increasing provided that f (ρ1 ) < f (ρ2 ) whenever ρ1 , ρ2 ∈ U and ρ1 < ρ2 . Similarly, we define the decreasing functions and the strictly decreasing functions. The inverse function of an invertible function f is denoted f (−1) , as opposed to the reciprocal of a real-valued function g, or the inverse of a matrix A, which are denoted g −1 and A−1 , respectively. The preimage of a set E under a function f is denoted by f ← (E). Let A be a matrix. Then At denotes the transpose matrix of A and Aij denotes the (i, j)-entry of A. If A is invertible, we set A−t ≡ (A−1 )t . Let n ∈ N \ {0}. We denote by Mn (K) the set of n × n matrices with entries in the field K. We denote by δl,j the Kronecker symbol. Namely,  1 if l = j , δl,j ≡ 0 if l = j . We denote by In or simply by I the identity matrix I ≡ (δlj )l,j=1,...,n . We denote by

On (K) ≡ A ∈ Mn (K) : At A = I = AAt the set of the orthogonal matrices of Mn (K). If S is a set, IS denotes the identity map from S to itself. If S is clear from the context, we simply write I instead of IS . We denote by suppf the support of a complex valued function f . If x ∈ R, we set

2.1 Basic Notation

13

⎧ ⎨ 1 if x > 0 , sgn(x) ≡ 0 if x = 0 , ⎩ −1 if x < 0 . For all ρ > 0, x ∈ Rn , xj denotes the j-th coordinate of x, |x| denotes the Euclidean modulus of x in Rn , i.e.,   n 2 |x| ≡ xj ∀x ≡ (x1 , . . . , xn ) ∈ Rn . j=1

Unless otherwise specified, we always equip Rn with the Euclidean norm | · |. Then we set Bn (x, ρ) ≡ {y ∈ Rn : |x − y| < ρ} . We often abbreviate the unit ball Bn (0, 1) as Bn . We denote by {e1 ,. . . , en } the canonical basis of Rn . We denote by a dot the scalar product of vectors. If (Z, M, μ) is a measured space, we retain standard abbreviations as μ − a.a. = μ − almost all ,

μ − a.e. = μ − almost everywhere .

We retain the standard notation for the Lebesgue space Lpμ (Z) of p-summable (equivalence classes of) measurable functions for p ∈ [1, +∞]. In particular, we set   1/p

f Lpμ (Z) ≡

|f |p dμ

∀f ∈ Lpμ (Z)

Z

if p ∈ [1, +∞[ and ∀f ∈ L∞ μ (Z)

f Lpμ (Z) ≡ ess supZ |f |

if p = +∞ and we abbreviate  · Lpμ (Z) as  · p whenever there is no ambiguity on the space Lpμ (Z). For basic inequalities such as the H¨older inequality for Lebesgue spaces, we refer to textbooks such as Folland [103, Chap. 6]. Also, if X is a vector subspace of L1μ (Z), we find convenient to set    f dμ = 0 . X0 ≡ f ∈ X :

(2.2)

Z

If A ∈ M and μ(A) ∈]0, +∞[, we set  − ≡ A

1 μ(A)

 . A

By the known inclusions of the lp (N) ≡ Lpμ (N) spaces with μ equal to the counting measure, we have the known inequality

14

2 Preliminaries

⎛ ⎝

∞ 

⎞1/q |aj |q ⎠

⎞1/p ⎛ ∞  ≤⎝ |aj |p ⎠

j=1

(2.3)

j=1

for all p, q ∈]0, +∞[ such that p ≤ q and for all sequences {aj }j∈N of real numbers such that the right-hand side is finite (cf., e.g., Folland [103, Prop. 6.11]). Instead the convexity of the p-power function for p ∈ [1, +∞[ implies that ⎛ ⎛ ⎞1/p ⎞ m m   ⎝ |aj |⎠ ≤ m1−(1/p) ⎝ |aj |p ⎠ j=1

(2.4)

j=1

for all m ∈ N \ {0} and (a1 , . . . , am ) ∈ Rm , and the concavity of the p power function for p ∈]0, 1] implies that ⎛ ⎛ ⎞1/p ⎞ m m   ⎝ |aj |⎠ ≥ m1−(1/p) ⎝ |aj |p ⎠ j=1

(2.5)

j=1

for all m ∈ N \ {0} and (a1 , . . . , am ) ∈ Rm . We denote by Ln the σ-algebra of Lebesgue measurable sets of Rn and by mn the Lebesgue measure in Rn . If μ is the Lebesgue measure in Rn , we normally omit the letter μ in the symbol Lpμ for the Lebesgue spaces and in abbreviations such as μ − a.a. and μ − a.e.. A subset M of Rn is a differential manifold of dimension s ∈ {1, . . . , n} and of p ∈ M , there exists an open neighborhood class C 1 embedded in Rn if, for every  n 1 W of p in R and a map ψ ∈ C Bs (0, 1), Rn such that ψ is a homeomorphism of Bs onto W ∩ M , ψ(0) = p, and the Jacobian matrix Dψ has rank s at all points of Bs (0, 1), i.e., a ψ is parametrization for M around p. By homeomorphism we understand a continuous bijection which has a continuous inverse. Instead, for the definition of a C 1 function on the closed set Bs (0, 1), we refer to Section 2.3. We denote by LM and mM (or simply ms ) the σ-algebra of Lebesgue surface measurable sets and the Lebesgue surface measure on a manifold M of dimension s embedded in Rn , respectively. We denote by dσ the area element of M (cf., e.g., Naumann and Simader [232]). If μ is the Lebesgue surface measure on M , we normally omit the subscript μ in the notation for the Lebesgue spaces and in abbreviations such as a.a. and a.e.. As is well known, if α ∈]0, +∞[ then the function e−t tα−1 is integrable in ]0, +∞[. Then the Euler Gamma function is defined by 

+∞

Γ (α) ≡ 0

By integrating by parts, we have

e−t tα−1 dt

∀α ∈]0, +∞[ .

2.2 Preliminaries of Linear Functional Analysis

15

∀α > 0 .

Γ (α + 1) = αΓ (α) By a simple inductive argument, we then have

∀n ∈ N .

Γ (n + 1) = n! By elementary Calculus, it is known that Γ (1/2) =

√ π.

We denote by sn the surface measure of ∂Bn (0, 1) and by ωn the n-dimensional measure of the ball Bn (0, 1). As is well known, ωn =

π n/2 , Γ ((n/2) + 1)

sn =

2π n/2 , Γ (n/2)

sn = nωn ,

for each n ∈ N \ {0} and 

1

tx−1 (1 − t)y−1 dt = 0

Γ (x)Γ (y) Γ (x + y)

∀x, y ∈]0, +∞[

(2.6)

(cf., e.g., Lebedev [187, §1.5]). We also mention that if x ∈ Rn and ρ ∈]0, +∞[, then   f (y) dσy = f (x + ρξ)ρn−1 dσξ ∂Bn (x,ρ)

∂Bn (0,1)

for each complex-valued integrable function f on ∂Bn (x, ρ).

2.2 Preliminaries of Linear Functional Analysis If X and Y a vector spaces on the field K, L(X, Y ) denotes the set of linear functions from X to Y . If X is a vector space on the field K, then a linear topology on X is a topology T on X such that both the sum from X × X to X and the product from K × X to X are continuous. We will mainly consider the case in which X is endowed by a norm  · X . As it is well known, ∞a norm on a space X generates a linear topology on X. A series j=0 xj with terms in a normed space X is said to be normally convergent provided that ∞  xj X < +∞ . j=0

If X is complete, then it is known that a normally convergent series is also convergent in X.

16

2 Preliminaries

Theorem 2.1. Let (X,  · X ) be a normed space. Let V be a closed subspace of X. Then the norm on the quotient X/V defined by  [x] X/V ≡ inf x + vX v∈V

∀[x] ∈ X/V

generates the quotient topology on X/V , i.e. the strongest topology on X/V such that the canonical projection π of X onto X/V is continuous. If X is complete, then (X/V,  · X/V ) is complete. If Y is a normed space and if T is a linear map from X/V to Y , then T is continuous if and only if T ◦ π is continuous. For a proof, we refer for example to Schaefer [255, p. 42]. Let X, Y be normed spaces. Then we endow the product space X × Y with the norm defined by (x, y)X×Y ≡ xX + yY

∀(x, y) ∈ X × Y ,

while we use the Euclidean norm for Rn . If T is a linear operator from X to Y , then we set Ker T ≡ {x ∈ X : T [x] = 0} for the kernel (or null space) of T and Im T ≡ {T [x] : x ∈ X} for the image (or range) of T . A linear operator T from X to Y is well known to be continuous if and only if it maps bounded subsets of X to bounded subsets of Y . It is customary to say that a linear operator T from X to Y is bounded provided that T maps bounded subsets of X to bounded subsets of Y . With such a terminology, a linear operator T from X to Y is continuous if and only if it is bounded. Then L(X, Y ) denotes the space of linear and continuous operators from X to Y , and we set T L(X,Y ) ≡

sup

T [x]Y

∀T ∈ L(X, Y ) ,

x∈BX (0,1)

where BX (p, r) ≡ {x ∈ X : x − pX < r}   for all p ∈ X and r ∈]0, +∞[. If Y is complete, then L(X, Y ),  · L(X,Y ) is well known to be a Banach space. We often refer to the following classical result. For a proof, we refer for example to Brezis [35, Theorem 2.6 and Corollary 2.7 p. 35]. Theorem 2.2 (of the Open Mapping). Let X, Y be Banach spaces. If T ∈ L(X, Y ) is surjective, then T is open, i.e., T maps open subsets of X to open subsets of Y .

2.2 Preliminaries of Linear Functional Analysis

17

In particular, if T ∈ L(X, Y ) is a bijection, then inverse operator T (−1) is continuous and accordingly T is a homeomorphism. We also note that if T ∈ L(X, Y ), then the Fundamental Theorem of Homomorphism implies that there exists a unique linear injective map T˜ from X/Ker T to Y such that T = T˜ ◦ π , where π denotes the canonical projection of X onto X/Ker T (cf., e.g., Kostrikin and Manin [145]). By Theorem 2.1, T˜ is continuous. If we further assume that T is surjective, then T˜ is a continuous bijection of X/Ker T onto Y and the Open Mapping Theorem implies that T˜ is a homeomorphism. As is well known a normed space of dimension n ∈ N is linearly homeomorphic to Kn (cf. e.g. Taylor and Lay [265, Chap. 2, Theorem 3.1]) and is therefore complete. Then we also have the following classical result (cf. e.g. Taylor and Lay [265, Chap. 2, Theorem 3.3]). Theorem 2.3. Let X be a normed space. If X1 is a finite dimensional subspace of X, then X1 is closed. Definition 2.4. Let X be a normed space. Let X1 be a subspace of X. We say that a linear map P1 from X to X is a projection of X onto X1 provided that P1 [X] ⊆ X1

P1 [ξ] = ξ

and

∀ξ ∈ X1 .

We note that if P is a projection onto X1 , then we necessarily have X1 = Im P1 . If X1 and X2 are subspaces of X and if X equals the algebraic direct sum of X1 and X2 , i.e., if X = X1 + X2

and

X1 ∩ X2 = {0} ,

then we write X = X1 ⊕ X2 , and we say that X2 is an algebraic direct complement (or algebraic supplement) of X1 in X. If X = X1 ⊕ X2 , then the map Φ from X1 × X2 to X that takes (x1 , x2 ) to x1 + x2 is a linear bijection, and if πj denotes the canonical projection of X1 × X2 onto Xj , then the map Pj ≡ πj ◦ Φ(−1) is a projection of X onto Xj for all j ∈ {1, 2} in the sense of Definition 2.4. Moreover, if x ∈ X, then (P1 [x], P2 [x]) is the only pair of X1 × X2 such that x = P1 [x] + P2 [x] . In particular, the pair of projections (P1 , P2 ) is uniquely determined by the condition that Pj is a projection of X onto Xj for all j ∈ {1, 2} and that P1 + P2 = IX . We say that P1 is a projection of X onto X1 along X2 .

18

2 Preliminaries

Definition 2.5. Let X be a normed space. Let X1 and X2 be subspaces of X such that X = X1 ⊕ X2 algebraically. Then we say that the direct sum X1 ⊕ X2 is topological provided that the corresponding projections P1 and P2 are continuous. If so, then we say that X2 is a topological direct complement (or topological supplement) of X1 in X. Then we have the following, which we exploit in the sequel. Theorem 2.6. Let X be a Banach space. Let X1 and X2 be linear subspaces of X such that X = X1 ⊕ X2 algebraically. Then the direct sum X1 ⊕ X2 is topological if and only if both X1 and X2 are closed in X. Proof. If the direct sum is topological, we note that X1 = {x ∈ X : P2 [x] = 0}. Hence, the continuity of P2 implies that X1 is closed in X. Similarly, X2 is also closed in X. Conversely, if both X1 and X2 are closed in X, then X1 and X2 are Banach spaces and X1 × X2 is also a Banach space. Then the Open Mapping Theorem implies that the linear and continuous bijection Φ from X1 × X2 to X that takes (x1 , x2 ) to x1 + x2 has a continuous inverse, and accordingly Pj = πj ◦ Φ(−1) is  continuous for each j ∈ {1, 2} and the direct sum X = X1 ⊕ X2 is topological.  Then we have the following immediate corollary. Corollary 2.7. Let X be a Banach space. Let X1 be a closed subspace of X of finite codimension, i.e., the dimension r of the quotient space X/X1 is finite. Then X1 has a topological direct complement (or topological supplement) X2 in X. Proof. Let π denote the canonical projection of X onto X/X1 . By assumption, there exist ξ1 ,. . . , ξr in X such that {π(ξ1 ), . . . , π(ξr )} is a basis for X/X1 . Then the subspace X2 of X generated by ξ1 ,. . . , ξr is finite dimensional and X equals the direct algebraic sum X1 ⊕X2 . Since finite dimensional subspaces of a normed space are closed (cf. Theorem 2.3), then both X1 and X2 are closed in X and Theorem 2.6 implies that the direct sum X1 ⊕ X2 is topological. Accordingly X2 is a topological   direct complement of X1 in X. Definition 2.8. Let X, Y be normed spaces. A linear map T form X to Y is said to be compact provided that T (A) is compact whenever A is a bounded subset of X. By exploiting the sequential compactness of the compact subsets of a normed space, one can prove that a map T ∈ L(X, Y ) is compact if and only if, whenever {xj }j∈N is a bounded sequence of X, the sequence {T [xj ]}j∈N has a subsequence which converges in Y . We often employ the following characterization of those bilinear maps which are continuous. For a proof, we refer to Brezis [35, p. 49].

2.3 Spaces of Classically Differentiable Functions

19

Proposition 2.9. Let X1 , X2 , Y be normed spaces. A bilinear map B from X1 ×X2 to Y is continuous if and only if there exists b ∈]0, +∞[ such that B[x1 , x2 ]Y ≤ bx1 X1 x2 X2

∀(x1 , x2 ) ∈ X1 × X2 .

2.3 Spaces of Classically Differentiable Functions Let m, n ∈ N \ {0}. Let Ω be an open subset of Rn . The space of m-times continuously differentiable R-valued functions on Ω is denoted by C m (Ω, R), or more m ∞ simply by C (Ω). We also set C (Ω) ≡ j∈N C j (Ω). r Let r ∈ N \ {0}. Let f ∈ (C m(Ω)). The s-th component of f is denoted fs , and s Df denotes the Jacobian matrix ∂f ∂xl s=1,...,r, . Instead, we denote the gradient of a l=1,...,n

function by ∇. For us the gradient is always a column vector. Let β ≡ (β1 , . . . , βn ) |β| be in Nn , |β| ≡ β1 + · · · + βn . Then Dβ f denotes β∂1 f βn , and we also use ∂x1 ...∂xn

∂ standard abbreviations as ∂xl ≡ ∂x . l m The subspace of C (Ω) of those functions f whose derivatives Dβ f of order |β| ≤ m can be extended with continuity to Ω is denoted by C m (Ω). If f ∈ C m (Ω) and |β| ≤ m, then we set

Dβ f (x) ≡ lim Dβ f (y) y→x

∀x ∈ ∂Ω .

The reader should be aware that this is not the only definition of C m (Ω) available in the literature. For other authors, C m (Ω) is defined as the space of restrictions to Ω of functions of class C m in an open neighorhood of Ω. Such a definition is easily seen to be more restrictive than the one adopted here. Then we consider the space

Cbm (Ω) ≡ f ∈ C m (Ω) : Dβ f ∈ B(Ω), ∀β ∈ Nn , |β| ≤ m endowed with the norm f C m (Ω) ≡ b



sup |Dβ f (x)| .

|β|≤m x∈Ω

  As is well known, the space Cbm (Ω),  · C m (Ω) is complete. If Ω is bounded, b

then we obviously have Cbm (Ω) = C m (Ω) (and in this case, we generally drop the subscript b). We note that the inclusion of Cbm (Ω) into Cb0 (Ω) has to be understood in the sense that each element f ∈ Cbm (Ω) has a uniquely determined continuous extension to Ω, which we still denote by f . Then by the definition of norm in Cbm (Ω) and in Cb0 (Ω), Cbm (Ω) is continuously imbedded into Cb0 (Ω).

20

2 Preliminaries

2.4 Distributions and Weak Derivatives Let n ∈ N \ {0}. Let K be a compact subset of Rn . Then we set D(K) ≡ Cc∞ (K) ≡ {ϕ ∈ C ∞ (Rn ) : supp ϕ ⊆ K} and pK,m (ϕ) ≡ sup sup |Dβ ϕ| |β|≤m K

∀ϕ ∈ Cc∞ (K)

for all m ∈ N. Then Cc∞ (K) is a vector space and {pK,m : m ∈ N} is a family of norms on Cc∞ (K) which generate a topology on Cc∞ (K). The topology on Cc∞ (K) generated by the family of norms {pK,m : m ∈ N} is by definition that generated by the family of open balls {ψ ∈ Cc∞ (K) : pK,m (ϕ − ψ) < r} with center at ϕ ∈ Cc∞ (K), radius r ∈]0, +∞[, and m ∈ N. The topology generated by the family of norms {pK,m : m ∈ N} is linear. Then one can characterize the linear functions from Cc∞ (K) to R which are continuous by means of the following (cf., e.g., Tr`eves [266]). Proposition 2.10. Let K be a compact subset of Rn . A linear function T from Cc∞ (K) to R is continuous if and only if there exist c > 0 and m ∈ N such that |T [ϕ]| ≤ cpK,m (ϕ)

∀ϕ ∈ Cc∞ (K) .

Now let Ω be an open subset of Rn . Then we set D(Ω) ≡ {ϕ ∈ Cc∞ (Rn ) : supp ϕ is compact , supp ϕ ⊆ Ω} . The space D(Ω) is said to be the space of the test functions in Ω and is a vector space. We now wish to introduce a linear topology on D(Ω) that has the property of being locally convex, i.e., such that each element of D(Ω) has a basis of convex neighborhoods. To do so, we note that  D(Ω) = D(K) , K is compact, K⊆Ω

we denote by iK the inclusion map from D(K) into D(Ω), and we state the following theorem. For a proof, we refer to Tr`eves [266]. Theorem 2.11. There exists a strongest locally convex separated topology T on D(Ω) such that the inclusion iK is continuous from D(K) to (D(Ω), T ) for each compact subset K of Ω. If T is a linear map from (D(Ω), T ) to R, then T is continuous if and only if the map T ◦ iK from D(K) to R is continuous for each compact subset K of Ω. We note that the maps T ◦ iK of the previous statement are precisely the restrictions of T to D(K).

2.4 Distributions and Weak Derivatives

21

Unless otherwise specified, we always retain D(Ω) as endowed with the topology T of Theorem 2.11. Definition 2.12. Let Ω be an open subset of Rn . We define the space of distributions in Ω as the topological dual D (Ω) ≡ {T ∈ L(D(Ω), R) : T is continuous} of D(Ω). If T ∈ D (Ω), it is customary to write T, ϕ ≡ T [ϕ]

∀ϕ ∈ D(Ω) .

We shall consider on D (Ω) the weak∗ topology. In particular, if {uj }j∈N is a se˜ ∈ D (Ω), then we have quence in D (Ω) and if u lim uj = u ˜

j→∞

in D (Ω)

if and only if lim uj , ϕ = ˜ u, ϕ

∀ϕ ∈ D(Ω) .

j→∞

Let x0 ∈ Ω. We denote by δx0 the linear map from D(Ω) to R defined by δx0 , ϕ ≡ ϕ(x0 )

∀ϕ ∈ D(Ω) .

The operator δx0 is said to be the ‘Dirac measure with mass concentrated at x0 ’. Then δx0 is a distribution in Ω. The linearity of δx0 is obvious. In order to prove the continuity of δx0 it suffices to show that δx0 is continuous on D(K) for each compact subset K of Ω. Such a continuity is an immediate consequence of Proposition 2.10 and of the following elementary inequality |δx0 , ϕ| ≤ sup |ϕ| = pK,0 (ϕ) K

∀ϕ ∈ D(K) .

Let L1loc (Ω) denote the space of locally integrable (equivalence classes of) functions from Ω to R, i.e., the space of measurable (equivalence classes of) functions from Ω to R such that f is integrable on each compact subset of Ω. If f ∈ L1loc (Ω), then one can associate to f a distribution J f by setting  f ϕ dx ∀ϕ ∈ D(Ω) . J f, ϕ ≡ Ω

By the linearity of the integration in Ω, J f is a linear map from D(Ω) to R. Then one can exploit Proposition 2.10 and prove that J f ◦ iK is actually continuous on D(K) for all compact subsets K of Ω. Then Theorem 2.11 implies that J f is continuous on D(Ω) and that accordingly J f is a distribution in Ω. Then one can assign to each f ∈ L1loc (Ω) the distribution J f and the linear map J from L1loc (Ω) to D (Ω) that takes f to J f is injective. Indeed, if f ∈ L1loc (Ω) and J f = 0, then we have

22

2 Preliminaries

 f ϕ dx = 0

∀ϕ ∈ D(Ω)

Ω

and thus f = 0 almost everywhere. The linear injection J identifies L1loc (Ω) with the subspace J [L1loc (Ω)] of D (Ω) and it is customary to write simply f instead of J f. Definition 2.13. Let Ω be an open subset of Rn . Let β ∈ Nn , u ∈ D (Ω). Then the Dβ -derivative of u in the sense of distributions is the element Dβ u of D (Ω) defined by ∀ϕ ∈ D(Ω) . Dβ u, ϕ ≡ (−1)|β| u, Dβ ϕ If m ∈ N \ {0}, f ∈ C m (Ω), then an argument based on the Divergence Theorem shows that the Dβ derivative of the distribution J f associated to f coincides with the distribution J Dβ f associated to Dβ f for each β ∈ Nn with |β| ≤ m. If f ∈ L1loc (Ω), then it is customary to say that Dβ J f , which we abbreviate as β D f , is the Dβ -weak derivative of f for each β ∈ Nn . If P (D) is a differential operator with constant coefficients, u ∈ D (Ω), and P (D)u = 0 , then it is customary to say that u is a solution in the sense of distributions (or a weak solution) of equation P (D)u = 0. We denote by S(Rn ) the Schwartz space of rapidly decreasing functions, i.e., the space S(Rn )   ∞ n 2 m/2 β n ≡ f ∈ C (R ) : sup (1 + |x| ) |D f (x)| < +∞ ∀m ∈ N, ∀β ∈ N . x∈Rn

We set pm,β (f ) ≡ sup (1 + |x|2 )m/2 |Dβ f (x)| x∈Rn

∀f ∈ S(Rn )

for all m ∈ N and β ∈ Nn and we set Prd ≡ {pm,β : m ∈ N, β ∈ Nn }. Then S(Rn ) is a linear space and Prd is a family of seminorms on S(Rn ) which generate a topology on S(Rn ). The topology on S(Rn ) generated by the family of seminorms Prd is by definition that generated by the family of open balls {ψ ∈ S(Rn ) : pm,β (f − ψ) < r} with center at f ∈ S(Rn ), radius r ∈]0, +∞[, m ∈ N, and β ∈ Nn . The topology generated by the family of seminorms Prd is linear, locally convex, metrizable, and complete, i.e., S(Rn ) is a Fr´echet space. We define the space of tempered distributions in Rn as the topological dual S  (Rn ) ≡ {T ∈ L(S(Rn ), R) : T is continuous} of S(Rn ). Then one can characterize the continuous linear functions from S(Rn ) to R by means of the following proposition (cf., e.g., Tr`eves [266]).

2.5 Real Analytic Functions and Spaces of Real Analytic Functions

23

Proposition 2.14. A linear function T from S(Rn ) to R is continuous if and only if there exist c > 0, m1 , . . . , ms ∈ N, and β1 , . . . , βs ∈ Nn such that |T, ψ| ≤ c sup pml ,βl (ψ)

∀ψ ∈ S(Rn ) .

l=1,...,s

We shall consider on S  (Rn ) the weak∗ topology. In particular, if {uj }j∈N is a ˜ ∈ S  (Rn ), then we have sequence in S  (Rn ) and if u lim uj = u ˜

j→∞

in S  (Rn )

if and only if lim uj , ϕ = ˜ u, ϕ

j→∞

∀ϕ ∈ S(Rn ) .

Finally, we observe that one could embed the space S  (Rn ) of tempered distributions into the space D (Rn ) of distributions in Rn and that accordingly one can define the derivatives in the sense of distributions also for the tempered distributions.

2.5 Real Analytic Functions and Spaces of Real Analytic Functions In Section A.6 of the Appendix, we define the notion of analytic function acting between Banach spaces. In the specific case of scalar functions defined on open subsets of Rn with n ∈ N such definition is equivalent to the following. Definition 2.15. Let Ω be an open subset of Rn . Let f ∈ C ∞ (Ω). Let xo in Ω. Then f is real analytic at xo provided that there exists a neighborhood Vxo of xo in Ω such that β o  (i) limj→∞ |β|≤j |D fβ!(x )| |(x − xo )β | exists in R for all x ∈ Vxo , β o  (ii) limj→∞ |β|≤j D fβ!(x ) (x − xo )β = f (x) for all x ∈ Vxo . We say that f is real analytic in Ω, provided that f is real analytic at all points of Ω. In what follows, we understand that ‘analytic’ always means ‘real analytic’. We denote by C ω (Ω) the set of analytic functions from Ω to R. Then we have C ω (Ω) ⊆ C ∞ (Ω) and the inclusion is proper whenever Ω is not empty. For example, the function f from R to itself defined by f (x) ≡ 0

if x ∈] − ∞, 0] ,

f (x) ≡ e−1/x

if x ∈]0, +∞[

is well known to belong to C ∞ (R) \ C ω (R). One could exploit this example to construct functions of C ∞ (Ω) \ C ω (Ω), whenever Ω is a nonempty subset of Rn .

24

2 Preliminaries

By the classical theorems on the sum and product with scalars of limits, the set C ω (Ω) is a vector space. Next we introduce the known Identity Principle for analytic functions (cf., e.g., John [130, p. 65]). Proposition 2.16. Let Ω be a connected open subset of Rn . Let f ∈ C ω (Ω). Then the following statements hold. (i) If there exists xo ∈ Ω such that Dα f (xo ) = 0 for all α ∈ Nn , then f is equal to zero at all points of Ω. (ii) If the set of zeros Z(f ) of f has at least an interior point, then f is equal to zero at all points of Ω. Next we introduce following known characterization of an analytic function in an open set (cf., e.g., John [130, Chap. 3]). Theorem 2.17. Let Ω be an open subset of Rn . Let f ∈ C ∞ (Ω). Then the following conditions are equivalent. (i) f is analytic in Ω. (ii) For each compact subset K of Ω, there exist MK , rK ∈]0, +∞[ such that sup |Dβ f (x)| ≤ MK

x∈K

|β|! |β|

rK

∀x ∈ K, β ∈ Nn .

Next, we turn to introduce the Roumieu classes. For all bounded open subsets Ω of Rn and ρ ∈]0, +∞[, we set 0 Cω,ρ (Ω) ≡

  ρ|β| Dβ uC 0 (Ω) < +∞ u ∈ C ∞ (Ω) : sup β∈Nn |β|!

and uC 0

ω,ρ (Ω)

≡ sup

β∈Nn

ρ|β| Dβ uC 0 (Ω) |β|!

(2.7)

0 ∀u ∈ Cω,ρ (Ω) .

0 (Ω) are real analytc By the characterization of Theorem 2.17, all functions of Cω,ρ   0 in Ω. As is well known, the Roumieu class Cω,ρ (Ω),  · C 0 (Ω) is a Banach ω,ρ space. By the definition of Roumieu norm, if τ ∈ Nn and ρ ∈]0, +∞[, then the operator 0 0 (Ω) to Cω,ρ Dτ that maps u in Dτ u is linear and continuous from Cω,ρ  (Ω) for all ρ ∈]0, ρ[.

Remark 2.18. Let Ω, Ω1 be open subsets of Rn . Let Ω be a bounded subset of Ω1 . If f ∈ C ω (Ω1 ), then by applying the condition of the characterization of the analytic functions of Theorem 2.17 (ii) to the compact set K ≡ Ω, we deduce that there 0 (Ω). exists ρ ∈]0, +∞[ such that f|Ω ∈ Cω,ρ Then we note that if we integrate a function in a Roumieu class in one of its variables, the resulting function is still in a Roumieu class. Indeed, the following proposition holds.

2.6 Spaces of H¨older and Lipschitz Continuous Functions

25

Proposition 2.19. Let n, n1 ∈ N \ {0}. Let Ω, Ω1 be bounded open subsets of 0 (Ω × Ω1 ), then the funcRn , Rn1 , respectively. Let ρ ∈]0, +∞[. If F ∈ Cω,ρ  0 0 tion Ω1 F (·, y) dy belongs to Cω,ρ (Ω) and the operator from Cω,ρ (Ω × Ω1 ) to  0 Cω,ρ (Ω) that takes F to the function Ω1 F (·, y) dy is linear and continuous. Proof. By the differentiability theorem for integrals depending on a parameter, we have          β β   D x F (x, y) dy  =  Dx F (x, y) dy   Ω1

Ω1

|β|! ≤ |β| mn1 (Ω1 )F C 0 (Ω×Ω1 ) ω,ρ ρ

∀x ∈ Ω

0 (Ω). for all β ∈ Nn . Thus the statement follows by the definition of norm in Cω,ρ

 

2.6 Spaces of H¨older and Lipschitz Continuous Functions Let ω be a function from [0, +∞[ to itself such that ω(0) = 0,

ω(r) > 0

∀r ∈]0, +∞[ ,

ω is increasing, lim+ ω(r) = 0 .

(2.8)

r→0

Let n, r ∈ N. If f is a function from a subset D of Rn to Rr , then we denote by |f : D|ω(·) the ω(·)-H¨older constant of f , which is delivered by the formula   |f (x) − f (y)| : x, y ∈ D, x = y . |f : D|ω(·) ≡ sup ω(|x − y|) If |f : D|ω(·) < ∞, we say that f is ω(·)-H¨older continuous. Sometimes, we simply write |f |ω(·) instead of |f : D|ω(·) . The subset of C 0 (D) whose functions are ω(·)-H¨older continuous is denoted by 0,ω(·) (D). By the triangular inequality, C 0,ω(·) (D) is a vector subspace of C 0 (D). C 0,ω(·) Cloc (D) denotes the space of those functions f ∈ C 0 (D) such that f|K belongs to C 0,ω(·) (K) for each compact subset K of D. 0,ω(·) Then we consider the space Cb (D) ≡ C 0,ω(·) (D) ∩ B(D) with the norm f C 0,ω(·) (D) ≡ sup |f (x)| + |f |ω(·) b

x∈D

0,ω(·)

If D is compact, then we obviously have Cb we generally drop the subscript b).

0,ω(·)

∀f ∈ Cb

(D) .

(D) = C 0,ω(·) (D) (and in this case,

26

2 Preliminaries

Remark 2.20. Let ω be as in (2.8). Let D be a subset of Rn . Let f be a bounded function from D to R, a ∈]0, +∞[. Then, 2 |f (x) − f (y)| ≤ sup |f | . ω(a) D x,y∈D, |x−y|≥a ω(|x − y|) sup

(x)−f (y)| Thus the difficulty of estimating the H¨older quotient |fω(|x−y|) of a bounded function f lies entirely in case 0 < |x − y| < a. Then we introduce the following elementary lemma, which enables to infer that the pointwiswe limit of a sequence of ω(·)-H¨older continuous functions is ω(·)H¨older continuous provided that the ω(·)-H¨older constants of the functions of the sequence are uniformly bounded. We exploit such an elementary lemma in the proof 0,ω(·) (D) is complete. of Theorem 2.22 below, where we prove that Cb

Lemma 2.21. Let ω be as in (2.8). Let n ∈ N. Let D be a subset of Rn . If a sequence {fj }j∈N of C 0,ω(·) (D) satisfies the inequality supl∈N |fl |ω(·) < +∞ and converges pointwise to a function f from D to R, then f ∈ C 0,ω(·) (D) and |f |ω(·) ≤ supl∈N |fl |ω(·) . Proof. It suffices to take the limit as j tends to infinity in the inequality |fj (x) − fj (y)| ≤ (sup |fl |ω(·) )ω(|x − y|)

∀x, y ∈ D

l∈N

which holds for all j ∈ N.

 

Next we prove the following. Theorem 2.22. Let ω be as in (2.8). Let n ∈ N. Let D be a subset of Rn . Then the 0,ω(·) space Cb (D) is complete. 0,ω(·)

Proof. Let {fj }j∈N be a Cauchy sequence in Cb

(D). Since

|fj (x) − fl (x)| ≤ fj − fl C 0,ω(·) (D) b

∀x ∈ D

for all j, l ∈ N, {fj (x)}j∈N is a Cauchy sequence in R for all x ∈ D. Then we can define a function f from D to R by setting f (x) ≡ lim fj (x)

∀x ∈ D .

j→∞

0,ω(·)

Since {fj }j∈N is a Cauchy sequence in Cb 0,ω(·) (D). Hence, we have bounded in Cb

(D), the sequence {fj }j∈N is

|f (x)| ≤ lim sup |fj (x)| ≤ sup fj C 0,ω(·) (D) j→∞

j∈N

b

∀x ∈ D

and f is bounded. Moreover, Lemma 2.21 implies that f ∈ C 0,ω(·) (D). Hence, 0,ω(·) 0,ω(·) f ∈ Cb (D). We now prove that {fj }j∈N converges to f in Cb (D). Let

2.6 Spaces of H¨older and Lipschitz Continuous Functions

27

 ∈]0, +∞[. Since {fj }j∈N is a Cauchy sequence, there exists k ∈ N such that j, l ≥ k ⇒ fj − fl C 0,ω(·) (D) ≤ /2 . b

We now fix an arbitrary j ∈ N such that j ≥ k, and we prove that fj − f C 0,ω(·) (D) ≤  . b

Since

|fj (x) − fl (x)| ≤ fj − fl C 0,ω(·) (D) ≤ /2

∀x ∈ D

b

for all l ∈ N such that l ≥ k, we can take the limit as l tends to infinity in the left and right-hand side and deduce that |fj (x) − f (x)| ≤ /2

∀x ∈ D .

Since {fj − fl }l∈N converges pointwise to fj − f , Lemma 2.21 implies that |fj − f |ω(·) ≤ sup |fj − fl |ω(·) ≤ sup fj − fl C 0,ω(·) (D) ≤ /2 . k≤l∈N

b

k≤l∈N

Hence, fj − f C 0,ω(·) (D) = sup |fj − f | + |fj − f |ω(·) ≤ (/2) + (/2) =  D

b

0,ω(·)

and thus Cb

 

(D) is complete.

Particularly important is the case in which ω(·) is the function rα for some fixed α ∈]0, 1] that is called H¨older exponent. In this case, we simply write |· : D|α α α instead of |· : D|rα , C 0,α (D) instead of C 0,r (D), Cb0,α (D) instead of Cb0,r (D), and we say that |f : D|α is the α-H¨older constant of f . If |f : D|α < +∞, we say that f is α-H¨older continuous. If α = 1, we say that f is Lipschitz continuous provided that f is 1-H¨older continuous and we set Lip(f ) ≡ |f : D|1 . Sometimes, we write Lip (D) instead of C 0,1 (D). We now prove the following embedding statement, which shows that the H¨older exponent is actually a smoothness parameter. Proposition 2.23. Let n ∈ N. Let D be a subset of Rn . Let α, β ∈]0, 1], β < α. Then Cb0,α (D) is continuously embedded into Cb0,β (D). Proof. The statement is an immediate consequence of the following elementary inequality |f (x) − f (y)| = |f (x) − f (y)|β/α |f (x) − f (y)|1−(β/α) β 1−(β/α) ≤ |f |β/α ≤ 21−(β/α) f C 0,α (D) |x − y|β ∀x, y ∈ D α |x − y| (2 sup |f |) D

b

which holds for all f ∈ Cb0,α (D). Next we prove the continuity of the pointwise product.

 

28

2 Preliminaries

Proposition 2.24. Let n ∈ N. Let D be a subset of Rn . Let α1 , α2 ∈]0, 1]. Then the 0,min{α1 ,α2 } pointwise multiplication from Cb0,α1 (D) × Cb0,α2 (D) to Cb (D) that takes a pair of functions (f, g) to the pointwise product f g is continuous. Proof. By the embedding Proposition 2.23, it suffices to consider case α ≡ α1 = α2 , a case which is an immediate consequence of the following elementary inequality |f (x)g(x) − f (y)g(y)| ≤ |f (x) − f (y)| |g(x)| + |f (y)| |g(x) − g(y)| ≤ |f |α |x − y|α (sup |g|) + (sup |f |) |g|α |x − y|α D

∀x, y ∈ D .

D

which holds for all (f, g) ∈ Cb0,α1 (D) × Cb0,α2 (D).

 

In particular, for all α ∈]0, 1] the space Cb0,α (D) is a Banach algebra with unity. Next we prove the following. Proposition 2.25. Let n ∈ N. Let D be a compact subset of Rn . Let α, α ∈]0, 1],  α < α. Then C 0,α (D) is compactly embedded into C 0,α (D). Proof. We must show that if {fj }j∈N is a bounded sequence of C 0,α (D), then there  exists a subsequence of {fj }j∈N which converges in C 0,α (D). By the Ascoli-Arzel`a Theorem (cf., e.g., Folland [103, Theorem 4.43, p. 137]) and by the inequalities |fj (x)| ≤ sup fl C 0,α (D) , l∈N

|fj (x) − fj (y)| ≤ (sup fl C 0,α (D) )|x − y|α

∀x, y ∈ D ,

l∈N

for all j ∈ N, there exists a subsequence {fjk }k∈N of {fj }j∈N which converges to f ∈ C 0 (D) uniformly in D. Then, by Lemma 2.21, we have f ∈ C 0,α (D).  We now show that {fjk }k∈N converges to f in the norm of C 0,α (D). Since 0 we already know that {fjk }k∈N converges to f in C (D), it suffices to show that lim supk→+∞ |fjk − f : D|α = 0. To do so, we fix an arbitrary  > 0 and we show that lim supk→+∞ |fjk − f : D|α ≤ . Now, |fjk − f : D|α has been defined as a supremum of a quotient for x, y ∈ D and the idea is to fix an appropriate δ > 0 and to analyze separately case |x − y| ≤ δ and case |x − y| > δ. We first try to determine an appropriate δ. By the definition of norm in C 0,α (D), we have |(fjk (x)−f (x))−(fjk (y)−f (y))| ≤ (sup fj −f C 0,α (D) )|x−y|α

∀x, y ∈ D ,

j∈N

for all k ∈ N. Thus if we set δ ≡



1/(α−α ) , (1+supj∈N fj −f C 0,α (D) )1/(α−α )

then we have

  |(fjk (x) − f (x)) − (fjk (y) − f (y))| : x, y ∈ D, 0 < |x − y| ≤ δ ≤  sup sup |x − y|α k∈N (2.9) and thus we have taken care of the points x, y ∈ D such that |x − y| ≤ δ. We now turn to consider the points x, y ∈ D such that |x−y| > δ. Since {fjk }k∈N converges to f uniformly in D, we have

2.6 Spaces of H¨older and Lipschitz Continuous Functions

 lim sup

k→+∞

29

|(fjk (x) − f (x)) − (fjk (y) − f (y))| : x, y ∈ D, δ < |x − y| |x − y|α

 =0 (2.10)

and thus we have taken care of the points x, y ∈ D such that |x − y| > δ. Then by combining the above inequality (2.9) and the above limiting relation (2.10), we conclude that   |(fjk (x) − f (x)) − (fjk (y) − f (y))| : x, y ∈ D, 0 < |x − y| ≤ . lim sup sup |x − y|α k→+∞ Since lim supk→+∞ fjk − f C 0 (D) = 0, we have lim sup fjk − f C 0,α (D) ≤  . k→+∞

Since  is arbitrary, we conclude that limk→+∞ fjk − f C 0,α (D) = 0.

 

Easily constructed examples show that if D is unbounded, the embedding of  into Cb0,α (D) for 0 < α < α ≤ 1 is not necessarily compact.

Cb0,α (D)

Example 2.26. Let D = R. Let f be a nonconstant bounded Lipschitz continuous function from R to itself with compact support. Let fj (x) ≡ f (x − j) for all x ∈ R and j ∈ N. Then supj∈N fj C 0,1 (R) = f C 0,1 (R) < +∞, but the sequence b

b



{fj }j∈N cannot have any convergent subsequence in Cb0,α (R) with α ∈]0, 1[. In deed, the convergence in Cb0,α (R) would imply the uniform convergence of the subsequence to its pointwise limit, which equals the 0 constant function. However, the 0 constant function cannot be a uniform limit for any subsequence of {fj }j∈N . Indeed, supR |fj | = supR |f | > 0 for all j ∈ N. We now turn to the question of whether a function of class C 1 on the closure of an open set is actually Lipschitz continuous. To do so, we need some preliminaries. We assume that Ω is an open connected subset of Rn . For every x, y ∈ Ω, there exists a path γx,y ∈ Lip ([0, 1], Rn ) such that γx,y ([0, 1]) ⊆ Ω,

γx,y (0) = x,

γx,y (1) = y.

(2.11)

The geodesic distance λΩ (x, y) of x and y in Ω is defined as λΩ (x, y) ≡ inf{length(γx,y ) : γx,y ∈ Lip ([0, 1], Rn ) and (2.11) holds} . (2.12) When no ambiguity can arise, we simply write λ instead of λΩ . By exploiting the definition of λΩ one can easily prove that λΩ is actually a distance in Ω. We observe that by exploiting the density of C 1 ([0, 1]) in the Sobolev space

W 1,1 (]0, 1[) ≡ f ∈ L1 (]0, 1[) : f  ∈ L1 (]0, 1[) , f W 1,1 (]0,1[) ≡ f L1 (]0,1[) + f  L1 (]0,1[)

∀f ∈ W 1,1 (]0, 1[)

30

2 Preliminaries

and the continuous embedding of W 1,1 (]0, 1[) into C 0 ([0, 1]) (cf., e.g., Brezis [35, Theorems 8.2, 8.7]), one could prove that the infimum in (2.12) does not change if we replace Lip ([0, 1], Rn ) with C 1 ([0, 1], Rn ). We say that   λΩ (x, y) : x, y ∈ Ω, x = y (2.13) c[Ω] ≡ sup |x − y| is the Whitney constant of Ω. Then we introduce the following definition. Definition 2.27. Let n ∈ N. Let Ω be an open subset of Rn . We say that Ω is regular in the sense of Whitney provided that Ω is connected and c[Ω] < +∞. Note that c[Ω] = 1 if Ω is convex. The following lemma shows the importance of the condition of regularity in the sense of Whitney. Lemma 2.28. Let n ∈ N. Let Ω be an open connected subset of Rn such that c[Ω] is finite. If f ∈ C 1 (Ω), then   |f (x) − f (y)| ≤ c[Ω] sup |∇f |

|x − y|

∀x, y ∈ Ω .

Ω

In particular, Cb1 (Ω) is continuously embedded into Cb0,1 (Ω). Proof. Since f and ∇f are uniquely determined by their values in Ω, it suffices to prove the inequality of the statement for x, y ∈ Ω. Since Ω is open and connected, there exists γx,y ∈ Lip ([0, 1], Rn ) as in (2.11). Then we have |f (x) − f (y)| = |f (γx,y (1)) − f (γx,y (0))|   1        = γx,y (s) · ∇f (γx,y (s)) ds ≤ sup |∇f | 0

Ω

1

 |γx,y (s)| ds .

0

Then, by taking the infimum on all γx,y ∈ Lip ([0, 1], Rn ) as in (2.11), we obtain   |f (x) − f (y)| ≤

sup |∇f | λΩ (x, y) . Ω

Then the definition of c[Ω] implies the validity of the inequality of the statement.   Classical examples show that if c[Ω] = +∞, then Cb1 (Ω) is not necessarily embedded into Cb0,1 (Ω), not even if Ω is bounded (cf., e.g., Gilbarg and Trudinger [107, p. 53]). We now introduce the following consequence of Lemma 2.28. Lemma 2.29. Let n ∈ N. Let Ω be a bounded open subset of Rn with a finite number of connected components, each of which is regular in the sense of Whitney. Then C 1 (Ω) is compactly embedded into C 0 (Ω).

2.6 Spaces of H¨older and Lipschitz Continuous Functions

31

Proof. We first prove the lemma in the case where Ω is connected and regular in the sense of Whitney. By Lemma 2.28, C 1 (Ω) is continuously embedded into C 0,1 (Ω) and, by Proposition 2.25, C 0,1 (Ω) is compactly embedded into C 0,1/2 (Ω). Since C 0,1/2 (Ω) is continuously embedded into C 0 (Ω) by definition of C 0,1/2 (Ω), the statement follows. Indeed, the composition of a linear and compact map with a linear and continuous map is linear and compact. Next we consider the case where Ω has more than one connected component. Let Ω1 , . . . , Ωκ + be the connected components of Ω and let Ψ be the κ + map from C 0 (Ω) to l=1 C 0 (Ωl ) that takes f to the κ + -tuple of restrictions (f|Ω1 , . . . , f|Ω + ). We can see that in general Ψ is not surjective. However, by κ

the definition of the involved norms, Ψ is a linear homeomorphism from C 0 (Ω) κ + onto its image Ψ (C 0 (Ω)) ⊆ l=1 C 0 (Ωl ) and restricts to a linear homeomorphism κ + from C 1 (Ω) onto Ψ (C 1 (Ω)) ⊆ l=1 C 1 (Ωl ). Since C 0 (Ω) is complete, it follows κ + that Ψ (C 0 (Ω)) is complete and therefore closed in l=1 C 0 (Ωl ). Now let {fj }j∈N be a bounded sequence in C 1 (Ω), then its Ψ -image {(fj|Ω1 , . . . , fj|Ω + )}j∈N is κ κ + bounded in l=1 C 1 (Ωl ). By the case in which Ω is connected, there exists a subsequence {(fjk |Ω1 , . . . , fjk |Ω + )}k∈N of {(fj|Ω1 , . . . , fj|Ω + )}j∈N that converges κ κ κ + to an element g = (g1 , . . . , gκ + ) in l=1 C 0 (Ωl ). Since Ψ (C 0 (Ω)) is closed in κ + 0 0 l=1 C (Ωl ), there exists f ∈ C (Ω) such that g = Ψ (f ). Since Ψ is a linear homeomorphism onto its image, we have lim fjk = lim Ψ (−1) (fjk |Ω1 , . . . , fjk ||Ω

k→∞

k→∞

κ+

) = Ψ (−1) (g) = f

and thus the proof is complete.

in C 0 (Ω)  

As the following example shows, the above Lemma 2.29 does not hold if Ω has an infinite number of connected components which are regular in the sense of Whitney. Example 2.30. Let Ωj ≡ Bn ((j + 1)−1 e1 , (j + 2)−3 ) for all j ∈ N. Since Ωj is convex, we have c[Ωj ] = 1 for all j ∈ N. Since the open balls Ωj are pairwise disjoint, the open set Ω ≡ j∈N Ωj has a countable family of connected components   each of which is regular in the sense of Whitney. Moreover, Ω = {0}∪ Ω . j j∈N Now let fl be the function from Ω to R defined by ! 1 if x ∈ 0≤j≤l Ωj ,  fl (x) = 0 if x ∈ {0} ∪ l 0 such that Bn (p, r) ⊆ Ω. Since both Bn (p, r) and Ωj are connected and Bn (p, r) ∩ Ωj = ∅, the union Bn (p, r) ∪ Ωj is connected. Since Bn (p, r) ∪ Ωj ⊆ Ω, the maximality of the connected component Ωj implies that Bn (p, r) ∪ Ωj ⊆ Ωj , and accordingly Bn (p, r) cannot intersect the complement of Ωj , a contradiction. Hence, p ∈ ∂Ω. We now show that Ωj is a local strict hypograph of class C 0 . If p ∈ ∂Ωj , then we have just shown that p ∈ ∂Ω and thus there exist R ∈ On (R), r, δ ∈]0, +∞[ such that C(p, R, r, δ) is a coordinate cylinder for Ω around p. Since Ω ∩ C(p, R, r, δ) is path connected and contained in Ω, it can intersect only one connected component of Ω, and it must be contained in such a component. Since p ∈ ∂Ωj , C(p, R, r, δ) must intersect Ωj . Accordingly Ω ∩ C(p, R, r, δ) ⊆ Ωj ⊆ Ω and thus Ω ∩ C(p, R, r, δ) = Ωj ∩ C(p, R, r, δ) . Hence, R(Ωj − p) ∩ (Bn−1 (0, r)×] − δ, δ[) = R(Ω − p) ∩ (Bn−1 (0, r)×] − δ, δ[) is the strict hypograph of some γ ∈ C 0 (Bn−1 (0, r), ] − δ, δ[) such that γ(0) = 0 and |γ(η)| < δ/2 for all η ∈ Bn−1 (0, r). Then C(p, R, r, δ) is a coordinate cylinder for Ωj around p and Ωj is a local strict hypograph of class C 0 . In order to show that the number of connected components of Ω is finite, we assume by contradiction that Ω has a sequence {Ωj }j∈N of (pairwise disjoint) connected components. Then we choose a point xj ∈ Ωj for each j ∈ N. By the compactness of Ω the sequence {xj }j∈N has a subsequence, which we still denote by the same symbol, which converges to a point x ˜ ∈ Ω. If x ˜ ∈ Ω, then x ˜ can belong to a single connected component of Ω and such a ˜ for all component is open. Hence, xj is contained in the very same component of x j large enough, a contradiction. Indeed, each connected component can contain xj only for one index j. Hence, x ˜ must belong to ∂Ω. x, R, r, δ) is a coordiThen there exist R ∈ On (R), r, δ ∈]0, +∞[ such that C(˜ nate cylinder for Ω around x ˜. Since Ω ∩ C(˜ x, R, r, δ) is path connected, there exists one and only one l ∈ N such that Ωl ∩ C(˜ x, R, r, δ) = Ω ∩ C(˜ x, R, r, δ)

38

2 Preliminaries

(see the above argument). Since C(˜ x, R, r, δ) is a neighborhood of x ˜ = limj→∞ xj , x, R, r, δ) should contain xj for all the set Ω ∩ C(˜ x, R, r, δ) and thus the set Ωl ∩ C(˜ j large enough and not only for the index j = l, a contradiction. Hence, the number of connected components of Ω is finite. + Next we prove that the sets of the family {Ωj }κ j=1 are pairwise disjoint. Assume that there exist indices j1 , j2 ∈ {1, . . . , κ + } and a point p ∈ Ω such that j1 = j2 and p ∈ Ωj1 ∩ Ωj2 . Since Ωj1 and Ωj2 are disjoint and open, we must have p ∈ ∂Ωj1 ∩ ∂Ωj2 . Since p ∈ ∂Ωj1 ∩ ∂Ωj2 ⊆ ∂Ω, there exist R ∈ On (R), r, δ ∈]0, +∞[ such that C(p, R, r, δ) is a coordinate cylinder for Ω around p. Next we note that C(p, R, r, δ) is a neighborhood of p and that p is a boundary point for both Ωj1 and Ωj2 . Then we must have Ωj1 ∩ C(p, R, r, δ) = ∅ and

Ωj2 ∩ C(p, R, r, δ) = ∅ .

Since Ω ∩ C(p, R, r, δ) is path connected, it must be contained in exactly one connected component of Ω and thus there exists a unique l ∈ N such that Ω ∩ C(p, R, r, δ) = Ωl ∩ C(p, R, r, δ) (see the above argument). Then the inclusion Ωj1 , Ωj2 ⊆ Ω implies that ∅ = Ωjs ∩ C(p, R, r, δ) ⊆ Ω ∩ C(p, R, r, δ) = Ωl ∩ C(p, R, r, δ)

∀s ∈ {1, 2}

and Ωjs ∩ Ωl = ∅ for all s ∈ {1, 2}. Since the connected components are pairwise disjoint, it follows that Ωj1 = Ωl = Ωj2 and j1 = j2 , a contradiction. Hence, Ωj1 ∩ Ωj2 = ∅. We now prove statement (ii). Let r ∈]0, +∞[ be such that Ω ⊆ Bn (0, r). Then we note that Rn \ Bn (0, r) is contained in Ω − and intersects all unbounded connected components of Ω − . Moreover, Rn \ Bn (0, r) is connected. Then any unbounded connected component of Ω − must contain Rn \ Bn (0, r). It follows that Ω − has only one unbounded connected component. Then, by the same argument used to prove statement (i), we conclude that statement (ii) holds true.  

2.8 Tangent Space to a Local Strict Hypograph Definition 2.39. Let n ∈ N \ {0}. Let M be a subset of Rn , p ∈ M . We say that the vector w ∈ Rn is semi-tangent to M at the point p provided that either w = 0 or there exists a sequence {xj }j∈N in M \ {p} which converges to p and such that xj − p w = lim . j→∞ |w| |xj − p| We say that the vector w ∈ Rn is tangent to M at the point p provided that both w and −w are semi-tangent to M at the point p.

2.8 Tangent Space to a Local Strict Hypograph

39

We denote by Tp M the set of all semi-tangent vectors to M at p. One can easily check that Tp M is a cone of Rn , i.e., that λw ∈ Tp M

whenever

(λ, w) ∈]0, +∞[×Tp M .

We say that Tp M is the cone of semi-tangent vectors to M at p. If Tp M is also a subspace of Rn , then we say that M has a tangent space at p, that Tp M is the tangent space to M at p and that p + Tp M is the affine tangent space to M at p. Lemma 2.40. Let n ∈ N \ {0}. Let M be a subset of Rn . Let γ be a function from an interval J of R to M . Let t˜ ∈ J. (i) If Jt˜+ ≡ J ∩ [t˜, +∞[ is nondegenerate and if γ is right differentiable at t˜ then its right derivative γr (t˜) is semi-tangent to M at γ(t˜). (ii) If Jt˜− ≡ J∩] − ∞, t˜] is nondegenerate, γ is left differentiable at t˜, and γl (t˜) denotes its left derivative, then the vector −γl (t˜) is semi-tangent to M at γ(t˜). Proof. If either γr (t˜) or γl (t˜) are equal to 0, then they are semi-tangent to M at γ(t˜) by definition. Thus we can assume that γr (t˜) = 0 and that γl (t˜) = 0. Now let {tj }j∈N be a sequence in J \ {t˜} which converges to t˜. In case (i) we assume that the terms of the sequence belong to Jt˜+ \ {t˜} and in case (ii) we assume that the terms of the sequence belong to Jt˜− \ {t˜}. Since the limit γr (t˜) or γl (t˜) of the incremental ratio of γ cannot be equal to zero, possibly neglecting a finite number of indices, we can assume that γ(tj ) − γ(t˜) = 0 for all j ∈ N. Then we have ⎧  γ (t˜) γ(tj )−γ(t˜) ⎪ ⎨ |γr (t˜)| in case (i) , ˜ γ(tj ) − γ(t) tj −t˜ r = lim γ(t )−γ(t˜) sgn (tj − t˜) = lim  j j→∞ |γ(tj ) − γ(t˜)| j→∞ ⎪ ⎩ − γl (t˜˜) in case (ii) . | t −t˜ | j |γ (t)| l

It follows that γr (t˜) is semi-tangent to M at γ(t˜) in case (i) and that −γl (t˜) is semitangent to M at γ(t˜) in case (ii).   Lemma 2.41. Let m, n ∈ N, 0 < m ≤ n. Let U be an open subset of Rm , x ∈ U . Let M be a subset of Rn . Let ϕ be a function from U to M . Let ϕ be differentiable at x. Then each vector of dϕ(x)[Rm ] is semi-tangent to M at p ≡ ϕ(x). Proof. Indeed, if v ∈ Rm , then the continuity of the product by a scalar number implies that there exists δ ∈]0, +∞[ such that x + tv ∈ U

∀t ∈ [−δ, δ]

and we can consider the map γ from [−δ, δ] to M defined by γ(t) ≡ ϕ(x + tv)

∀t ∈ [−δ, δ] .

40

2 Preliminaries

By the chain rule

γ  (0) = dϕ(x)[v] ,

and the previous Lemma 2.40 implies that γ  (0) is semi-tangent to M at the point γ(0) = ϕ(x) = p.   Proposition 2.42. Let n ∈ N \ {0, 1}. Let Ω be an open local strict hypograph of class C 0 of Rn . Let p ∈ ∂Ω. Let R ∈ On (R), r, δ ∈]0, +∞[ be such that C(p, R, r, δ) is a coordinate cylinder for Ω around p. Let the function γ in C 0 (Bn−1 (0, r)) represent ∂Ω as a graph in C(p, R, r, δ). Let η˜ ∈ Bn−1 (0, r). Then the following statements hold. (i) If γ is differentiable at η˜, then the cone of semi-tangent vectors Tψp (˜η) (∂Ω) to ∂Ω at the point   η˜ t η) ≡ p + R ψp (˜ γ(˜ η) is a vector subspace of Rn of dimension n − 1 and the unit vector   1 −∇γ(˜ η) t # νΩ (ψp (˜ η )) ≡ R 1 1 + |∇γ(˜ η )|2 is normal to Tψp (˜η) (∂Ω). η , rη˜) ⊆ Bn−1 (0, r) and (ii) If there exists rη˜ ∈]0, +∞[ such that Bn−1 (˜ qη˜,1 (γ) ≡

|γ(η) − γ(˜ η )| < +∞ , |η − η˜| η∈Bn−1 (˜ η ,rη˜ )\{˜ η} sup

and if the cone Tψp (˜η) (∂Ω) of semi-tangent vectors to ∂Ω at the point ψp (˜ η ) is a vector subspace of Rn of dimension n − 1, then γ is differentiable at η˜ and the η )) of statement (i) is normal to Tψp (˜η) (∂Ω). unit vector νΩ (ψp (˜ Proof. (i) Since the map from Rn to itself that takes w to p + Rt w is an affine isometry, it suffices to show that the cone T(˜η,γ(˜η)) graph(γ) of semi-tangent vectors is a subspace of Rn of dimension (n − 1) and that   1 −∇γ(˜ η) # η , γ(˜ η )) ≡ νgraph(γ) (˜ 1 1 + |∇γ(˜ η )|2 is normal to T(˜η,γ(˜η)) graph(γ). We set g(η, y) ≡ y − γ(η) for all (η, y) in η , γ(˜ η )), and Bn−1 (0, r)×] − δ, δ[. Then g is continuous, is differentiable at (˜   −∇γ(˜ η) ∇g(˜ η , γ(˜ η )) =

= 0 . 1 Then the kernel Ker dg(˜ η , γ(˜ η )) of the differential dg(˜ η , γ(˜ η )) of g at (˜ η , γ(˜ η )) has dimension (n − 1). We now show that

2.8 Tangent Space to a Local Strict Hypograph

41

T(˜η,γ(˜η)) graph(γ) ⊆ Ker dg(˜ η , γ(˜ η )) .

(2.19)

To do so, we take v ∈ T(˜η,γ(˜η)) graph(γ) and we show that ∇g(˜ η , γ(˜ η )) · v = 0. We can clearly assume that v = 0. Since v is semi-tangent to graph(γ) at the point η } such that (˜ η , γ(˜ η )), there exists a sequence {ηj }j∈N in Bn−1 (0, r) \ {˜ (˜ η , γ(˜ η )) = lim (ηj , γ(ηj )) , j→∞

v (ηj , γ(ηj )) − (˜ η , γ(˜ η )) = lim . |v| j→∞ |(ηj , γ(ηj )) − (˜ η , γ(˜ η ))| Since g is differentiable at (˜ η , γ(˜ η )), we have lim

j→∞

|g(ηj , γ(ηj )) − g(˜ η , γ(˜ η )) − dg(˜ η , γ(˜ η ))[ηj − η˜, γ(ηj ) − γ(˜ η )]| = 0. |(ηj − η˜, γ(ηj ) − γ(˜ η ))|

Moreover, g(ηj , γ(ηj )) = g(˜ η , γ(˜ η )) = 0 for all j ∈ N and thus we have $ % (ηj − η˜, γ(ηj ) − γ(˜ η )) η , γ(˜ η )) lim dg(˜ = 0. j→∞ |(ηj − η˜, γ(ηj ) − γ(˜ η ))| Since the argument of dg(˜ η , γ(˜ η )) tends to v/|v| as j tends to infinity and the linear map dg(˜ η , γ(˜ η )) is continuous, we deduce that $ % v dg(˜ η , γ(˜ η )) = 0. |v| Accordingly, v ∈ Ker dg(˜ η , γ(˜ η )). Hence, we have proved that inclusion (2.19) holds true. Next we set ϕ(η) ≡ (η, γ(η))

∀η ∈ Bn−1 (0, r) .

By the previous Lemma 2.41, we have dϕ(˜ η )[Rn−1 ] ⊆ T(˜η,γ(˜η)) graph(γ) and thus we have dϕ(˜ η )[Rn−1 ] ⊆ T(˜η,γ(˜η)) graph(γ) ⊆ Ker dg(˜ η , γ(˜ η )) .   In−1 has rank n − 1, the map dϕ(˜ η ) is injective and thus Since Dϕ(˜ η) = Dγ(˜ η) η , γ(˜ η )) is also dϕ(˜ η )[Rn−1 ] is a vector space of dimension (n − 1). Since Ker dg(˜ a vector space of dimension (n − 1), we conclude that dϕ(˜ η )[Rn−1 ] = T(˜η,γ(˜η)) graph(γ) = Ker dg(˜ η , γ(˜ η )) and that T(˜η,γ(˜η)) graph(γ) is a vector space of dimension (n − 1). Since the vector

42

2 Preliminaries



−∇γ(˜ η) 1



is normal to Ker dg(˜ η , γ(˜ η )), we conclude that νgraph(γ) (˜ η , γ(˜ η )) is normal to the space T(˜η,γ(˜η)) graph(γ). We now turn to the proof of statement (ii). Since the map from Rn to itself that takes w to p + Rt w is an affine isometry, the cone T(˜η,γ(˜η)) graph(γ) of semitangent vectors is a subspace of Rn of dimension (n − 1). Let the unit vector ν ≡ (ν1 , . . . , νn ) be orthogonal to T(˜η,γ(˜η)) graph(γ). We first show that νn = 0. Assume by contradiction that νn = 0. Then we have νn = ν · en = 0 and accordingly en ∈ T(˜η,γ(˜η)) graph(γ). Then there exists a sequence {ηj }j∈N in η , rη˜) \ {˜ η } which converges to η˜ and such that Bn−1 (˜ lim (ηj , γ(ηj )) = (˜ η , γ(˜ η )) ,

j→∞

(ηj , γ(ηj )) − (˜ η , γ(˜ η )) = en . j→∞ |(ηj , γ(ηj )) − (˜ η , γ(˜ η ))| lim

In particular,

ηj − η˜ = 0. j→∞ |(ηj , γ(ηj )) − (˜ η , γ(˜ η ))| lim

By assumption, we have |(ηj , γ(ηj )) − (˜ η , γ(˜ η ))| |ηj − η˜| # & |ηj − η˜|2 + |γ(ηj ) − γ(˜ η )|2 = ≤ 1 + qη2˜,1 |ηj − η˜|

∀j ∈ N ,

with qη˜,1 as in statement (ii) of the proposition. Thus the above limiting relation implies that lim

j→∞

ηj − η˜ |ηj − η˜| = lim

j→∞

ηj − η˜ η , γ(˜ η ))| |(ηj , γ(ηj )) − (˜ = 0, |(ηj , γ(ηj )) − (˜ η , γ(˜ η ))| |ηj − η˜|

a contradiction. Hence, νn = 0. Next we turn to show that γ is differentiable at η˜ and that 1 ∀v ∈ Rn−1 . dγ(˜ η )[v] = − v · (ν1 , . . . , νn−1 ) νn Clearly, the operator in the right-hand side is linear and continuous in v. Thus, we have to show that lim

v→0

γ(˜ η + v) − γ(˜ η) +

1 νn v

|v|

· (ν1 , . . . , νn−1 )

= 0.

2.8 Tangent Space to a Local Strict Hypograph

43

It suffices to show that if {vj }j∈N is a sequence in Bn−1 (0, rη˜) \ {0} which converges to 0, then each subsequence {vjk }k∈N of {vj }j∈N has a further subsequence {vjkl }l∈N such that γ(˜ η + vjkl ) − γ(˜ η) +

lim

1 ν n v jk l

· (ν1 , . . . , νn−1 )

|vjkl |

l→∞

= 0.

Since the sequence 

((˜ η + vjk ) − η˜, γ(˜ η + vjk ) − γ(˜ η )) |(˜ η + vjk ) − η˜, γ(˜ η + vjk ) − γ(˜ η ))|

 k∈N

is bounded, then there exists a further subsequence {vjkl }l∈N and a unit vector w in T(˜η,γ(˜η)) graph(γ) such that lim

l→∞

((˜ η + vjkl ) − η˜, γ(˜ η + vjkl ) − γ(˜ η )) |(˜ η + vjkl ) − η˜, γ(˜ η + vjkl ) − γ(˜ η ))|

= w.

Then, by taking the scalar product with ν, we have lim

vjkl · (ν1 , . . . , νn−1 ) + (γ(˜ η + vjkl ) − γ(˜ η ))νn |(vjkl , γ(˜ η + vjkl ) − γ(˜ η ))|

l→∞

= w · ν = 0.

Next we note that & |(vjkl , γ(˜ η + vjkl ) − γ(˜ η ))| |vjkl |

=

η + vjkl ) − γ(˜ η ))2 |vjkl |2 + (γ(˜ |vjkl |

≤

&

1 + qη2˜,1

for all l ∈ N, and thus the above limiting relation implies that lim

vjkl · (ν1 , . . . , νn−1 ) + (γ(˜ η + vjkl ) − γ(˜ η ))νn |vjkl |

l→∞

= lim

l→∞

×

vjkl · (ν1 , . . . , νn−1 ) + (γ(˜ η + vjkl ) − γ(˜ η ))νn |(vjkl , γ(˜ η + vjkl ) − γ(˜ η ))|

|(vjkl , γ(˜ η + vjkl ) − γ(˜ η ))| |vjkl |

=0

and γ is differentiable at η˜. Then statement (i) implies that the unit vector νΩ (ψp (˜ η ))   of statement (i) is normal to Tψp (˜η) (∂Ω) and the proof is complete. Definition 2.43. Let Ω be an open subset of Rn . Let x ∈ ∂Ω. Let v ∈ Rn \ {0}. (i) We say that v points to the interior of Ω at x provided that there exists ςx,v > 0 such that x + tv ∈ Ω ∀t ∈]0, ςx,v [ .

44

2 Preliminaries

(ii) We say that v points to the exterior of Ω at x provided that there exists ςx,v > 0 such that ∀t ∈]0, ςx,v [ . x + tv ∈ Rn \ Ω Then we have the following proposition, which gives a sufficient condition in order that a vector points to the interior or to the exterior of a local strict hypograph at a boundary point at which the tangent space does exist. Proposition 2.44. Let n ∈ N \ {0, 1}. Let Ω be an open local strict hypograph of class C 0 . Let p ∈ ∂Ω. Let R ∈ On (R), r, δ ∈]0, +∞[ be such that C(p, R, r, δ) is a coordinate cylinder for Ω around p. Let the function γ ∈ C 0 (Bn−1 (0, r)) represent ∂Ω as a graph in C(p, R, r, δ). If there exists r0 ∈]0, r[ such that q1 (γ) ≡

|γ(η)| < +∞ η∈Bn−1 (0,r0 )\{0} |η| sup

and if the cone Tp (∂Ω) of semi-tangent vectors to ∂Ω at the point p is a vector subspace of Rn of dimension n − 1, then γ is differentiable at 0, the unit vector   1 −∇γ(0) # νΩ (p) ≡ Rt 1 1 + |∇γ(0)|2 is normal to Tp (∂Ω) and points to the exterior of Ω, and −νΩ (p) points to the interior of Ω. If v ∈ Rn and if v · νΩ (p) > 0, then v points to the exterior of Ω. If instead v · νΩ (p) < 0, then v points to the interior of Ω. Proof. By Proposition 2.42 (ii) we already know that γ is differentiable at 0 and that νΩ (p) is normal to Tp (∂Ω). So, it suffices to prove the last statement on v and then take v = νΩ (p) to deduce the rest. Let ξ ≡ (ξ  , ξn ) ≡ Rv ,

νγ (p) ≡ RνΩ (p) .

Since the map that takes z to p + Rt z is the composition of a translation with an orthogonal transformation, it suffices to show that if ξ · νγ (p) > 0, then ξ points to the exterior of hypographs (γ) at (0, γ(0)) and if ξ · νγ (p) < 0 then ξ points to the interior of hypographs (γ) at (0, γ(0)). We first consider the case in which ξ · νγ (p) > 0. Since γ is differentiable at 0, then we have γ(tξ  ) − γ(0) − ξn = ∇γ(0) · ξ  − ξn t→0 t # ∇γ(0) · ξ  = 1 + |∇γ(0)|2 # − ξn 1 + |∇γ(0)|2 # = −ξ · νγ (p) 1 + |∇γ(0)|2 . lim

If ξ · νγ (p) > 0, then there exists tξ ∈]0, +∞[ such that

2.9 Lipschitz Subsets of Rn

γ(tξ  ) − γ(0) < tξn

45

∀t ∈]0, tξ [ ,

i.e., such that (0, γ(0)) + tξ ∈ Rn \ hypographs (γ) for all t ∈]0, tξ [. Hence, ξ points to the exterior of hypographs (γ) at (0, γ(0)). A proof in case ξ · νγ (x) < 0 can be effected by the same argument.  

2.9 Lipschitz Subsets of Rn Definition 2.45. Let n ∈ N \ {0, 1}. We say that an open subset Ω of Rn is a Lipschitz subset of Rn provided that for every p ∈ ∂Ω, there exist R ∈ On (R) and r, δ ∈]0, +∞[ such that C(p, R, r, δ) is a coordinate cylinder for Ω around p and that the corresponding function γ which represents ∂Ω as a graph in C(p, R, r, δ) is Lipschitz continuous. By definition, a Lipschitz subset of Rn is also a local strict hypograph of class C . If a Lipschitz subset Ω of Rn is also bounded, then Lemma 2.38 implies that Ω and of Ω − have a finite number of connected components which are also Lipschitz subsets of Rn . We now consider the question of the Whitney regularity of Lipschitz sets and we first introduce the following. 0

Lemma 2.46. Let n ∈ N \ {0, 1}, r, δ, δ1 ∈]0, +∞[, δ1 < δ. Let γ belong to Lip (Bn−1 (0, r), ] − δ, δ[) and |γ(η)| < δ1 for all η ∈ Bn−1 (0, r). Then the strict hypograph hypographs (γ) ≡ {(η, y) ∈ Bn−1 (0, r)×] − δ, δ[: y < γ(η)} is regular in the sense of Whitney (cf. Definition 2.16). Proof. Let (˜ η , y˜), (η  , y  ) ∈ hypographs (γ). In order to define a path which joins η ) − y˜, γ(η  ) − y  }[ the two points (˜ η , y˜) and (η  , y  ), we fix δ  ∈]0, min{δ − δ1 , γ(˜ and we define the function ϕ by setting   ∀t ∈ [0, 1] . ϕ(t) ≡ (a(t), b(t)) ≡ η˜ + t(η  − η˜), γ(˜ η + t(η  − η˜)) − δ  Clearly, ϕ(0) = (˜ η , γ(˜ η) − δ ) , ϕ(1) = (η  , γ(η  ) − δ  ) ,  −δ < −δ1 − δ ≤ γ(˜ η + t(η  − η˜)) − δ  < γ(˜ η + t(η  − η˜))

∀t ∈ [0, 1] .

In particular, ϕ(t) ∈ hypographs (γ) for all t ∈ [0, 1]. Since ϕ is a composition of Lipschitz continuous maps, ϕ is Lipschitz continuous. The length of ϕ satisfies the following inequality

46

2 Preliminaries



1

length (ϕ) ≤

&

|˜ η − η  |2 + (Lip(γ))2 |˜ η − η  |2 dt = |˜ η −η  |

#

1 + (Lip(γ))2 .

0

If the set B ≡ {t ∈ [0, 1] : y˜ + t(y  − y˜) ≥ b(t)} is empty, then y˜ + t(y  − y˜) < b(t) < γ(˜ η + t(η  − η˜))

∀t ∈ [0, 1]

and the segment with end points (˜ η , y˜) and (η  , y  ) lies entirely in hypographs (γ). Then we have the following equality for the geodesic distance λ ≡ λhypographs (γ) in hypographs (γ) λ((˜ η , y˜), (η  , y  )) = |(˜ η , y˜) − (η  , y  )| .

(2.20)

If instead B = ∅, then we set t1 ≡ min B ,

t2 ≡ max B .

Since δ  is smaller than γ(˜ η ) − y˜ and γ(η  ) − y  , we have 0, 1 ∈ / B. Then we have 0 < t1 < t2 < 1, and the continuity of γ implies that y˜ + tr (y  − y˜) = b(tr )

r ∈ {1, 2} .

Since the points (˜ η , y˜) and (η  , y  ) can be joined by the piecewise Lipschitz path η , y˜) and ϕ(t1 ) by a segment, ϕ(t1 ) and in hypographs (γ) obtained by joining (˜ ϕ(t2 ) by the restriction of ϕ to [t1 , t2 ], and ϕ(t2 ) and (η  , y  ) by a segment, then the following inequality for the geodesic distance λ((˜ η , y˜), (η  , y  )) holds λ((˜ η , y˜), (η  , y  )) ≤ |(˜ η , y˜) − ϕ(t1 )| + |˜ η−η | 

#

(2.21) 1+

(Lip(γ))2

≤ 2|(˜ η , y˜) − (η , y )| + |(˜ η , y˜) − (η , y )| 







#

+ |ϕ(t2 ) − (η , y )| 



1 + (Lip(γ))2 .

By (2.20) and (2.21), we conclude that the Whitney constant c[hypographs (γ)] of hypographs (γ) satisfies the following inequality # c[hypographs (γ)] ≤ 2 + 1 + (Lip(γ))2 .   Thus we are ready to prove the following. Proposition 2.47. Let n ∈ N \ {0, 1}. A bounded open connected Lipschitz subset Ω of Rn is regular in the sense of Whitney. Proof. As usual, we denote by λΩ (p, q) the geodesic distance in Ω between two points p and q of Ω. Assume by contradiction that Ω is not regular in the sense of Whitney. Then there exists a sequence {(pj , qj )}j∈N in Ω × Ω such that

2.9 Lipschitz Subsets of Rn

pj = qj

47

∀j ∈ N ,

λΩ (pj , qj ) = +∞ . j→∞ |pj − qj | lim

(2.22)

Since Ω × Ω is compact, there exist a subsequence of {(pj , qj )}j∈N which we still denote by the same symbol and (P, Q) ∈ Ω × Ω such that lim pj = P ,

j→∞

lim qj = Q .

(2.23)

j→∞

We now define a set A(x) for each point x of Ω. If x ∈ Ω, then we denote by A(x) an open ball with center at x and closure contained in Ω. Since A(x) is convex, A(x) is regular in the sense of Whitney. If instead x ∈ ∂Ω, then we denote by A(x) the intersection of Ω with a coordinate cylinder for Ω around x. Since A(x) is a translation of a rotation of a strict hypograph, Lemma 2.46 implies that A(x) is regular in the sense of Whitney. Hence, A(x) is regular in the sense of Whitney for all x ∈ Ω. In particular, we have c[A(P )] < +∞ ,

c[A(Q)] < +∞ .

(2.24)

Now let P1 ∈ A(P )\{P }, Q1 ∈ A(Q)\{Q}. We can clearly assume that P1 = Q1 . By (2.23) there exists j0 ∈ N such that pj ∈ A(P ) , qj ∈ A(Q)

∀j ∈ N, j ≥ j0 .

By (2.24) there exist a path γpj ,P1 ∈ Lip ([0, 1], A(P )) and a path γQ1 ,qj ∈ Lip ([0, 1], A(Q)) such that γpj ,P1 (0) = pj , γpj ,P1 (1) = P1 , γQ1 ,qj (0) = Q1 , γQ1 ,qj (1) = qj , and length(γpj ,P1 ) ≤ (c[A(P )] + 1)|pj − P1 | , length(γQ1 ,qj ) ≤ (c[A(Q)] + 1)|Q1 − qj | for each j ≥ j0 . Since Ω is connected, there exists a path γP1 ,Q1 ∈ Lip ([0, 1], Ω) such that γP1 ,Q1 (1) = Q1 . γP1 ,Q1 (0) = P1 , Then we have |pj − P1 | λΩ (pj , qj ) ≤ (c[A(P )] + 1) |pj − qj | |pj − qj | |Q1 − qj | length(γP1 ,Q1 ) + (c[A(Q)] + 1) + |pj − qj | |pj − qj | Since

j ∈ N, j ≥ j0 .

48

2 Preliminaries

lim |pj − P1 | = |P − P1 | ∈]0, +∞[ and lim |Q1 − qj | = |Q1 − Q| ∈]0, +∞[ ,

j→∞

j→∞

(2.25) the limiting relation in (2.22) implies that limj→∞ |pj − qj | = 0 and accordingly that P = Q. Since pj , qj ∈ A(P ) = A(Q) for j ≥ j0 and A(P ) is regular in the sense of Whitney and contained in Ω, we must have λA(P ) (pj , qj ) λΩ (pj , qj ) ≤ ≤ c[A(P )] |pj − qj | |pj − qj |

∀j ≥ j0 ,  

in contradiction with (2.22). Next we prove the following technical lemma.

Lemma 2.48. Let n ∈ N \ {0, 1}, r ∈]0, +∞[. Then Rn \ Bn (0, r) is regular in the sense of Whitney. Proof. Since Rn \Bn (0, r) is pathwise connected, it suffices to show that there exists c ∈ [1, +∞[ such that, for all given x, y ∈ Rn \ Bn (0, r) with x = y, we can find a path γx,y ∈ Lip ([0, 1], Rn ) as in (2.11) with Ω = Rn \ Bn (0, r) such that length (γx,y ) ≤ c|x − y| . √ We claim that we can take c = π/ 2 (although we are not saying that this is the best constant). Indeed, let Π be the two-dimensional plane that contains x, y, and the origin of Rn . Since Π is linearly isometric to C, which we identify with R2 , and orthogonal transformations in C are isometries, it suffices to show that if ξ = ρ1 and η = ρ2 eiθ with ρ1 , ρ2 , and θ real numbers such that r < ρ2 ≤ ρ1 and θ ∈ [0, π], then we can find a path γ˜ξ,η ∈ Lip ([0, 1], C) as in (2.11) with Ω = C \ B2 (0, r) such that √ length (˜ γξ,η ) ≤ (π/ 2)|ξ − η| . To do so, we join ξ and η with the path γ˜ξ,η that is the composite of t ∈ [0, 1] → ρ1 + t(ρ2 − ρ1 ) ∈ C and t ∈ [0, 1] → ρ2 eitθ ∈ C . We can verify that γ˜ξ,η is a Lipschitz path, that γ˜ξ,η is contained in C \ Bn (0, r), and that we have length(˜ γξ,η ) = (ρ1 − ρ2 ) + ρ2 θ . To compare length(˜ γξ,η ) with |ξ − η|, we note that |ξ − η|2 = ρ21 + ρ22 − 2ρ1 ρ2 cos θ = (ρ1 − ρ2 )2 + 2ρ1 ρ2 (1 − cos θ) , that

and we compute

1 − cos θ = sin2 (θ/2) ≥ (θ/π)2 2

∀θ ∈ [0, π] ,

2.9 Lipschitz Subsets of Rn

49

2

(length(˜ γξ,η )) ≤ 2(ρ1 − ρ2 )2 + 2(ρ2 θ)2 ≤ 2(ρ1 − ρ2 ) + 2ρ1 ρ2 π 2

Accordingly,

2



1 − cos θ 2

 ≤

π2 |ξ − η|2 . 2

π length(˜ γξ,η ) ≤ √ |ξ − η| 2  

and the proof is complete. Then we can prove the following.

Proposition 2.49. Let n ∈ N \ {0, 1}. If Ω is a bounded open Lipschitz subset of Rn , then all the connected components of Ω and of Ω − are Lipschitz sets and are regular in the sense of Whitney. Proof. Since every bounded open Lipschitz subset of Rn has at most a finite number of connected components, each of which is a Lipschitz subset of Rn , we conclude that each one of such components is regular in the sense of Whitney. Since the exterior Ω − of Ω has also at most a finite number of connected components, each of which is a Lipschitz subset of Rn , all the bounded connected components of the exterior Ω − are regular in the sense of Whitney, and thus it suffices to show that the unbounded connected component (Ω − )0 of Ω − is regular in the sense of Whitney. Now let r ∈]1, +∞[ be such that Ω ⊆ Bn (0, r − 1). Then both (Ω − )0 ∩ Bn (0, r)

and

(Ω − )0 ∩ Bn (0, r + 1)

are bounded open connected Lipschitz subsets of Rn and are accordingly regular in the sense of Whitney. We also note that the sets (Ω − )0 ∩ (Rn \ Bn (0, r − 1)) = Rn \ Bn (0, r − 1) , (Ω − )0 ∩ (Rn \ Bn (0, r)) = Rn \ Bn (0, r) are regular in the sense of Whitney by Lemma 2.48. Now let x, y ∈ (Ω − )0 . If both x and y belong to (Ω − )0 ∩ Bn (0, r), then we have λ(Ω − )0 (x, y) ≤ λ(Ω − )0 ∩Bn (0,r) (x, y) ≤ c[(Ω − )0 ∩ Bn (0, r)]|x − y|. If both x and y belong to (Ω − )0 ∩ (Rn \ Bn (0, r)), then both x and y belong to (Ω − )0 ∩ (Rn \ Bn (0, r − 1)) and thus we have λ(Ω − )0 (x, y) ≤ λ(Ω − )0 ∩(Rn \Bn (0,r−1)) (x, y) ≤ c[(Ω − )0 ∩ (Rn \ Bn (0, r − 1))]|x − y| . If instead x ∈ (Ω − )0 ∩ Bn (0, r) and y ∈ (Ω − )0 ∩ (Rn \ Bn (0, r)), then there exists a point ξ in the segment [x, y] such that ξ ∈ ∂Bn (0, r). On the other hand ∂Bn (0, r) ⊆ (Ω − )0 and thus ∂Bn (0, r) ⊆ (Ω − )0 ∩ Bn (0, r + 1). Hence, λ(Ω − )0 (x, ξ) ≤ λ(Ω − )0 ∩Bn (0,r+1) (x, ξ) ≤ c[(Ω − )0 ∩ Bn (0, r + 1)]|x − ξ| .

50

2 Preliminaries

Since ∂Bn (0, r) ⊆ (Ω − )0 ∩ (Rn \ Bn (0, r − 1)), we also have λ(Ω − )0 (ξ, y) ≤ λ(Ω − )0 ∩(Rn \Bn (0,r−1)) (ξ, y) ≤ c[(Ω − )0 ∩ (Rn \ Bn (0, r − 1))]|ξ − y| . Since ξ ∈ [x, y], we have |x − y| = |x − ξ| + |ξ − y| and thus λ(Ω − )0 (x, y) ≤ λ(Ω − )0 (x, ξ) + λ(Ω − )0 (ξ, y) ' ( ≤ max c[(Ω − )0 ∩ Bn (0, r + 1)], c[(Ω − )0 ∩ (Rn \ Bn (0, r − 1))] ×(|x − ξ| + |ξ − y|) ( ' = max c[(Ω − )0 ∩ Bn (0, r + 1)], c[(Ω − )0 ∩ (Rn \ Bn (0, r − 1))] |x − y| . Then the above inequalities imply that (Ω − )0 is regular in the sense of Whitney.

 

We note that if γ is the Lipschitz continuous map associated to a coordinate cylinder C(p, R, r, δ) around a boundary point p of a Lipschitz subset Ω of Rn , then the Rademacher Theorem A.12 implies that γ is differentiable at almost every point of its domain Bn−1 (0, r) and the area element corresponding to the parametrization   η ∀η ∈ Bn−1 (0, r) ψp (η) ≡ p + Rt γ(η) is given by & # det ((Dψp (η))t (Dψp (η))) = 1 + |∇γ(η)|2

for a.e. η ∈ Bn−1 (0, r) .

Then we can define the (n − 1)-dimensional measure on the image of the map ψp by saying that a subset E of the image of the map ψp is measurable if and only if ψp← (E) is measurable in Bn−1 (0, r) and has measure  # mψp (Bn−1 (0,r)) (E) ≡ 1 + |∇γ(η)|2 dη . ψp← (E)

Then, by exploiting a partition of unity, we can define the (n − 1)-dimensional measure on ∂Ω (cf., e.g., Naumann and Simader [232]). We denote by L∂Ω and by m∂Ω (or simply mn−1 ) the σ-algebra of measurable subsets of ∂Ω and the (n − 1)dimensional measure on ∂Ω, respectively. Theorem 2.50 (of Rademacher for Lipschitz Sets). Let n ∈ N \ {0, 1}. Let Ω be an open Lipschitz subset of Rn . Then there exists a subset N of ∂Ω of m∂Ω -measure zero such that if p ∈ (∂Ω) \ N , then the cone Tp (∂Ω) of semi-tangent vectors is a vector space of dimension (n−1), i.e., ∂Ω has a tangent space of dimension (n−1) at p. Proof. Since Ω is a Lipschitz set, for every point x ∈ ∂Ω there exist rx , δx in ]0, +∞[, Rx ∈ On (R) and a coordinate cylinder C(x, Rx , rx , δx ) for Ω around x.

2.9 Lipschitz Subsets of Rn

51

Then ∂Ω ⊆



C(x, Rx , rx , δx ) .

x∈∂Ω

Now each cylinder C(x, Rx , rx , δx ) is a neighborhood of x and accordingly it contains an open cube Qx which contains x and which equals the cartesian product of intervals with rational endpoints. Since the set of rational numbers is countable, the set of cubes {Qx }x∈∂Ω is either finite or countable and there exists a sequence {xj }j∈N in ∂Ω such that {Qx }x∈∂Ω = {Qxj }j∈N . Since {Qx }x∈∂Ω covers ∂Ω, we have   Qx j ⊆ C(xj , Rxj , rxj , δxj ) . ∂Ω ⊆ j∈N

j∈N

Let the Lipschitz continuous function γxj from Bn−1 (0, rj ) to ]−δxj , δxj [ represent ∂Ω in C(xj , Rxj , rxj , δxj ) as a graph, for each j ∈ N. Since γxj is Lipschitz continuous, the Rademacher Theorem implies that there exists a subset Nj of measure zero of Bn−1 (0, rj ) such that γxj is differentiable at all points of Bn−1 (0, rj ) \ Nj . Now let ψxj be the parametrization corresponding to γxj . By definition of (n − 1)dimensional measure on ∂Ω, the set ψxj (Nj ) has m∂Ω -measure equal to zero. Moreover, Proposition 2.42 (i) implies that the cone of semi-tangent vectors to ∂Ω at the points of ψxj (Bn−1 (0, rj ) \ Nj ) is a vector space of dimension (n − 1). Now let  ψxj (Nj ) . N≡ j∈N

Since N is a finite or countable union of sets of m∂Ω -measure zero, then m∂Ω (N ) equals 0. Moreover, if p ∈ ∂Ω \ N , then the cone of semi-tangent vectors to ∂Ω at p is a vector space of dimension (n − 1). Thus the proof is complete.   Proposition 2.51. Let n ∈ N \ {0, 1}. Let Ω be an open Lipschitz subset of Rn . If p ∈ ∂Ω and if ∂Ω has a tangent space Tp (∂Ω) at p of dimension (n − 1), then there exists one and only one unit vector νΩ (p) which is normal to Tp (∂Ω) and such that νΩ (p) points to the exterior of Ω and −νΩ (p) points to the interior of Ω. Proof. Since Ω is a Lipschitz subset of Rn , then there exist r, δ ∈]0, +∞[ such that C(p, R, r, δ) is a coordinate cylinder for Ω around p and the function γ from Bn−1 (0, r) to ] − δ, δ[ which represents ∂Ω in C(p, R, r, δ) is Lipschitz continuous. Since Tp (∂Ω) is a vector space of dimension (n − 1) and γ is Lipschitz continuous, Proposition 2.44 implies that γ is differentiable at 0, that   1 −∇γ(0) t # νΩ (p) ≡ R 1 1 + |∇γ(0)|2

52

2 Preliminaries

is normal to Tp (∂Ω) and points to the exterior of Ω, and that −νΩ (p) points to the interior of Ω. Since the unit vectors which are normal to a vector subspace of dimension (n − 1) are precisely 2 and Ω ∩ Ω − = ∅. We conclude that νΩ (p) is unique and the proof is complete.   We say that the unit vector νΩ (p) of the previous statement is the outward unit normal to ∂Ω at p. By the Rademacher Theorem 2.50, νΩ (p) exists at almost all p ∈ ∂Ω. If no ambiguity can arise, then we simply write ν(p) instead of νΩ (p).

2.10 Elementary Inequalities on the Boundary of a Lipschitz Subset of Rn We first introduce the following elementary lemma, which collects either known inequalities or variants of known inequalities (cf., e.g., Cialdea [53, §8], and the paper [89, Lemma 3.2] with Dondi). We exploit such inequalities in the next chapters of this book. Lemma 2.52. Let n ∈ N, λ ∈ R. Then the following statements hold. (i) 1  |x − y| ≤ |x − y| ≤ 2|x − y| 2 for all x , x ∈ Rn , x = x , y ∈ Rn \ Bn (x , 2|x − x |). (ii)

|x − y|λ ≤ 2|λ| |x − y|λ ,

|x − y|λ ≤ 2|λ| |x − y|λ

for all x , x ∈ Rn , x = x , y ∈ Rn \ Bn (x , 2|x − x |). (iii) There exists qλ ∈]0, +∞[ such that ||x − y|λ − |x − y|λ | ≤ qλ |x − x | |x − y|λ−1 for all x , x ∈ Rn , x = x , y ∈ Rn \ Bn (x , 2|x − x |). Proof. Statement (i) is an immediate consequence of the triangular inequality. Indeed, 1 1 |x − y| ≥ |x − y| − |x − x | ≥ |x − y| − |x − y| = |x − y| , 2 2 1 |x − y| ≤ |x − x | + |x − y| ≤ |x − y| + |x − y| ≤ 2|x − y| . 2 The first inequality in (ii) for λ ≥ 0 follows by raising inequality |x −y| ≤ 2|x −y| of statement (i) to the power λ. Instead, for λ < 0 the same inequality follows by raising inequality |x −y| ≤ 2|x −y| of statement (i) to the power λ. Since statement (i) implies that |x − y| ≤ 2|x − y| ,

|x − y| ≤ 2|x − y| ,

2.10 Elementary Inequalities on the Boundary of a Lipschitz Subset of Rn

53

the second inequality of (ii) can be proved by interchanging the roles of x and x . To prove (iii), we follow Cialdea [53, §8]. We first assume that |x −y| < |x −y|. By the Lagrange’s Mean Value Theorem, there exists ζ ∈ [|x − y|, |x − y|] such that ||x − y|λ − |x − y|λ | ≤ |λ|ζ λ−1 | |x − y| − |x − y| | . If λ ≥ 1, inequality ζ ≤ |x − y| and (i) imply that ζ λ−1 ≤ |x − y|λ−1 ≤ 2|λ−1| |x − y|λ−1 . If λ < 1, inequality ζ ≥ |x − y| implies that ζ λ−1 ≤ |x − y|λ−1 ≤ 2|λ−1| |x − y|λ−1 . Then we have ||x − y|λ − |x − y|λ | |λ−1|

≤ |λ|2

(2.26) 





| |x − y| − |x − y| | |x − y|

λ−1

,

which implies the validity of (iii). Similarly, in case |x − y| > |x − y|, we can prove that (2.26) holds with x and x interchanged. Then (ii) implies the validity of (iii).   Then we have the following classical lemma. Lemma 2.53. Let n ∈ N \ {0, 1}. Let Ω be a bounded open Lipschitz subset of Rn . Then the following statements hold. (i) If λ ∈] − ∞, n − 1[, then cΩ,λ ≡ sup

x∈∂Ω

 ∂Ω

dσy |x − y|λ

is finite. (ii) If λ ∈] − ∞, n − 1[, then cΩ,λ ≡

 sup

sλ−(n−1)

x ∈∂Ω,s∈]0,+∞[

Bn (x ,s)∩∂Ω

dσy − y|λ

|x

is finite. (iii) If λ ∈]n − 1, +∞[, then c Ω,λ ≡

 sup x ∈∂Ω,s∈]0,+∞[

sλ−(n−1) ∂Ω\Bn (x ,s)

dσy |x − y|λ

is finite. Proof. For each x ∈ ∂Ω, there exists a coordinate cylinder C(x, Rx , rx , δx ) around x, and we can assume that rx +δx < diam(∂Ω) so that (∂Ω)\C(x, Rx , rx , δx ) = ∅.

54

2 Preliminaries

Since {C(x, Rx , rx /2, δx ) ∩ ∂Ω}x∈∂Ω is an open cover of ∂Ω and ∂Ω is compact, there exist finite families {x(j) }qj=1 , {Rj }qj=1 , {rj }qj=1 , {δj }qj=1 such that ∂Ω ⊆

q 

(C(x(j) , Rj , rj /2, δj ) ∩ ∂Ω) .

j=1

We denote by {γj }qj=1 and {ψx(j) }qj=1 , the corresponding families of functions which represent ∂Ω in the cylinders {C(x(j) , Rj , rj , δj )}qj=1 as a graph and which parametrize ∂Ω around the points {x(j) }qj=1 , respectively. Let ω ∈]0, +∞[ be a Lebesgue number for the above finite cover of ∂Ω (cf. Theorem A.6). There is no loss of generality in assuming that ω
(n − 1) ≥ 0 and x ∈ Bn (x , s) ∩ ∂Ω ⊆ Aj , we have  |x − y|−λ dσy (∂Ω)\ψx(j) (Bn−1 (0,rj ))



.

(2.29)

(2.30)

dσ(ω  )−λ ≤ mn−1 (∂Ω)(ω  )−λ ,

≤ (∂Ω)\ψx(j) (Bn−1 (0,rj ))

1 an inequality which enables to estimate the integral of |x −y| λ for y in the first set 1 in the right-hand side of equality (2.28). We now estimate the integral of |x −y| λ for y in the second set in the right-hand side of equality (2.28). If y = ψx(j) (η) with η ∈ Bn−1 (0, rj ) and |ψx(j) (η  ) − ψx(j) (η)| ≥ s, then the elementary inequality & |ψx(j) (η  ) − ψx(j) (η)| = |η − η  |2 + |γj (η  ) − γj (η)|2 & ≤ 1 +  |∇γj | 2∞ |η − η  |

implies that |η − η  | ≥ # Since s ≤

ω 6

≤

1 12

s . 1 +  |∇γj | 2∞

minj=1,...,s rj , we have

58

2 Preliminaries

2rj ≥ 2s ≥ #

s . 1 +  |∇γj | 2∞

Moreover, if ξ ∈ Bn−1 (0, rj ), then we have 1 rj + rj < 2rj . 2

|η  − ξ| ≤ |η  | + |ξ| ≤ Hence, 

ψx(j) (Bn−1 (0,rj ))\{ψx(j) (η): η∈Bn−1 (0,rj ),|ψx(j) (η  )−ψx(j) (η)|≥s}

≤ ≤

&

&



1 +  |∇γj | 2∞

 Bn−1 (0,rj )\Bn−1

η ,

√

 1 +  |∇γj | 2∞

& ≤ 1 +  |∇γj | 2∞ sn−1

#

 Bn−1



(η  ,2r

j )\Bn−1

2rj

√

s

|η − η  |−λ dη

1+ |∇γj | 2 ∞

√

 s

|η − η  |−λ dη

1+ |∇γj | 2 ∞

ρ−λ ρn−2 dρsn−1

1+ |∇γj | 2 ∞

 |∇γj | 2∞

η ,

 s

|x − y|−λ dσy

⎡



1+ s ⎣(2rj )(n−1)−λ − # (n − 1) − λ 1 +  |∇γj | 2∞ # λ−(n−1) sn−1 1 +  |∇γj | 2∞ & ≤ 1 +  |∇γj | 2∞ s(n−1)−λ . λ − (n − 1) =

(2.31) (n−1)−λ ⎤ ⎦

Then inequalities (2.30) and (2.31) imply the validity of statement (kk).

 

2.11 Schauder Spaces in Open Subsets of Rn Let m ∈ N \ {0}, n ∈ N, α ∈]0, 1]. Let Ω be an open subset of Rn . Then the Schauder space of exponents m, α in Ω is defined as

C m,α (Ω) ≡ f ∈ C m (Ω) : Dη f ∈ C 0,α (Ω), ∀η ∈ Nn , |η| = m . As we have already said in Section 2.3, C m (Ω) is the subspace of C m (Ω) of those functions f from Ω to R whose derivatives Dη f of order |η| ≤ m can be extended with continuity to Ω, and if x ∈ ∂Ω, then we set Dη f (x) ≡ lim Dη f (y) . y→x

2.11 Schauder Spaces in Open Subsets of Rn

59

m,α Then Cloc (Ω) denotes the space of those functions f ∈ C m (Ω) such that m,α f|Ω∩Bn (0,ρ) belongs to C m,α (Ω ∩ Bn (0, ρ)) for all ρ ∈]0, +∞[ while Cloc (Ω) de-

notes the space of those functions f ∈ C m (Ω) such that f|Ω  belongs to C m,α (Ω  ) for each bounded open subset Ω  of Ω such that Ω  ⊆ Ω. We set Cbm,α (Ω) ≡ Cbm (Ω) ∩ C m,α (Ω)

and we equip Cbm,α (Ω) with the norm f C m,α (Ω) ≡ f C m (Ω) + b

b



|Dη f |α

∀f ∈ Cbm,α (Ω) .

|η|=m

By exploiting the completeness of Cbm (Ω) and of Cb0,α (Ω), we can show that Cbm,α (Ω) is complete. If Ω is bounded, then we obviously have Cbm,α (Ω) = C m,α (Ω) (and in this case we may drop the subscript b). We note that the inclusion of Cbm,α (Ω) into Cb0 (Ω) has to be understood in the sense that each element f ∈ Cbm,α (Ω) has a uniquely determined continuous extension to Ω, which we still denote f . Then by the definition of norm in Cbm,α (Ω) and in Cb0 (Ω), Cbm,α (Ω) is continuously imbedded into Cb0 (Ω). In case the exponent m is replaced with 0 and α ∈]0, 1], the Schauder space of exponents 0, α in Ω is defined as the H¨older space C 0,α (Ω) that we had already introduced. If m ∈ N\{0}, then the definition of norm in Cbm,α (Ω) implies that the following two elementary inequalities hold ∂xr uC m−1,α (Ω) ≤ uC m,α (Ω) b

b

uC m,α (Ω) ≤ uC 0 (Ω) + b

n 

b

∀r ∈ {1, . . . , n} ,

(2.32)

∂xj uC m−1,α (Ω) ≤ nuC m,α (Ω) b

b

j=1

for all u ∈ Cbm,α (Ω). In particular, ∂xr is linear and continuous from Cbm,α (Ω) to Cbm−1,α (Ω) for all r ∈ {1, . . . , n}. In order to compactify our notation when we deal with Schauder spaces in the closure of an open set, we find convenient to set C 0,0 ≡ C 0 ,

C m,0 ≡ C m

if m ∈ N \ {0} .

(2.33)

Then we have the following. Proposition 2.54. Let m, n ∈ N. Let Ω be an open subset of Rn . Then the following statements hold. (i) If α, α ∈]0, 1], α > α , then Cbm,α (Ω) is continuously embedded into  Cbm,α (Ω). (ii) If Ω is regular in the sense of Whitney, then Cbm+1 (Ω) is continuously embedded into Cbm,1 (Ω) (cf. Definition 2.27).

60

2 Preliminaries

(iii) If Ω is regular in the sense of Whitney and if α1 , α2 ∈]0, 1], then the pointwise m,min{α1 ,α2 } multiplication from Cbm,α1 (Ω) × Cbm,α2 (Ω) to Cb (Ω) that takes a pair of functions (f, g) to the pointwise product f g is continuous. (iv) If Ω is bounded and if Ω has a finite number of connected components, each of which is regular in the sense of Whitney, and if α, α ∈]0, 1], α > α , then  C m,α (Ω) is compactly embedded into C m,α (Ω). Proof. Statement (i) follows by the definition of norm in Cbm,α (Ω) and by the con tinuity of the embedding of Cb0,α (Ω) into Cb0,α (Ω) (cf. Proposition 2.23). Since Ω is regular in the sense of Whitney, statement (ii) follows by the inequality  uC m,1 (Ω) = uC m (Ω) + |Dη u|1 b

b

|η|=m

≤ uC m (Ω) + b



c[Ω] sup |∇(Dη u)| Ω

|η|=m



≤ uC m (Ω) + nc[Ω] b

sup |Dη u| ≤ nc[Ω]uC m+1 (Ω) , b

|η|=m+1 Ω

for all u ∈ Cbm+1 (Ω) (cf. Lemma 2.28). We now consider statement (iii). By statement (i), it suffices to consider case α ≡ α1 = α2 . We proceed by induction on m. Case m = 0 follows by Proposition 2.24. Next we assume that the statement holds for m and we prove it for m + 1. If u, v ∈ Cbm+1,α (Ω), then we have uvC m+1,α (Ω) b

≤ uvC 0 (Ω) +

n 

(∂xj u)vC m,α (Ω) +

n 

b

b

j=1

u(∂xj v)C m,α (Ω) , b

j=1

uvC 0 (Ω) ≤ uC 0 (Ω) vC 0 (Ω) ≤ uC m+1,α (Ω) vC m+1,α (Ω) b

b

b

b

b

(cf. (2.32)). Since ∂xj is continuous from Cbm+1,α (Ω) to Cbm,α (Ω) and Cbm+1,α (Ω) is continuously embedded into Cbm,α (Ω), the inductive assumption implies the validity of statement (iii) (cf. Proposition 2.9). Next we consider statement (iv). Again, we argue by induction on m. If m = 0, then the statement holds by Proposition 2.25. Next we assume that the statement holds for m and we prove it for m + 1. Let {fl }l∈N be a bounded sequence in C m+1,α (Ω). The continuity of first order partial derivatives from C m+1,α (Ω) to C m,α (Ω) implies that the sequence {∂xj fl }l∈N is bounded in C m,α (Ω) for all j ∈ {1, . . . , n}. Then the inductive assumption implies that there exist a subsequence  {∂xj flk }k∈N of {∂xj fl }l∈N and gj ∈ C m,α (Ω) such that lim ∂xj flk = gj

k→∞



in C m,α (Ω) ,

(2.34)

2.11 Schauder Spaces in Open Subsets of Rn

61

for all j ∈ {1, . . . , n}. Since {flk }k∈N is bounded in C 1 (Ω), which is compactly embedded into C 0 (Ω), possibly selecting a subsequence, we can assume that {flk }k∈N has a uniform limit f ∈ C 0 (Ω) (cf. Lemma 2.29). Then the limiting relation (2.34) and classical differentiation theorems for sequences of differentiable functions imply that ∂ xj f = g j

∀j ∈ {1, . . . , n} .

in Ω



Accordingly f ∈ C m+1,α (Ω). Moreover, the inequality flk −f C m+1,α (Ω) ≤ flk −f C 0 (Ω) +

n 

∂xj flk −∂xj f C m,α (Ω)

∀k ∈ N

j=1 

implies that {flk }k∈N converges to f in C m+1,α (Ω) (cf. (2.32)).

 

Remark 2.55. Statement (ii) can be shown to hold in case Ω has a finite number of Whitney regular connected components with pairwise disjoint closures. Instead if we assume that Ω has a finite number of Whitney regular connected components whose closures are not pairwise disjoint the embedding of statement (ii) is not necessarily true (to construct an example, cf., e.g., Gilbarg and Trudinger [107, p. 53]). Also statement (iii) can be shown to hold in case Ω has a finite number of Whitney regular connected components with pairwise disjoint closures. Remark 2.56. As Example 2.30 shows, statement (iv) is no longer true for m > 0 if we remove the assumption that Ω has a finite number of connected components. Corollary 2.57. Let n ∈ N \ {0, 1}. Let m ∈ N. Let Ω be a bounded open Lipschitz subset of Rn . Then the following statements hold. (i) C m+1 (Ω) is continuously embedded into C m,1 (Ω). (ii) If α1 , α2 ∈]0, 1], then the pointwise multiplication from C m,α1 (Ω)×C m,α2 (Ω) to C m,min{α1 ,α2 } (Ω) that takes a pair of functions (f, g) to the pointwise product f g is continuous.  (iii) If α, α ∈]0, 1], α > α , then C m,α (Ω) is compactly embedded into C m,α (Ω). Proof. Since Ω is a bounded open Lipschitz subset of Rn , it has a finite number of connected components which are bounded open Lipschitz sets and which have pairwise disjoint closures. Hence, Ω has a finite number of Whitney regular bounded open connected components with pairwise disjoint closures. Then the statement is an immediate consequence of Proposition 2.54 and of the above Remark 2.55.   We conclude the present section with an elementary inequality for functions in Schauder spaces. Lemma 2.58. Let m, n ∈ N, α ∈]0, 1]. Let Ω be a convex open subset of Rn . Then we have

62

2 Preliminaries

    m j    d f (x) j f (y) − [y − x]   j!   j=0 ≤ |y − x|m+α |dm f (·) : Ω|α

Γ (α + 1) Γ (m + α + 1)

∀x, y ∈ Ω

for all f ∈ Cbm,α (Ω), where Γ is the Euler Gamma function and  |d f (·) : Ω|α ≡ m

dm f (ξ  ) − dm f (ξ  )L(m) (Rn ,R) : ξ  , ξ  ∈ Ω, ξ  = ξ  |ξ  − ξ  |α

 ,

where dm f (ξ  ) − dm f (ξ  )L(m) (Rn ,R) ≡ |f (ξ  ) − f (ξ  )| 

∀ξ  , ξ  ∈ Ω

if m = 0 ,



d f (ξ ) − d f (ξ )L(m) (Rn ;R) m

=

m

sup (u1 ,...,um )∈Bn (0,1)m

|[dm f (ξ  ) − dm f (ξ  )][u1 , . . . , um ]| ∀ξ  , ξ  ∈ Ω if m > 0

(see Appendix A.6 for the notation relative to high order differentials and for the corresponding norms). Proof. Let x, y ∈ Ω. If m = 0, then the inequality follows by the definition of α-H¨older continuous function. If m ≥ 1, then the Taylor formula with remainder term in integral form implies that f (y) =

m−1 

dj f (x) [y − x]j j! j=0  1 1 + (1 − t)m−1 dm f (x + t(y − x))[y − x]m dt (m − 1)! 0

and thus     m j    d f (x) j f (y) − [y − x]   j!   j=0   1  1 ≤  (1 − t)m−1 dm f (x + t(y − x))[y − x]m dt (m − 1)! 0   dm f (x) [y − x]m  − m!    1   1 m−1 m m m  = (1 − t) [d f (x + t(y − x)) − d f (x)][y − x] dt (m − 1)! 0  1 1 ≤ (1 − t)m−1 (m − 1)! 0 ×dm f (x + t(y − x)) − dm f (x)L(m) (Rn ,R) |y − x|m dt

2.12 Composition of Functions in Schauder Spaces

≤

1 (m − 1)!



63

1

(1 − t)m−1 |dm f (·) : Ω|α |y − x|m |t(y − x)|α dt 0

 1 1 = |x − y| |d f (·) : Ω|α t(α+1)−1 (1 − t)m−1 dt Γ (m) 0 1 Γ (α + 1)Γ (m) , = |x − y|m+α |dm f (·) : Ω|α Γ (m) Γ (m + α + 1) m+α

m

 

and thus the proof is complete (cf. (2.6)).

2.12 Composition of Functions in Schauder Spaces We now wish to prove that the composition of two functions in a Schauder space is still a function in a Schauder space. Thus we introduce the following. Theorem 2.59. Let n, n1 ∈ N \ {0}. Let m ∈ N. Let α, β ∈ [0, 1]. Let  αβ if m = 0 , μm (α, β) ≡ min{α, β} if m ≥ 1 . Let Ω be a bounded open subset of Rn . We assume that Ω is regular in the sense of Whitney. Let Ω1 be a bounded open subset of Rn1 . Then the following statements hold. (i) If (f, g) ∈ C 0,α (Ω1 ) × C 0,β (Ω, Ω1 ), then the composite function T [f, g] ≡ f ◦ g belongs to C 0,μ0 (α,β) (Ω). If m ≥ 1 and if (f, g) ∈ C m,α (Ω1 ) × C m,β (Ω, Ω1 ) and g(Ω) ⊆ Ω1 , then the composite function T [f, g] belongs to C m,μm (α,β) (Ω). (ii) The (nonlinear) map T from C 0,α (Ω1 ) × C 0,β (Ω) to C 0,μ0 (α,β) (Ω), that takes (f, g) to T [f, g], maps bounded sets to bounded sets. If m ≥ 1, then the (nonlinear) map T from

(f, g) ∈ C m,α (Ω1 ) × C m,β (Ω, Ω1 ) : g(Ω) ⊆ Ω1 to C m,μm (α,β) (Ω), that takes (f, g) to T [f, g], maps bounded sets to bounded sets. Proof. First we choose a norm on C m,β (Ω, Rn1 ) and we set gC m,β (Ω,Rn1 ) ≡

n1 

gl C m,β (Ω)

∀g ≡ (g1 , . . . , gn1 ) ∈ C m,β (Ω, Rn1 ) .

l=1

We proceed by induction on m. If m = 0, then we have

64

2 Preliminaries

sup |f ◦ g| ≤ sup |f | ≤ f C m,α (Ω1 ) . Ω

Ω1

If μ0 (α, β) > 0, then we have |f ◦ g(x) − f ◦ g(y)| ≤ |f : Ω1 |α |g(x) − g(y)|α n1  ≤ |f : Ω1 |α |gl (x) − gl (y)|α l=1

≤ |f : Ω1 |α

n1 

αβ |gl : Ω|α β |x − y|

∀x, y ∈ Ω

l=1

for all (f, g) ∈ C m,α (Ω1 ) × C m,β (Ω, Ω1 ) (see inequality (2.3)). Thus (i) and (ii) hold true for m = 0. We now assume that (i) and (ii) hold true for m ≥ 0 and we prove them for m + 1. If (f, g) ∈ C m+1,α (Ω1 ) × C m+1,β (Ω, Ω1 ) and g(Ω) ⊆ Ω1 , then  n1   ∂gl ∂f ∂ f ◦ g(x) = ◦ g(x) (x) ∀x ∈ Ω ∂xj ∂yl ∂xj l=1

for all j ∈ {1, . . . , n}. Indeed, if x ∈ Ω, we have g(x) ∈ Ω1 and the equality follows by the chain rule. If instead x ∈ ∂Ω, then there exists a sequence {xs }s∈N in Ω which converges to x and the continuity of g at x implies that the sequence {g(xs )}s∈N of Ω1 converges to g(x). Since the above equality holds at the points ∂f ∂gl at g(x) and the continuity of ∂x at x imply that the above xs , the continuity of ∂y j l equality holds at x. Then we observe that Proposition 2.54 (ii) implies that C m+1,β (Ω) is continuously embedded into C m,1 (Ω) ,

(2.35)

that the inductive assumption implies that T maps bounded subsets of

(f, g) ∈ C m,α (Ω1 ) × C m,1 (Ω, Ω1 ) : g(Ω) ⊆ Ω1 to bounded subsets of C m,μm (α,1) (Ω) = C m,α (Ω) ,

(2.36)

and that Proposition 2.54 (iii) implies that the pointwise product is continuous from C m,α (Ω) × C m,β (Ω) to C m,μm+1 (α,β) (Ω) . (2.37)   ∂f ∂gl Then ∂y ◦ g ∂x ∈ C m,μm+1 (α,β) (Ω) for all l ∈ {1, . . . , n1 } and j ∈ {1, . . . , n} j l and accordingly f ◦ g ∈ C m+1,μm+1 (α,β) (Ω). Next we denote by cm,α,β the norm of the (bilinear) pointwise product of (2.37) (for the norm of a multilinear operator, see (A.3) of the Appendix). Then we have f ◦ gC m+1,μm+1 (α,β) (Ω)

2.13 Local Strict Hypographs of a Schauder Class

0  n1 0 n   0 ∂f ∂gl 0 0 0 ≤ sup |f ◦ g| + 0 ∂yl ◦ g ∂xj 0 Ω

j=1 l=1 n1 n  

65

C m,μm+1 (α,β) (Ω)

0 0 0 0 0 ∂f 0 ∂gl 0 0 0 0 0 cm,α,β 0 ◦ g 0 ∂yl 0 0 0 m,α C (Ω) ∂xj C m,β (Ω) j=1 l=1 0 0 n1 n   0 ∂f 0 0 ≤ f C m+1,α (Ω1 ) + cm,α,β 0 ◦ g0 gC m+1,β (Ω) 0 m,α ∂yl C (Ω) j=1 ≤ f C m+1,α (Ω1 ) +

l=1

for all (f, g) ∈ C m+1,α (Ω1 ) × C m+1,β (Ω, Ω1 ) such that g(Ω) ⊆ Ω1 . Then statements (2.35), (2.36) imply the validity of statement (ii) and the proof is complete.   We note that one cannot expect the operator T of Theorem 2.59 to be continuous at all points. For a discussion on the regularity properties of T and for corresponding references, we refer to Appell and Zabreiko [14] and to [157].

2.13 Local Strict Hypographs of a Schauder Class Definition 2.60. Let n ∈ N \ {0, 1}. Let m ∈ N, α ∈ [0, 1]. We say that an open subset Ω of Rn is a local strict hypograph of class C m,α provided that, for every point p ∈ ∂Ω, there exist R ∈ On (R) and r, δ ∈]0, +∞[ such that C(p, R, r, δ) is a coordinate cylinder for Ω around p and that the corresponding function γ, which represents ∂Ω as a graph in C(p, R, r, δ) is of class C m,α (Bn−1 (0, r)) Since the function γ that represents ∂Ω as a graph in C(p, R, r, δ) is defined on Bn−1 (0, r), the membership of γ in C m,α (Bn−1 (0, r)) in case m = 0 of Definition 2.60 has to be understood in the sense that γ has a continuous extension to Bn−1 (0, r) and that such extension is of class C m,α (Bn−1 (0, r)). Since C m,α (Bn−1 (0, r)) is a subspace of C m,α1 (Bn−1 (0, r)), which in turn is a subspace of C h,β (Bn−1 (0, r)), for all α1 ∈ [0, α], h ∈ {0, . . . , m − 1}, β ∈ [0, 1], m ∈ N \ {0}, a local strict hypograph of class C m,α is also of class C m,α1 and of class C h,β for all α1 ∈ [0, α], h ∈ {0, . . . , m − 1}, β ∈ [0, 1], m ∈ N \ {0}. If Ω is a local strict hypograph of class C m,α , then Lemma 2.38 implies that the connected components of both Ω and Ω − are of class C m,α . We note that if Ω is a local strict hypograph of class C 1 , p ∈ ∂Ω, and C(p, R, r, δ) is a coordinate cylinder for Ω around p with a corresponding function γ, then the map   η t ψp (η) ≡ p + R ∀η ∈ Bn−1 (0, r) γ(η) is a homeomorphism of Bn−1 (0, r) onto ψp (Bn−1 (0, r)) = (∂Ω) ∩ C(p, R, r, δ) and

66

2 Preliminaries

& det ((Dψp (η))t (Dψp (η))) =

#

1 + |∇γ(η)|2

∀η ∈ Bn−1 (0, r) .

for (∂Ω) ∩ C(p, R, r, δ). If η ∈ Bn−1 (0, r), In particular, ψpis a parametrization  −∇γ(η) √ 1 is normal to the graph of γ at the point then the vector 1+|∇γ(η)|2 1 (η, γ(η)) and the vector   1 −∇γ(η) t # νΩ (ψp (η)) = R 1 1 + |∇γ(η)|2 is the outward unit normal to ∂Ω at p (cf. Propositions 2.44 and 2.51). Thus, in case ∇γ(0) equals 0, we have νΩ (p) = νΩ (ψp (0)) = Rt en . Namely, the vector Rt en , which has the direction of the axis of the cylinder C(p, R, r, δ), coincides with νΩ (p). We find convenient to introduce the following definition. Definition 2.61. Let n ∈ N \ {0, 1}. Let Ω be an open subset of Rn . Let p ∈ ∂Ω, R ∈ On (R), r, δ ∈]0, +∞[. We say that a coordinate cylinder C(p, R, r, δ) for Ω around p is normal provided that the corresponding function γ which represents ∂Ω in C(p, R, r, δ) as a graph is differentiable at 0 and ∇γ(0) = 0 . Next we note that if Ω is a local strict hypograph of class C m,α with m ≥ 1 and α ∈ [0, 1], then for each p ∈ ∂Ω, there exists a coordinate cylinder C(p, R, r, δ) around p and thus   Ω ∩ p + Rt (Bn−1 (0, r)×] − δ, δ[) = p + Rt {(η, y) ∈ Bn−1 (0, r)×] − δ, δ[: y < γ(η)}  = x ∈ p + Rt {(η, y) ∈ Bn−1 (0, r)×] − δ, δ[} :  R(x − p) ∈ hypographs (γ)   t = x ∈ p + R {(η, y) ∈ Bn−1 (0, r)×] − δ, δ[} : h(R(x − p)) < 0 , where h(η, y) ≡ y − γ(η) for all (η, y) ∈ Bn−1 (0, r)×] − δ, δ[. Then, if we set W ≡ p + Rt (Bn−1 (0, r)×] − δ, δ[) , g(x) ≡ h(R(x − p)) ∀x ∈ W , we know that W is an open and convex neighborhood of p, that g ∈ C m,α (W ), that

2.13 Local Strict Hypographs of a Schauder Class

67

Ω ∩ W = {x ∈ W : g(x) < 0} , and that ∇g(x) = (∇h(R(x − p)))t R = 0 ∀x ∈ W .   −∇γ Indeed, ∇h = cannot vanish. Hence, a local strict hypograph of class 1 C m,α is an open set of class C m,α in the sense of the following definition in the form with the local constraints. Definition 2.62. Let n ∈ N \ {0, 1}. Let m ∈ N \ {0}, α ∈ [0, 1]. We say that an open subset Ω of Rn is of class C m,α (in the form with the local constraints), if for every p ∈ ∂Ω, there exist an open neighborhood W of p in Rn and g ∈ C m,α (W ) such that Ω ∩ W = {x ∈ W : g(x) < 0} , ∇g(x) = 0

∀x ∈ W .

The function g is said to be a local constraint for Ω around p. We can verify that if Ω, p, W , g are as in the above definition, then (∂Ω) ∩ W = {x ∈ W : g(x) = 0} and νΩ (x) =

∇g(x) |∇g(x)|

∀x ∈ (∂Ω) ∩ W .

Conversely, we now prove that if Ω is of class C m,α with m ≥ 1 in the sense of the above definition in the form with the local constraints, then Ω is a local strict hypograph of class C m,α . Actually, we prove that we can choose the coordinate cylinder, so that ∇γ(0) = 0. Moreover, we show that if Ω is bounded, then we can make a uniform choice for r, δ and we can take ∇γ as small as we please. These facts are presented in the following Lemma 2.63. For a weaker version in case m = 0 we refer for example to [170, Lemma 10.1]. Lemma 2.63 (of the Uniform Cylinders). Let n ∈ N \ {0, 1}. Let m ∈ N \ {0}, α ∈ [0, 1]. Let Ω be an open subset of Rn of class C m,α (in the form with the local constraints of Definition 2.62). Let r∗ , δ∗ ∈]0, +∞[, ς ∈]0, 1[. (i) If p ∈ ∂Ω, then there exist rp ∈]0, r∗ [, δp ∈]0, δ∗ [ with rp < δp , and Rp ∈ On (R) such that C(p, Rp , rp , δp ) is a coordinate cylinder for Ω around p, and such that the corresponding function γp is of class C m,α (Bn−1 (0, rp )) and satisfies the condition ∇γp (0) = 0 and the inequality |∇γp (η)| ≤ ς

∀η ∈ Bn−1 (0, rp ) .

(ii) If we further assume that Ω is bounded, then there exist r ∈]0, r∗ [, δ ∈]0, δ∗ [ with r < δ such that if p ∈ ∂Ω, then there exists Rp ∈ On (R) such that

68

2 Preliminaries

C(p, Rp , r, δ) is a coordinate cylinder for Ω around p and the corresponding function γp which represents ∂Ω in C(p, Rp , r, δ) satisfies the condition ∇γp (0) = 0 and the inequalities |∇γp (η)| ≤ ς

∀η ∈ Bn−1 (0, r) ,

sup γp C m,α (Bn−1 (0,r)) < +∞ .

p∈∂Ω

Proof. If p ∈ ∂Ω, then there exists ρp ∈]0, min{r∗ , δ∗ , ς, 1}[ , and a local constraint gp ∈ C m,α (Bn (p, ρp )) for Ω around p. In particular, Ω ∩ Bn (p, ρp ) = {x ∈ Bn (p, ρp ) : gp (x) < 0} ,

∇gp (x) = 0 ∀x ∈ Bn (p, ρp ) .

Possibly shrinking ρp , we can assume that

inf |∇gp (x) · νΩ (y)| : (x, y) ∈ Bn (p, ρp ) × ((∂Ω) ∩ Bn (p, ρp )) > 0 . (2.38) Indeed, ∇gp (p) · νΩ (p) = |∇gp (p)| > 0 . We first prove statement (ii) and then we turn to consider the proof of statement (i), which can be deduced by a straightforward modification of that of statement (ii). Since ∂Ω is compact, there exist a natural number k and, for each j ∈ {1, . . . , k}, a point p(j) ∈ ∂Ω, a real number ρp(j) ∈]0, min{r∗ , δ∗ , ς, 1}[, and a function gp(j) in C m,α (Bn (p(j) , ρp(j) )) as above such that ∂Ω ⊆

k 

Bn (p(j) , ρp(j) /2) .

(2.39)

j=1

Next we choose ι ∈]0, 1/2[ and c ∈]0, +∞[ such that ι < inf |∇gp(j) (x) · νΩ (y)| :

(x, y) ∈ Bn (p(j) , ρp(j) ) × ((∂Ω) ∩ Bn (p(j) , ρp(j) )) ,

gp(j) C m,α (B

n (p

(j) ,ρ

p(j)

))

≤ c,

(2.40)

for all j ∈ {1, . . . , k}. Since the functions ∇gp(j) are uniformly continuous, there exists an increasing function ω from ]0, +∞[ to itself such that limt→0+ ω(t) = 0 and |∇gp(j) (x ) − ∇gp(j) (x )| ≤ ω(|x − x |) for all j ∈ {1, . . . , k}. Next we set

∀x , x ∈ Bn (p(j) , ρp(j) ) , (2.41)

2.13 Local Strict Hypographs of a Schauder Class

ρ≡

min

j∈{1,...,k}

69

1 ρ (j) 2 p

(2.42)

and we choose an arbitrary q ∈ ∂Ω. By the inclusion in (2.39), there exists jq ∈ {1, . . . , k} such that q ∈ Bn (p(jq ) , ρp(jq ) /2) and accordingly Bn (q, ρ) ⊆ Bn (q, ρp(jq ) /2) ⊆ Bn (p(jq ) , ρp(jq ) ) .

(2.43)

gq ≡ gp(jq ) |Bn (q,ρ)

(2.44)

Then the map is a local constraint for Ω around q. Next we turn to define a matrix Rq ∈ On (R) so that the map   η n t R  (η, y) → q + Rq ∈ Rn y maps Bn (0, ρ) onto Bn (q, ρ) and the hyperplane Rn−1 × {0} onto the affine tangent space q + Tq (∂Ω) to ∂Ω at q, which coincides with q + Ker dgq (q). To do so, we choose an orthonormal basis v (1) , . . . , v (n−1) of Ker dgq (q), and we define Rq by the equalities ∀l ∈ {1, . . . , n − 1} ,

Rq v (l) = el Then we set

 fq (η, y) = gq q +

Rqt

  η y

Rq νΩ (q) = en .

(2.45)

∀(η, y) ∈ Bn (0, ρ) .

(2.46)

By the definition of fq , we have fq (0, 0) = 0 , (∇fq (0, 0))t = (∇gq (q))t Rqt , ∂fq (0, 0) = ∇fq (0, 0) · el ∂ηl = (Rq ∇gq (q)) · el = ∇gq (q) · v (l) = 0

(2.47)

∀l ∈ {1, . . . , n − 1} ,

∂fq (0, 0) = ∇fq (0, 0) · en = (Rq ∇gq (q)) · en ∂y = ∇gq (q) · νΩ (q) = |∇gq (q)| > 0 . Next we note that ι < inf |∇gp(jq ) (x) · νΩ (y)| : (x, y) ∈ Bn (p(jq ) , ρp(jq ) ) × ((∂Ω) ∩ Bn (p(jq ) , ρp(jq ) )) ≤ inf |∇gp(jq ) (x) · νΩ (q)| x∈Bn (q,ρ)

=

inf (η,y)∈Bn (0,ρ)

       t ∇g (jq ) q + Rqt η · Rq en   p y



70

2 Preliminaries

    ∂fq   = inf  ∂y (η, y) . (η,y)∈Bn (0,ρ)

(2.48)

Since all entries of the orthogonal matrices are bounded by 1, the definition of norm in C m,α (Bn (0, ρ)) implies that sup (q,R)∈(∂Ω)×On (R)

q + Rt (η, y)t C m,α (Bn (0,ρ)) < +∞ .

Then inequality (2.40), equalities (2.44), (2.46), and Theorem 2.59 on the composition of functions in Schauder spaces with Ω = Bn (0, ρ) and with Ω1 = B(p(j) , ρp(j) ) for each j ∈ {1, . . . , k} ensure the existence of σ ∈]ι, +∞[ such that ∀q ∈ ∂Ω . (2.49) fq C m,α (Bn (0,ρ)) ≤ σ Moreover, |∇fq (η  , y  ) − ∇fq (η  , y  )|     = ∇gp(jq ) (q + Rqt (η  , y  )t ) − ∇gp(jq ) (q + Rqt (η  , y  )t )

(2.50)

≤ ω(|Rqt (η  , y  )t − Rqt (η  , y  )t |) = ω(|(η  , y  ) − (η  , y  )|)

∀(η  , y  ), (η  , y  ) ∈ Bn (0, ρ)

(cf. (2.41)). Next we choose δ, r ∈]0, ρ[ such that the following conditions hold r(n − 1) < δ/2 < δ ,

r + δ < ρ,

ω(r + δ) < ςι < ι .

(2.51)

In particular, we note that the inequality r + δ < ρ of (2.51) ensures that Bn−1 (0, r) × [−δ, δ] ⊆ Bn (0, ρ) . Possibly taking a smaller δ and r, we can also assume that δ is as in Corollary A.22 of the Implicit Function Theorem with Ω = Bn (0, ρ), (xo , y o ) = (0, 0), δ ∗ = δ∗ , ω1,0 = ω and that r < r1 , where r1 is as in the same Corollary A.22. Then inequalities (2.48), (2.49), and Corollary A.22 of the Implicit Function Theorem with Ω = Bn (0, ρ), (xo , y o ) = (0, 0), δ ∗ = δ∗ , ω1,0 = ω imply that there exist M ∈]0, +∞[ and for each q ∈ ∂Ω a function γq in C m,α (Bn−1 (0, r), ] − δ, δ[) such that {(η, y) ∈ Bn−1 (0, r)×] − δ, δ[: fq (η, y) = 0} = {(η, γq (η)) : η ∈ Bn−1 (0, r)} ,

γq C m,α (Bn−1 (0,r)) ≤ M .

In particular, γq (0) = 0. We also note that the known formula for the partial derivatives of an implicitly defined function implies that

2.13 Local Strict Hypographs of a Schauder Class

71

∂f

q ∂γl ∂η (0, γq (0)) (0) = − ∂fql =0 ∂ηl ∂y (0, γq (0))      ∂f   ∂γl   ∂ηql (η, γq (η))    ∂fq −1  ∂fq   = −  ≤ ι (η) (η, γ (η)) − (0, γ (0))  q q  ∂ηl   ∂fq   ∂ηl ∂ηl (η, γq (η)) 

∂y

≤ι

−1

ω(|(η, γq (η)|) ≤ ι−1 ω(δ + r) ≤ ι−1 ςι = ς

(2.52)

for all l ∈ {1, . . . , n − 1} and for all q ∈ ∂Ω (cf. the third inequality of (2.51)). Moreover, the Mean Value inequality and the following conditions γq (0) = 0 ,

r(n − 1)ς < ςδ/2 < δ/2

(cf. the first inequality of (2.51)) imply that γq (Bn−1 (0, r)) ⊆] − δ/2, δ/2[ (cf. inequality (2.52)). A simple topological argument shows that the sets {(η, y) ∈ Bn−1 (0, r)×] − δ, δ[: y < γq (η)} , {(η, y) ∈ Bn−1 (0, r)×] − δ, δ[: y > γq (η)} , are path connected. Since the zeros of fq are located on the graph of γq , then fq does not vanish on such sets. Accordingly, the sign of fq is constant on such ∂f sets. Since fq (0, 0) = 0 and ∂ynq (0, 0) > 0, we conclude that fq is negative on {(η, y) ∈ Bn−1 (0, r)×] − δ, δ[: y < γq (η)} and positive on {(η, y) ∈ Bn−1 (0, r)×] − δ, δ[: y > γq (η)} and that accordingly {(η, y) ∈ Bn−1 (0, r)×] − δ, δ[: fq (y, η) < 0} = {(η, y) ∈ Bn−1 (0, r)×] − δ, δ[: y < γq (η)} . Since gq is a local constraint for Ω around q and q + Rqt (Bn−1 (0, r)×] − δ, δ[) ⊆ Bn (0, ρ) , we have   q + Rqt (Bn−1 (0, r)×] − δ, δ[) ∩ Ω

= x ∈ q + Rqt (Bn−1 (0, r)×] − δ, δ[) : gq (x) < 0 = q + Rqt ({(η, y) ∈ Bn−1 (0, r)×] − δ, δ[: fq (y, η) < 0}) = q + Rqt (hypographs (γq )) .

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2 Preliminaries

Hence, C(q, Rq , r, δ) is a coordinate cylinder for Ω around q and the proof of statement (ii) is complete. In order to prove statement (i), we fix p ∈ ∂Ω, we consider a local constraint gp as in (i), we define a corresponding Rp as in (2.45), and we deduce the existence of γp ∈ C m,α (Bn−1 (0, rp ), ] − δp , δp [) by means of the Implicit Function Theorem. Then the same argument above shows that C(p, Rp , rp , δp ) is a coordinate cylinder for Ω around p.   By the remarks right before Definition 2.62 of set of class C m,α in the form with the local constraints and by the Lemma 2.63 of the uniform cylinders, we have the following. Proposition 2.64. Let n ∈ N \ {0, 1}. Let m ∈ N \ {0}, α ∈]0, 1]. Let Ω be an open subset of Rn . Then Ω is of class C m,α (in the form with the local constraints) if and only if Ω is a local strict hypograph of class C m,α . We also introduce the following form of the definition of open set of class C m,α . Definition 2.65. Let n ∈ N \ {0, 1}. Let m ∈ N \ {0}, α ∈ [0, 1]. We say that an open subset Ω of Rn is of class C m,α (in the form with the diffeomorphisms) if, for every p ∈ ∂Ω, there exist an open neighborhood U of 0 in Rn , an open neighborhood W of p in Rn , and a homeomorphism φp from U onto W such that   (−1)   ∈ C m,α W , Rn , and φp ∈ C m,α U , Rn , φp φp ({x ∈ U : xn < 0}) = W ∩ Ω , φp ({x ∈ U : xn = 0}) = W ∩ ∂Ω , φp (0) = p .

(2.53)

The map φp is said to be a parametrization of Ω around p. Remark 2.66. Since φp is continuous, possibly shrinking the open neighborhood U , we can take the open neighborhood W to be as small as we wish. Since a diffeomorphism is a locally Lipschitz homeomorphism, Proposition 2.31 implies that possibly shrinking U in the above definition, we can assume that U is an open ball and that W = φp (U ) is regular in the sense of Whitney. Possibly shrinking the radius of the ball we can assume that φp extends to a homeomorphism from U onto W . Then by taking the composition of φp with a dilation, we can assume that U = Bn (0, 1). Then a classical argument based on the Implicit Function Theorem shows the validity of the following. Proposition 2.67. Let n ∈ N \ {0, 1}. Let m ∈ N \ {0}, α ∈ [0, 1]. Let Ω be an open subset of Rn . Then Ω is of class C m,α (in the form with the diffeomorphisms) if and only if it is a local strict hypograph of class C m,α . Proof. Let Ω be of class C m,α (in the form with the diffeomorphisms). Let p ∈ ∂Ω. Let U , W , φp be as in Definition 2.65. Then we define gp to be the n-th component

2.13 Local Strict Hypographs of a Schauder Class

73

(−1)

of the map φp . We observe that (∇gp )t coincides with the n-th row of the Ja(−1) cobian matrix of φp , whose determinant cannot vanish at any point of W . Then ∇gp cannot be equal to zero at any point of W and the map gp is a local constraint for Ω around p. Thus Ω is of class C m,α in the form with the local constraints. Then Proposition 2.64 implies that Ω is a local strict hypograph of class C m,α . We now prove the converse. Let Ω be a local strict hypograph of class C m,α . Let p ∈ ∂Ω. Then there exist R ∈ On (R), r, δ ∈]0, +∞[ such that C(p, R, r, δ) is a coordinate cylinder for Ω around p, and the corresponding function γ which represents ∂Ω in C(p, R, r, δ) as a graph is of class C m,α (Bn−1 (0, r)). Then the function φp from U ≡ Bn−1 (0, r)×] − δ/2, δ/2[ onto W ≡ p + Rt (U ) defined by φp (η, y) ≡ p + Rt (η, y + γ(η))t

∀(η, y) ∈ U ,

can be verified to be a parametrization of Ω around p of class C m,α . Indeed, φp is injective, satisfies (2.53), and its inverse function φ(−1) (ξ) p ≡ (πRn−1 ◦ R(ξ − p), πn ◦ R(ξ − p) − γ ◦ πRn−1 ◦ R(ξ − p))

∀ξ ∈ W

is of class C m,α . Here πRn−1 denotes the projection of the first (n − 1) components   and πn denotes the projection of the last component. By Propositions 2.64 and 2.67 we can see that the so far introduced different definitions of sets of class C m,α are equivalent when m ≥ 1. Namely, we have the following. Remark 2.68. Let n ∈ N \ {0, 1}. If m ∈ N \ {0}, α ∈ [0, 1], and Ω is an open subset of Rn , then the following statements are equivalent: (i) Ω is of class C m,α in the form with the local constraints, (ii) Ω is of class C m,α in the form with the diffeomorphisms, (iii) Ω is a local strict hypograph of class C m,α . Thus, from now on we simply say that Ω is of class C m,α for a certain m ≥ 1 and α ∈ [0, 1], without specifying in which of the above forms. Moreover, we say that Ω is of class C m if Ω is of class C m,0 , and that Ω is of class C ∞ provided that Ω is of class C m for all m ≥ 1. If instead m = 0, then we understand that a set Ω is of class C 0,α when it is a local strict hypograph of class C 0,α . By Lemma 2.63 on the uniform coordinate cylinders, the following statement holds. Corollary 2.69. Let n ∈ N \ {0, 1}. Let Ω be a bounded open subset of Rn of class C 1 . Then there exists t0 ∈]0, +∞[ such that x + tνΩ (x) ∈ Ω x + tνΩ (x) ∈ Ω −

∀t ∈] − t0 , 0[ , ∀t ∈]0, t0 [ ,

74

2 Preliminaries

for all x ∈ ∂Ω. Proof. By Lemma 2.63, there exist r, δ ∈]0, +∞[ such that, for each p ∈ ∂Ω, there exists Rp ∈ On (R) such that C(p, Rp , r, δ) is a coordinate cylinder for Ω around p. Then it suffices to take t0 = δ/2. Indeed, p + tνΩ (p) = p + tRpt en belongs to C(p, Rp , r, δ) ∩ Ω for all t ∈] − δ, 0[ and belongs to C(p, Rp , r, δ) ∩ Ω − for all t ∈]0, δ[.   Then we have the following classical lemma, which is often useful. Lemma 2.70. Let n ∈ N \ {0, 1}. Let Ω be an open subset of Rn . Let K be a compact subset of Ω. Then there exists an open bounded subset Ω1 of Ω of class C ∞ such that K ⊆ Ω1 ⊆ Ω1 ⊆ Ω . If we further assume that K is connected, then we can take Ω1 to be connected. Proof. Here we follow an argument suggested by G. De Marco.1 By the classical Lemma A.36, there exists a cutoff function φ ∈ D(Ω) such that 0 ≤ φ ≤ 1 in Ω, and φ = 1 on K. By Sard’s Theorem A.31, there exists a regular value c ∈]0, 1[ for φ, i.e., a constant c ∈]0, 1[ such that ∇φ(x) = 0

∀x ∈ {ξ ∈ Ω : φ(ξ) = c} .

Then the open and bounded subset Ω1 ≡ {x ∈ Ω : φ(ξ) > c} of Ω is of class C ∞ and contains K. Moreover Ω1 ⊆ supp φ ⊆ Ω . If we further assume that K is connected, then K must be contained in a single connected component of Ω1 . Then we can replace Ω1 with the only connected component of Ω1 which contains K. Indeed, Proposition 2.64 ensures that Ω1 is a local strict hypograph of class C ∞ , Lemma 2.38 ensures that its connected components   are also local strict hypographs of class C ∞ .

2.14 Extendibility of Functions of Schauder Spaces on an Open Subset of Class C m,α In this section, we plan to extend functions of the Schauder class C h,α (Ω) for some h ∈ {0, . . . , m} to the whole of Rn in case Ω is of class C m,α with m ≥ 1 and α ∈ [0, 1]. Since locally around the boundary points, sets of class C m,α can be mapped diffeomorphically onto a half-ball, we begin with the following technical 1

Professor at the University of Padua, Italy.

2.14 Extendibility of Functions of Schauder Spaces on an Open Subset of Class C m,α

75

extension lemma for functions defined on a half-ball. We present here a proof based on an argument of Hestenes [123]. Lemma 2.71. Let n ∈ N \ {0, 1}, h ∈ N, α ∈ [0, 1] (cf. (2.33)). Let ρ ∈]0, +∞[. Let n B+ n (0, ρ) ≡ Bn (0, ρ) ∩ {x ∈ R : xn > 0} , n B− n (0, ρ) ≡ Bn (0, ρ) ∩ {x ∈ R : xn < 0} .

(2.54)

Then there exists a linear and continuous extension operator E from − {f ∈ C h,α (B− n (0, ρ)) : supp (f ) ⊆ Bn (0, ρ/2)}

(2.55)

{g ∈ C h,α (Bn (0, ρ)) : supp (g) ⊆ Bn (0, ρ/2)}

(2.56)

to such that E[f ]|B− (0,ρ) = f n

for all f ∈ C

h,α

(B− n (0, ρ))

such that supp (f ) ⊆ B− n (0, ρ/2) (cf. (2.33)).

Proof. The Vandermonde determinant associated to the real numbers (−1), . . . , (−h − 1) , is not equal to zero and accordingly there exists a unique solution (λ1 , . . . , λh+1 ) of the linear system h+1 

λj (−j)s = 1

∀s ∈ {0, . . . , h} .

(2.57)

j=1

Then we set E[f ](x1 , . . . , xn ) ! if x ∈ Bn (0, ρ), xn ≤ 0 , f (x1 , . . . , xn ) ≡ h+1 λ f (x , . . . , x , −jx ) if x ∈ Bn (0, ρ), xn > 0 , j 1 n−1 n j=1 − for all f ∈ C h,α (B− n (0, ρ)) such that supp (f ) ⊆ Bn (0, ρ/2). In the definition of E[f ], we understand that f has value 0 at all points outside of B− n (0, ρ). We also note that if |(x1 , . . . , xn )| > ρ/2 and xn > 0, then |(x1 , . . . , xn−1 , −jxn )| > ρ/2 and thus f (x1 , . . . , xn−1 , −jxn ) = 0 for all j ∈ {1, . . . , h + 1}. − If f ∈ C h,α (B− n (0, ρ)) and supp (f ) ⊆ Bn (0, ρ/2), then by the definition of E[f ], we can see that the support of E[f ] is contained in Bn (0, ρ/2). Moreover, we can verify that the function E[f ] and its derivatives up to order h are continuous at all points of Bn (0, ρ) with xn = 0. Finally, by the equalities (2.57) and by a straightforward computation, we can prove that E[f ] and its derivatives up to order

76

2 Preliminaries

h are continuous at all points of Bn (0, ρ) with xn = 0. Hence, E[f ] is of class C h and has support in Bn (0, ρ/2). We now prove that E is linear and continuous from the space in (2.55) to the space in (2.56). To do so, we estimate the C h,α norm of E[f ] in terms of the h+1 s C h,α norm of f . By the inequality (2.57), we have 1 ≤ j=1 j |λj | for all s ∈ {0, . . . , h}. Then the chain rule implies that ⎛ ⎞ h+1  |Dβ (E[f ])(x)| ≤ ⎝ j βn |λj |⎠ sup |Dβ f | ∀x ∈ Bn (0, ρ) (2.58) j=1

B− n (0,ρ))

− n for all f ∈ C h,α (B− n (0, ρ)) such that supp (f ) ⊆ Bn (0, ρ/2) and β ∈ N such that n |β| ≤ h. Now let α > 0. If β ∈ N and |β| = h, then α |Dβ (E[f ])(x) − Dβ (E[f ])(y)| ≤ |Dβ f : B− n (0, ρ)|α |x − y|

for all x, y ∈ B− n (0, ρ). By Theorem 2.59 on the composition of functions in Shauder spaces and by the continuity of the pointwise product in Schauder spaces, there exists cβ > 1 such that α |Dβ (E[f ])(x) − Dβ (E[f ])(y)| ≤ cβ |Dβ f : B− n (0, ρ)|α |x − y| − + for all x, y ∈ B+ n (0, ρ). If x ∈ Bn (0, ρ) and y ∈ Bn (0, ρ), then there exists ξ in the + segment [x, y] such that ξn = 0 and [x, ξ] ⊆ B− n (0, ρ), [ξ, y] ⊆ Bn (0, ρ). Then we have

|Dβ (E[f ])(x) − Dβ (E[f ])(y)| − α β α ≤ |Dβ f : B− n (0, ρ)|α |x − ξ| + cβ |D f : Bn (0, ρ)|α |ξ − y| α α ≤ cβ |Dβ f : B− n (0, ρ)|α (|x − ξ| + |ξ − y| ) α α ≤ cβ |Dβ f : B− n (0, ρ)|α (|x − y| + |x − y| )

and we deduce that α |Dβ (E[f ])(x) − Dβ (E[f ])(y)| ≤ 2cβ |Dβ f : B− n (0, ρ)|α |x − y|

(2.59)

for all x, y ∈ Bn (0, ρ). Inequalities (2.58) and (2.59) imply the continuity of the linear map E and thus the proof is complete.   Then we are ready to prove the following extension theorem, which we formulate as in Troianiello [268, Theorem 1.3]. Theorem 2.72. Let n ∈ N \ {0, 1}. Let m ∈ N \ {0}, h ∈ {0, . . . , m}, α ∈ [0, 1]. Let Ω be a bounded open subset of Rn of class C m,α . Let Ω  be a bounded open subset of Rn such that Ω ⊆ Ω  . Then there exists a linear and continuous

2.14 Extendibility of Functions of Schauder Spaces on an Open Subset of Class C m,α

77

extension operator FΩ  from C h,α (Ω) to C h,α (Ω  ) such that FΩ  [f ]|Ω = f and supp FΩ  [f ] ⊆ Ω  for all f ∈ C h,α (Ω) (cf. (2.33)).   Proof. Let {φa }a=1,...,b with φa ∈ C m,α Bn (0, 1), Rn be parametrizations of Ω around points of ∂Ω such that ∂Ω ⊆

b 

φa (Bn (0, 1/2))

a=1

(see Definition 2.65). By Remark 2.66, we can assume that φa (Bn (0, 1)) ⊆ Ω  and that φa (Bn (0, 1)) is regular in the sense of Whitney for all a ∈ {1, . . . , b}. By Lemma A.35 on the existence of the partition of unity, there exists a finite family {θa }a∈{1,...,b} of functions of class Cc∞ (Rn ) such that supp θa ⊆ φa (Bn (0, 1/2))

b 

∀a ∈ {1, . . . , b} ,

θa (x) = 1

∀x ∈ ∂Ω .

a=1

    Next we note that φa ∈ C m,α Bn (0, 1), Rn ⊆ C h,1 Bn (0, 1), Rn when h < m. Then, by Proposition 2.54 (iii) on the continuity of the pointwise product, by the inclusion φa (B− n (0, 1)) ⊆ Ω, and by Theorem 2.59 on the composition of functions in Schauder spaces, there exists c1 > 0 such that (θa f ) ◦ φa C h,α (B− (0,1)) ≤ c1 f C h,α (Ω) n

∀f ∈ C h,α (Ω) ,

for all a ∈ {1, . . . , b} both in case h < m and h = m. We also note that supp (θa f ) ◦ φa ⊆ B− n (0, 1/2)

∀f ∈ C h,α (Ω)

for all a ∈ {1, . . . , b}. Let E be the extension operator for the unit ball of Lemma 2.71. Then we have supp E[(θa f ) ◦ φa ] ⊆ Bn (0, 1/2) ⊆ Bn (0, 1) for all f ∈ C h,α (Ω) and for all a ∈ {1, . . . , b}. Since φa (Bn (0, 1)) is regular in the (−1) sense of Whitney and φa ∈ C m,α (φa (Bn (0, 1))) ⊆ C h,1 (φa (Bn (0, 1))) when h < m, the composition Theorem 2.59 for functions in a Schauder space implies that there exists c2 > 0 such that E[(θa f ) ◦ φa ] ◦ φ(−1) C h,α (φa (Bn (0,1))) a ≤ c2 E[(θa f ) ◦ φa ]C h,α (Bn (0,1)) ≤ c1 c2 f C h,α (Ω)

∀f ∈ C h,α (Ω)

for all a ∈ {1, . . . , b} both in case h < m and h = m. The operator Ga from {g ∈ C h,α (φa (Bn (0, 1))) : supp (g) ⊆ φa (Bn (0, 1/2))}

78

2 Preliminaries

to C h,α (Ω  ), that takes a function g to  g(y) if y ∈ φa (Bn (0, 1/2)) , Ga [g](y) ≡ 0 if y ∈ Ω  \ φa (Bn (0, 1/2)) , is linear and continuous for all a ∈ {1, . . . , b}. Since the support of the function (−1) is contained in φa (Bn (0, 1/2)) ⊆ Ω  for all f ∈ C h,α (Ω) E[(θa f ) ◦ φa ] ◦ φa and for all a ∈ {1, . . . , b}, then the operator FΩ  defined by FΩ  [f ] ≡

b 

1 2 Ga E[(θa f ) ◦ φa ] ◦ φ(−1) a

∀f ∈ C h,α (Ω)

a=1

 

satisfies the properties of the statement.

2.15 On the Extendibility of Continuous Functions to the Closure of Open Sets of Class C 1 We first introduce the following technical elementary lemma. Lemma 2.73. Let n ∈ N \ {0, 1}. Let M be a (n − 1)-dimensional manifold of class C 1 embedded into Rn . If x ∈ Rn \ M and if x ˜ ∈ M satisfies the equality |x − x ˜| = inf |x − y| , y∈M

then the vector (x − x ˜) is orthogonal to the tangent space Tx˜ M of M at x ˜. Proof. Let f be the function from M to R defined by f (y) ≡ (y − x) · (y − x) for all y ∈ M . Clearly, f has a local minimum at y = x ˜ and ∇f (˜ x) = 2(˜ x − x). By the Lagrange Multiplier Theorem, ∇f (˜ x) must equal a multiple of the gradient of a local constraint for M around x ˜. Since the gradient of a local constraint for M x) = 2(˜ x − x) is orthogonal to around x ˜ is orthogonal to Tx˜ M , we deduce that ∇f (˜   Tx˜ M . We are now ready to introduce a sufficient condition for the extendibility to the boundary of a continuous function on an open subset of class C 1 . Lemma 2.74. Let n ∈ N \ {0, 1}. Let Ω be a bounded open subset of Rn of class C 1 . Then the following statements hold. (i) If u ∈ C 0 (Ω) and if

lim u(x + tνΩ (x)) ≡ f (x)

t→0−

exists uniformly in x ∈ ∂Ω, then f ∈ C 0 (∂Ω) and the function u ˜ from Ω to R defined by  u(x) if x ∈ Ω , u ˜(x) ≡ f (x) if x ∈ ∂Ω ,

2.15 On the Extendibility of Continuous Functions to the Closure of Open Sets of Class C 1

79

is continuous. (ii) If u ∈ C 0 (Ω − ) and if lim u(x + tνΩ (x)) ≡ f (x)

t→0+

exists uniformly in x ∈ ∂Ω, then f ∈ C 0 (∂Ω) and the function u ˜− from Ω − to R defined by  u(x) if x ∈ Ω − , − u ˜ (x) ≡ f (x) if x ∈ ∂Ω , is continuous. Proof. We consider for example statement (i). Since f is a uniform limit of continuous functions, then f is continuous. We now fix x0 ∈ ∂Ω and prove the continuity of u ˜ at x0 . Let  > 0. By the uniform continuity of the function f on the compact set ∂Ω, there exists δ > 0 such that |f (x1 ) − f (x2 )| ≤ /2

if

x1 , x2 ∈ ∂Ω , |x1 − x2 | ≤ δ .

Possibly shrinking δ, we can assume that y −tνΩ (y) ∈ Ω and that y +tνΩ (y) ∈ Ω − for all t ∈]0, δ] and for all y ∈ ∂Ω (see Corollary 2.69). We now assume that x ∈ Ω u(x) − u ˜(x0 )| = |f (x) − f (x0 )| ≤ . and that |x − x0 | ≤ δ/2. If x ∈ ∂Ω, we have |˜ We now consider case x ∈ Ω. Since ∂Ω is compact, there exists x ˜ ∈ ∂Ω such that |x − x ˜| = inf |x − y| y∈∂Ω

and we have |x − x0 | ≥ |x − x ˜| > 0 ,

(x − x ˜) is orthogonal to Tx˜ (∂Ω)

(cf. Lemma 2.73), and |˜ x − x0 | ≤ |˜ x − x| + |x − x0 | ≤

δ δ + = δ. 2 2

Since (x − x ˜) is orthogonal to Tx˜ (∂Ω), there exists λ ∈ {1, −1} such that νΩ (˜ x) = x−˜ x . If λ = 1, then x = x ˜ + |x − x ˜ |ν (˜ x ). Since |x − x ˜ | ≤ |x − x | ≤ δ, our λ |x−˜ Ω 0 x| − x) ∈ Ω contrary to our assumption choice of δ implies that x = x ˜ + |x − x ˜|νΩ (˜ x−˜ x x ∈ Ω. Hence, λ = −1 and νΩ (˜ x) = − |x−˜ . ˜ − |x − x ˜|νΩ (˜ x) and x| Then x = x |˜ u(x) − u ˜(x0 )| = |u(x) − f (x0 )| ≤ |u(x) − f (˜ x)| + |f (˜ x) − f (x0 )| ≤ |u(˜ x − |x − x ˜|νΩ (˜ x)) − f (˜ x)| + /2 . By assumption, we have lim u(x + tνΩ (x)) ≡ f (x)

t→0−

∀x ∈ ∂Ω ,

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2 Preliminaries

uniformly in x ∈ ∂Ω. Accordingly, there exists δ  ∈]0, δ/2[ such that |u(x + tνΩ (x)) − f (x)| ≤ /2

∀x ∈ ∂Ω ,

for all t ∈] − δ  , 0[. In particular, |u(˜ x − |x − x ˜|νΩ (˜ x)) − f (˜ x)| ≤ /2 , if |x − x0 | ≤ δ  . Then the inequality above implies that if |x − x0 | ≤ δ  ,

|˜ u(x) − u ˜(x0 )| ≤ /2 + /2 = 

 

and u ˜ is continuous at x0 .

2.16 A Consequence of the Rule of Change of Variables for Diffeomorphisms We now introduce the following consequence of the chain rule and of the rule of change of variables in Lebesgue integrals. For the convenience of the reader, we include a proof. Lemma 2.75. Let n ∈ N \ {0, 1}. Let Ω be a bounded open subset of Rn of class C 1 . Let p ∈ Rn ,  ∈ R \ {0}. Then the set Ωp, ≡ p + Ω is of class C 1 and the following statements hold. (i) νΩp, (p + ξ) = sgn ()νΩ (ξ) (ii) If ω ∈ L1 (∂Ωp, ), then 

∀ξ ∈ ∂Ω .

(2.60)

ω(p + η) dση

(2.61)

 ω(y) dσy = ||

p+∂Ω

n−1 ∂Ω

for all ω ∈ L1 (p + ∂Ω). Proof. Since dilations and translations are homeomorphisms in Rn , we can easily see that ∂Ωp, = p + ∂Ω. Now let ξ ∈ ∂Ω, x ≡ p + ξ. Since Ω is of class C 1 , there exist rξ > 0 and a local constraint gξ ∈ C 1 (Bn (ξ, rξ )) for Ω around ξ. Then the function gx (y) ≡ gξ ((y − p)/)

∀y ∈ p + Bn (ξ, rξ )

is easily seen to be a local constraint for x. Hence, Ωp, is of class C 1 . We now prove the formula of (i). Since gξ is a local constraint for Ω around ξ and gx is a local constraint for Ωp, around x, we have

2.17 A Fundamental Inequality of the Unit Normal on the Boundary of a Set of Class C 1,α

νΩ (ξ) =

∇gξ (ξ) , |∇gξ (ξ)|

νΩp, (p + ξ) =

∇gx (p + ξ) |∇gx (p + ξ)|

81

.

By the chain rule, we have (∇gx (x)) = (∇gξ )((x − p)/)−1 = (∇gξ )(ξ)−1 . Hence the formula in (i) holds true. We now consider statement (ii). If ψ ∈ C 1 (Bn−1 (0, 1), ∂Ω) is a local parametrization for the C 1 manifold ∂Ω, then p + ψ is a parametrization for the C 1 manifold p + ∂Ω = ∂Ωp, and the area element corresponding to p + ψ is given by #

det ((D(p + ψ)(η))t (D(p + ψ)(η)))    ∂(p + ψ) ∂(p + ψ)   ¯ ...∧ ¯ (η)∧ (η) = ∂η1 ∂ηn−1    ∂ψ  ∂ψ ¯ ...∧ ¯ = ||n−1  (η)∧ (η) ∂η1 ∂ηn−1 # n−1 t = || det (Dψ(η)) (Dψ(η))) ∀η ∈ Bn−1 (0, 1) ,

¯ ...∧ ¯ · denotes the vector product of (n − 1) vectors in Rn (which is a where ·∧ vector orthogonal to the space generated by its factors). Then statement (ii) can be verified by exploiting the definition of integral over ∂Ω and ∂Ωp, , the corresponding formulas for the area element, and the rule of change of variables in integrals. We leave the details to the reader.  

2.17 A Fundamental Inequality of the Unit Normal on the Boundary of a Set of Class C 1,α We now introduce the following two lemmas, which can be verified by elementary Calculus and by exploiting the Lebesgue number of a cover (see Theorem A.6). In particular, the following classical lemma is relevant in the analysis of singular integrals. Lemma 2.76. Let n ∈ N \ {0, 1}. Let Ω be a bounded open subset Rn of class C 1 . Then the following statements hold. (i) Let x ∈ ∂Ω. Let C(x, R, r, δ) be a normal coordinate cylinder for Ω around x. Let the function γ ∈ C 1 (Bn−1 (0, r), ] − δ, δ[) represent ∂Ω in the cylinder as a graph. Let ωγ be a function from [0, +∞[ to itself such that ωγ (0) = 0 ,

ωγ (t) > 0

∀t ∈]0, +∞[ ,

ωγ is increasing, and lim+ ωγ (t) = 0 , t→0

(2.62)

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2 Preliminaries

and such that     ∂γ ∂γ  ≤ ωγ (|η − ξ|)  (η) − (ξ)  ∂ηl ∂ηl 

∀η, ξ ∈ Bn−1 (0, r)

(2.63)

for all l ∈ {1, . . . , n − 1}. Then   n−1  ∂γ   (ηl − ξl ) (η) − (γ(η) − γ(ξ)) ≤ (n − 1)ωγ (|ξ − η|)|ξ − η|    ∂ηl l=1

for all ξ, η ∈ Bn−1 (0, r). (ii) The function ωΩ from [0, +∞[ to itself defined by   |(y − x) · νΩ (y)| : x, y ∈ ∂Ω , 0 < |x − y| ≤ t ωΩ (t) ≡ sup |x − y| ∀t ∈]0, +∞[ , ωΩ (0) ≡ 0 satisfies the conditions in (2.62) and the inequality |(y − x) · νΩ (y)| ≤ ωΩ (|x − y|)|x − y|

∀x, y ∈ ∂Ω .

(iii) If α ∈]0, 1] and if we further assume that Ω is of class C 1,α , then   |(y − x) · νΩ (y)| : x, y ∈ ∂Ω , x = y < +∞ . cΩ,α ≡ sup |x − y|1+α Proof. We first note that ωγ as in (2.62) and (2.63) certainly exists. Indeed, the func∂γ is uniformly continuous in Bn−1 (0, r) and it suffices to take the supremum tion ∂η l ∂γ of the moduli of continuity of the functions ∂η in Bn−1 (0, r) as l ∈ {1, . . . , n − 1}. l We now prove the inequality in statement (i). If ξ, η ∈ Bn−1 (0, r), then we can assume that ξ = η and the segment [ξ, η] with endpoints ξ and η is contained in Bn−1 (0, r). Thus the Mean Value Theorem implies that there exists τ ∈]ξ, η[ such that n−1  ∂γ (τ )(ηl − ξl ) . γ(η) − γ(ξ) = (η − ξ) · ∇γ(τ ) = ∂ηl l=1

Then we deduce that      n−1  n−1 ∂γ ∂γ ∂γ     (ηl − ξl ) (η) − (γ(η) − γ(ξ)) =  (ηl − ξl ) (η) − (τ )       ∂ηl ∂ηl ∂ηl l=1

l=1

≤ (n − 1)ωγ (|η − τ |)|η − ξ| ≤ (n − 1)ωγ (|η − ξ|)|η − ξ| and thus the proof of statement (i) is complete.

2.17 A Fundamental Inequality of the Unit Normal on the Boundary of a Set of Class C 1,α

83

We now consider statement (ii). For each x ∈ ∂Ω, there exists a coordinate cylinder C(x, Rx , rx , δx ) around x. Then {C(x, Rx , rx , δx ) ∩ ∂Ω}x∈∂Ω is an open cover of ∂Ω. Possibly shrinking rx , we can assume that the function which represents ∂Ω in C(x, Rx , rx , δx ) is of class C 1 on Bn−1 (0, rx ). Since ∂Ω is compact, m m m m there exist finite families {x(j) }m j=1 , {Rj }j=1 , {rj }j=1 , {δj }j=1 , {γj }j=1 , such (j) (j) that every C(x , Rj , rj , δj ) is a coordinate cylinder around x , γj represents ∂Ω in C(x(j) , Rj , rj , δj ), and ∂Ω =

m 

(C(x(j) , Rj , rj , δj ) ∩ ∂Ω) .

j=1

Let Λ be a Lebesgue number for such a finite cover of ∂Ω (cf. Theorem A.6). Let ωγj be as in (i) for the coordinate cylinder C(x(j) , Rj , rj , δj ) for all j ∈ {1, . . . , m}. Then we set ω ˜ (t) ≡ (n − 1)

sup

ωγj (t)

∀t ∈]0, +∞[ .

j∈{1,...,m}

Since each ωγj satisfies the conditions in (2.62), then also ω ˜ satisfies the conditions in (2.62). We now estimate ωΩ . By the Schwarz inequality, we have    (y − x) · νΩ (y)   ≤ 1 < +∞ .  (2.64) M1 ≡ sup   |x − y| (x,y)∈(∂Ω)2 ,|x−y|≥Λ Then we turn to show that |(y − x) · νΩ (y)| ≤ ω ˜ (|x − y|)|x − y|

∀(x, y) ∈ (∂Ω)2 , |x − y| ≤ Λ . (2.65)

Let x, y ∈ ∂Ω be such that |x − y| ≤ Λ. Since Λ is a Lebesgue number, there exists j ∈ {1, . . . , m} such that {x, y} ⊆ C(x(j) , Rj , rj , δj ) ∩ ∂Ω. Now let ξ, η in Bn−1 (0, rj ) be such that x = ψx(j) (ξ)

y = ψx(j) (η) ,

where ψx(j) is the parametrization of ∂Ω in C(x(j) , Rj , rj , δj ) associated to γj . Then we have |(y − x) · νΩ (y)|       η ξ =  x(j) + Rjt − x(j) − Rjt γj (η) γj (ξ)      1 −∇γj (η) t  # · Rj 1 1 + |∇γj (η)|2         1 η−ξ −∇γj (η)   # = ·  2 1  γj (η) − γj (ξ) 1 + |∇γj (η)| 

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2 Preliminaries

= |−(η − ξ) · ∇γj (η) + (γj (η) − γj (ξ))| #

1 . 1 + |∇γj (η)|2

Now by statement (i), we have |−(η − ξ) · ∇γj (η) + (γj (η) − γj (ξ))| #

1 1 + |∇γj (η)|2

≤ (n − 1)ωγj (|ξ − η|)|ξ − η| ≤ (n − 1)ωγj (|ψx(j) (ξ) − ψx(j) (η)|)|ψx(j) (ξ) − ψx(j) (η)| = (n − 1)ωγj (|x − y|)|x − y| ≤ ω ˜ (|x − y|)|x − y| . Indeed, |ξ − η| ≤ |(ξ − η, γj (ξ) − γj (η))| = |Rjt (ξ − η, γj (ξ) − γj (η))t | = |ψx(j) (ξ) − ψx(j) (η)| . Hence, inequality (2.65) holds true. By inequalities (2.64) and (2.65), we have ωΩ (t) ≤ ω ˜ (t)

∀t ∈]0, Λ] ,

ωΩ (t) ≤ sup{M1 , ω ˜ (t)}

∀t ∈ [Λ, +∞[ .

In particular, ωΩ (t) < +∞ for all t ∈]0, +∞[. By definition of ωΩ , the function ˜ (t) = 0, we have limt→0 ωΩ (t) = 0 and thus ωΩ ωΩ is increasing. Since limt→0 ω satisfies all conditions in (2.62). Finally, we prove statement (iii). If Ω is of class C 1,α , then each function γj is of cass C 1,α and we can take    ∂γj   : Bn−1 (0, rj ) tα ∀t ∈]0, +∞[ ωγj (t) = max  l=1,...,n−1 ∂ηl α for all j ∈ {1, . . . , m}. Then we have    ∂γj   ω ˜ (t) = (n − 1) max  : Bn−1 (0, rj ) tα , l=1,...,n−1, ∂ηl α

ωΩ (t) ≤ ω ˜ (t)

j=1,...,m

for all t ∈]0, Λ], and the above proof of statement (ii) ensures that ! )   M1 ω ˜ (|x − y|) M1 , sup , ω ˜ (1) . ≤ max cΩ,α ≤ max Λα (x,y)∈(∂Ω)2 ,0 1 − ϑ, ∂Ω 2 |a(x) · (y − x)| ≤ ϑ|x − y| ∀x, y ∈ ∂Ω , |x − y| < τ .

inf a · νΩ > 1 −

Proof. Since νΩ ∈ C 0 (∂Ω, Rn ), the Tietze Extension Theorem 2.33 (see also Theorem 2.85 below) implies that there exists ν˜ ∈ Cc0 (Rn ) such that ν˜|∂Ω = νΩ

on ∂Ω .

Then the set {x ∈ Rn : |˜ ν (x)| > ϑ} is an open neighborhood of ∂Ω and Lemma 2.70 implies that there exists an open bounded subset U of Rn of class C ∞ such that ν (x)| > ϑ} . ∂Ω ⊆ U ⊆ U ⊆ {x ∈ Rn : |˜ Let {ϕ }>0 be a family of standard mollifiers. Then we have lim ϕ ∗ ν˜ = ν˜

→0

uniformly on U (cf. Proposition A.33). Then there exists 1 ∈]0, 1[ such that    ϕ ∗ ν˜ ν˜   < ϑ/4 on U  |ϕ ∗ ν˜| − |˜ ν|  for all  ∈]0, 1 ] and we can set a(x) ≡

ϕ1 ∗ ν˜(x) |ϕ1 ∗ ν˜(x)|

∀x ∈ U .

Since |a(x) − νΩ (x)| < ϑ/4 < ϑ for all x ∈ ∂Ω, we have |a(x)|2 + |νΩ (x)|2 − 2a(x) · νΩ (x) < ϑ2

∀x ∈ ∂Ω

and accordingly 2 − ϑ2 < 2a(x) · νΩ (x)

∀x ∈ ∂Ω .

2

Hence, min∂Ω a · νΩ > 1 − ϑ2 and the second last inequality in (2.66) holds true. By Lemma 2.76, the function ωΩ from [0, +∞[ to itself is increasing and satisfies the equality ωΩ (0) = 0 and the inequality |νΩ (x) · (y − x)| ≤ ωΩ (|x − y|)|x − y|

∀x, y ∈ ∂Ω .

Then we have |a(x) · (y − x)| ≤ |a(x) − νΩ (x)||y − x| + |νΩ (x) · (y − x)|

2.18 Existence of Tubular Neighborhoods of the Boundary of Bounded Open Sets

≤ ((ϑ/4) + ωΩ (|x − y|))|x − y|

87

∀x, y ∈ ∂Ω .

Since limt→0 ωΩ (t) = 0, there exists τ ∈]0, +∞[ such that ∀t ∈ [0, τ ]

ωΩ (t) < ϑ/2

 

and thus all the conditions in (2.66) are satisfied.

We note that Fichera [98, pp. 207-208] already recognized the importance of the existence of a vector field a such that the essential infimum of a · νΩ is bounded away from 0 for sets with a piecewise smooth boundary in the analysis of boundary value problems for systems of partial differential equations, and in particular for the analysis of boundary value problems with unilateral constraints (see also Fichera [99, p. 413]). For the introduction of a vector field such that the essential infimum of a · νΩ is bounded away from 0 in the case of Lipschitz sets, we mention Grisvard [112, Lemma 1.5.1.9]. A unit vector field a as in the previous lemma can play the role of a normal in the definition of a tubular neighborhood of ∂Ω, as the following statement shows (see also reference [186] with Rossi for a related result which holds under stronger assumptions on Ω). Lemma 2.79. Let n ∈ N \ {0, 1}. Let m ∈ N \ {0}. Let Ω be a bounded open subset of Rn of class C 1 . If an open neighborhood U of ∂Ω and a ∈ C 1 (U , Rn ) satisfy conditions (2.66) for some ϑ ∈]0, 1[ and τ ∈]0, +∞[, then there exists t1 ∈]0, +∞[ such that the following statements hold. (i) The map Ψ from (∂Ω)×] − t1 , t1 [ to Rn defined by Ψ (x, t) = x + ta(x)

∀(x, t) ∈ (∂Ω)×] − t1 , t1 [

is injective, and the set A(t2 ) ≡ Ψ ((∂Ω)×] − t2 , t2 [) is an open neighborhood of ∂Ω for all t2 ∈]0, t1 ]. (ii) We have x + ta(x) ∈ Ω

∀t ∈] − t1 , 0[

and

x + ta(x) ∈ Ω −

∀t ∈]0, t1 [ .

Proof. By Lemma 2.63 of the uniform cylinders, there exist r, δ ∈]0, +∞[ such that r < δ ≤ τ /2 , and such that, for each x ∈ ∂Ω, there exists Rx ∈ On (R) such that C(x, Rx , r, δ) is a coordinate cylinder for Ω around x. Possibly shrinking r we can assume that the function γx which represents ∂Ω in C(x, Rx , r, δ) as a graph is of class C 1 (Bn (0, r), ] − δ/2, δ/2[). We first show the existence of t1 as in (i). Assume by contradiction that Ψ is not injective for any choice of t1 . Then for each j ∈ N there exist (xj , tj ), (xj , tj ) ∈ ∂Ω×] − 2−j , 2−j [ such that

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2 Preliminaries

(xj , tj ) = (xj , tj ) , In particular,

xj = xj

Ψ (xj , tj ) = Ψ (xj , tj ) . ∀j ∈ N .

Indeed, if xj = xj , then a(xj ) = a(xj ) and the equality xj + tj a(xj ) = xj + tj a(xj ) implies that

tj a(xj ) = tj a(xj )

and thus tj = tj , contrary to (xj , tj ) = (xj , tj ). Possibly selecting a subsequence, we can assume that there exist x , x ∈ ∂Ω such that x = lim xj , j→∞

x = lim xj . j→∞

Since a is continuous and limj→∞ tj = limj→∞ tj = 0, we have x = lim xj + tj a(xj ) = lim xj + tj a(xj ) = x j→∞

j→∞

and we set x ˜ ≡ x = x . By our contradiction assumption, we have the equality 0 = xj + tj a(xj ) − xj − tj a(xj )

∀j ∈ N

which we rewrite as 0 = (xj − xj ) + (tj − tj )a(xj ) + tj (a(xj ) − a(xj ))

∀j ∈ N

(2.67)

and, by dividing by |xj − xj |, we obtain tj − tj a(xj ) − a(xj ) xj − xj   a(x − t ) = − j j |xj − xj | |xj − xj | |xj − xj | Then the triangular inequality implies that    t − t   j j     ≤ 1 + |tj |Lip(a)  |xj − xj | 

∀j ∈ N .

∀j ∈ N .

(2.68)

Next, we go back to the above equality (2.67), we take the scalar product with  x j −xj  |2 , |x −x j j

0=1+

and obtain

a(xj ) − a(xj ) xj − xj xj − xj tj − tj   a(x + t ·  ) · j j |xj − xj | |xj − xj | |xj − xj | |xj − xj |

∀j ∈ N . (2.69)

In order to exploit the last inequality in (2.66), we choose j0 ∈ N so that

2.18 Existence of Tubular Neighborhoods of the Boundary of Bounded Open Sets

|xj − xj | ≤ |xj − x ˜| + |˜ x − xj | < τ

89

∀j0 ≤ j ∈ N

and thus inequality (2.68) and equality (2.69) imply that 1 ≤ (1 + |tj |Lip(a))ϑ + |tj |Lip(a)

∀j0 ≤ j ∈ N .

Since limj→∞ tj = 0, we obtain 1 ≤ ϑ, a contradiction. Hence, there exists t1 in ]0, +∞[ such that Ψ is injective on (∂Ω)×] − t1 , t1 [. We now turn to show that A(t2 ) is open for all t2 ∈]0, t1 ]. Let t2 ∈]0, t1 ] and (x , t ) ∈ A(t2 ). Since Ψ is injective, then the composition Ψ  of Ψ with the continuous and injective map (x + Rxt (η, γ(η))t , s) of the variable (η, s), i.e., the map Ψ  (η, s) ≡ Ψ (x + Rxt (η, γ(η))t , s) = x + Rxt (η, γ(η))t + sa(x + Rxt (η, γ(η))t ) ∀(η, s) ∈ Bn−1 (0, r)×] − t2 , t2 [ , is continuous and injective. Then the Theorem of Invariance of Domain A.32 implies that Ψ  (Bn−1 (0, r)×] − t2 , t2 [) is an open subset of Rn . Since (x , t ) ∈ Ψ  (Bn−1 (0, r)×] − t2 , t2 [) ⊆ A(t2 ), it follows that (x , t ) is interior to A(t2 ). Hence, A(t2 ) is open. Next we turn to prove statement (ii). Let x ∈ ∂Ω. Then we set A− (x, t1 ) ≡ {x+ta(x) : t ∈]0, t1 [} .

A+ (x, t1 ) ≡ {x+ta(x) : t ∈]−t1 , 0[} , Since Ψ is injective, we have

Ψ ({x}×] − t1 , t1 [) ∩ ∂Ω = {x} . Indeed, if there exists t ∈] − t1 , t1 [ such that y = x + ta(x) ∈ ∂Ω , then Ψ (y, 0) = Ψ (x, t) and the injectivity of Ψ implies that x = y, t = 0. Hence, A± (x, t1 ) ⊆ Rn \ ∂Ω . Since the sets A± (x, t1 ) are connected, they cannot intersect both Ω and Ω − . Indeed, the connected components of Ω and Ω − are disjoint. We now show that A+ (x, t1 ) ⊆ Ω

and

A− (x, t1 ) ⊆ Ω − .

By the third condition of (2.66) and by Proposition 2.44, −a(x) points to the interior of Ω and thus there exists ςx,−a(x) > 0 such that

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2 Preliminaries

x + ta(x) ∈ Ω

∀t ∈] − ςx,−a(x) , 0[ ,

and thus in particular for all t ∈] − min{ςx,−a(x) , t1 }, 0[. Hence, the connected set A+ (x, t1 ) contains points of Ω and thus it cannot contain points of Ω − . As a consequence, we must have A+ (x, t1 ) ⊆ Ω. The inclusion A− (x, t1 ) ⊆ Rn \ Ω can be proved similarly.  

2.19 A Sufficient Condition for the H¨older Continuity of Continuously Differentiable Functions, in the Wake of the Work of Carlo Miranda In this section we plan to prove a sufficient condition for a real valued continuously differentiable function f in an open subset Ω of Rn to be α-H¨older continuous in Ω for some α ∈]0, 1]. If Ω =]0, 1[, then an elementary sufficient condition is that |y|1−α |f  (y)| is bounded for y ∈]0, 1[. If Ω = Bn−1 (0, 1)×]0, 1[, then a sufficient condition of the α-H¨older continuity of f ∈ C 1 (Ω) is that |y|1−α |∇f (η, y)|

is bounded in (η, y) ∈ Bn−1 (0, 1)×]0, 1[

(cf. Agmon, Douglis and Nirenberg [2, p. 717]). If γ ∈ C 1 (Bn−1 (0, r), ] − δ, δ[) for some r, δ ∈]0, +∞[ and if Ω = hypographs (γ) , then Miranda [212, p. 308] has shown that a sufficient condition for the α-H¨older continuity of f ∈ C 1 (Ω) is that |γ(η) − y|1−α |∇f (η, y)|

is bounded in (η, y) ∈ hypographs (γ) .

Then in order to obtain a sufficient condition for a function defined in an open subset Ω of Rn of class C 1 , one can exploit the above result of Miranda and a standard localization argument based on the existence of a finite cover of ∂Ω consisting of coordinate cylinders and of a corresponding partition of unity. However, later in the book it will be convenient to have a condition that is independent from the choice of a specific atlas of ∂Ω. For this reason, we prove a global condition in the following Proposition 2.82, where we exploit the approximate outer unit normal a of Lemma 2.78. For a treatment in the case where Ω is a Lipschitz set, we refer to [170]. To prove Proposition 2.82, we need the following two preliminary technical Lemmas 2.80 and 2.81. Lemma 2.80. Let n ∈ N \ {0, 1}. Let ϑ ∈]0, 1[. If v, w ∈ Rn and if |v · w| ≤ ϑ|v| |w| ,

2.19 A Sufficient Condition for the H¨older Continuity of Continuously...

91

then |v + w|2 ≥ (1 − ϑ)(|v|2 + |w|2 ) + ϑ(|v| − |w|)2 . Proof. It suffices to note that |v + w|2 = |v|2 + |w|2 + 2v · w ≥ |v|2 + |w|2 − 2|v · w| ≥ |v|2 + |w|2 − 2ϑ|v||w| = (1 − ϑ)(|v|2 + |w|2 ) + ϑ(|v|2 + |w|2 − 2|v||w|) = (1 − ϑ)(|v|2 + |w|2 ) + ϑ(|v| − |w|)2 .   Lemma 2.81. Let n ∈ N \ {0, 1}. Let Ω be a bounded open subset of Rn of class C 1 . Let ϑ ∈]0, 1[. Let U be an open neighborhood of ∂Ω of class C ∞ . Let a in C ∞ (U , Rn ) satisfy the conditions in (2.66) for some ϑ ∈]0, 1[ and τ ∈]0, +∞[. Let t1 ∈]0, +∞[ be as in Lemma 2.79. Let p ∈ ∂Ω. Let r, δ ∈]0, +∞[, 2δ < min{τ /2, t1 } ,

r < δ,

and Rp ∈ On (R) be such that C(p, Rp , r, δ) is a coordinate cylinder for Ω around p. Let γp ∈ C 1 (Bn−1 (0, r)) represent ∂Ω in C(p, Rp , r, δ) as a graph. Let C(p, Rp , r, δ) ∩ Ω ⊆ A+ (t1 ) ≡ {x + ta(x) : t ∈] − t1 , 0[, x ∈ ∂Ω} . Let

%



r (1 − ϑ)1/2 t1 , t2 ∈ 0, min , √ 4 2 2(Lip(a) + 1) 2

$ .

If α ∈]0, 1], then there exists B ∈]0, +∞[ such that |f : [p + Rpt (Bn−1 (0, r/4)×] − δ/4, δ/4[)] ∩ A+ (t2 )|α ≤ BMt1 ,α (f )

(2.70)

for all f ∈ C 1 (Ω) such that Mt1 ,α (f ) ≡ sup{|t|1−α |∇f (x + ta(x))| : (x, t) ∈ (∂Ω)×] − t1 , 0[} < +∞ . (2.71) Proof. Let p , p ∈ [p + Rpt (Bn−1 (0, r/4)×] − δ/4, δ/4[)] ∩ A+ (t2 ). We plan to estimate the difference |f (p ) − f (p )| and show that it is smaller than a constant times Mt1 ,α (f )|p − p |α . In order to exploit condition (2.71) on f , we plan to to define a C 1 path in A+ (t1 ) that joins p and p . Lemma 2.79 implies that there exist x , x ∈ ∂Ω, t , t ∈] − t2 , 0[ such that p = x + t a(x ) ,

p = x + t a(x ) .

By the triangular inequality and by the equality |a(x )| = 1, and by the inequality t2 < r/4 < δ/4, we have x ∈ Bn (p , t2 ) ⊆ [p + Rpt (Bn−1 (0, r/4)×] − δ/4, δ/4[)] + Bn (0, t2 )

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2 Preliminaries

⊆ [p + Rpt (Bn−1 (0, r/2)×] − δ/2, δ/2[)] and similarly

x ∈ [p + Rpt (Bn−1 (0, r/2)×] − δ/2, δ/2[)] .

Next we set ψp (η) ≡ p + Rpt (η, γp (η))t

∀η ∈ Bn−1 (0, r) .

Since x , x ∈ ∂Ω, then there exist ξ  , ξ  ∈ Bn−1 (0, r/2) such that x = ψp (ξ  ) = p + Rpt (ξ  , γp (ξ  ))t , x = ψp (ξ  ) = p + Rpt (ξ  , γp (ξ  ))t . Then we note that the convexity of Bn−1 (0, r/2) implies that y(s) ≡ (1 − s)ξ  + sξ  ∈ Bn−1 (0, r/2)

∀s ∈ [0, 1]

and that the convexity of ] − t2 , 0[ implies that (1 − s)t + st ∈] − t2 , 0[

∀s ∈ [0, 1] .

Then we have G(s) ≡ ψp (y(s)) +[(1 − s)t + st ]a(ψp (y(s))) ∈ A+ (t2 ) ∀s ∈ [0, 1] , t G(s) ∈ [p + Rp (Bn−1 (0, r/2)×] − δ/2, δ/2[)] + Bn (0, t2 ) ⊆ [p + Rpt (Bn−1 (0, 3r/4)×] − 3δ/4, 3δ/4[)] 

∀s ∈ [0, 1] ,



G(0) = p ,

G(1) = p .

Moreover, 

1

length(G) ≤

&

|ξ  − ξ  |2 + Lip2 (γp )|ξ  − ξ  |2 ds + |t − t |

0

 1& +|t2 |Lip (a) |ξ  − ξ  |2 + Lip2 (γp )|ξ  − ξ  |2 ds 0 & & ≤ |ξ  − ξ  | 1 + Lip2 (γp ) + |t − t | + |t2 |Lip (a)|ξ  − ξ  | 1 + Lip2 (γp ) & ≤ |x − x | 1 + Lip2 (γp )(1 + |t2 |Lip (a)) + |t − t | . We now wish to estimate |x − x | and |t − t | in terms of |p − p |. There is no loss of generality in assuming that t < t .

2.19 A Sufficient Condition for the H¨older Continuity of Continuously...

By assumption, we know that     a(x) · y − x  ≤ ϑ  |y − x| 

93

∀x, y ∈ C(p, Rp , r, δ) ∩ (∂Ω) , x = y .

Indeed, |x−y| ≤ 2r +δ < τ for all x, y ∈ C(p, Rp , r, δ)∩(∂Ω). Then Lemma 2.80 implies that |p − p | = |(x + t a(x )) − (x + t a(x ))| = |(x − x ) + (t − t )a(x ) − t (a(x ) − a(x ))| ≥ |(x − x ) + (t − t )a(x )| − |t | |(a(x ) − a(x ))| # ≥ (1 − ϑ)1/2 |x − x |2 + |t − t |2 |a(x )|2 − t2 Lip(a)|x − x | . By the elementary inequality √ a1 + a2 ≤ 2(a21 + a22 )1/2

∀a1 , a2 ∈ [0, +∞[ ,

we have (1 − ϑ)1/2

#

|x − x |2 + |t − t |2 |a(x )|2 − t2 Lip(a)|x − x |

(1 − ϑ)1/2  √ (|x − x | + |t − t |) − t2 Lip(a)|x − x | 2   (1 − ϑ)1/2 (1 − ϑ)1/2  √ √ − t2 Lip(a) |x − x | + |t − t | = 2 2 (1 − ϑ)1/2  (1 − ϑ)1/2  √ √ |x − x | + |t − t | . ≥ 2 2 2 ≥

Indeed, t2 ≤

(1−ϑ)1/2 √ . 2 2(1+Lip(a))

Hence, we deduce that

√

2 |p − p | , |t − t | ≤ (1 − ϑ)1/2 



√ 2 2 |x − x | ≤ |p − p | . (2.72) (1 − ϑ)1/2 



At this point we may try to use the path G to estimate the difference |f (p ) − f (p )| for a function f ∈ C 1 (Ω) that satisfies condition (2.71). Indeed, we have |f (p ) − f (p )| ≤



1

|∇f (G(s))| |G (s)| ds

0



1

≤

Mt1 ,α (f ) |(1 − s)t − st |α−1 |G (s)| ds

0

≤ Mt1 ,α (f ) |t |α−1 length(G) ≤ Mt1 ,α (f ) |t |α−1

94

2 Preliminaries

 ×

 √ √ & 2 2 2 2 1 + Lip (γp )(1 + |t2 |Lip (a)) + |p − p | . (1 − ϑ)1/2 (1 − ϑ)1/2

However, we cannot estimate |t | from below and, as a consequence, the factor |t |α−1 may be arbitrarily large, making useless the inequality above. To overcome this difficulty we have to modify the path G. Then, we take d1 ≡ min{|p − p |, r/4} and we define a path with endpoints x + (t − d1 )a(x ) ,

x + (t − d1 )a(x ) .

To do so, we set Gd1 (s) ≡ G(s) − d1 a(ψp (y(s)))

∀s ∈ [0, 1] .

Since (1 − s)t + st − d1 ∈] − t2 − d1 , 0[ ⊆] − (t1 /2) − (r/4), 0[⊆] − (t1 /2) − (t1 /8), 0[⊆] − t1 , 0[

∀s ∈]0, 1[ ,

we have Gd1 (s) ∈ A+ (t1 )

∀s ∈]0, 1[ .

Then we have Gd1 (s) ∈ [p + Rpt (Bn−1 (0, 3r/4)×] − 3δ/4, 3δ/4[)] + Bn (0, d1 ) ⊆ C(p, Rp , r, δ) for all s ∈ [0, 1] and Gd1 (0) = x + (t − d1 )a(x ) ,

Gd1 (1) = x + (t − d1 )a(x ) .

Then we have the following inequality for the length of Gd1 , length(Gd1 )

 1& |ξ  − ξ  |2 + Lip2 (γp )|ξ  − ξ  |2 ds ≤ length(G) + d1 Lip(a) 0 $ % & ≤ (1 + d1 Lip(a)) |x − x | 1 + Lip2 (γp )(1 + |t2 |Lip (a)) + |t − t | ≤ (1 + d1 Lip(a)) * + √ √ & 2 2 2 2 1 + Lip (γp )(1 + |t2 |Lip (a)) + |p − p |. × (1 − ϑ)1/2 (1 − ϑ)1/2

2.19 A Sufficient Condition for the H¨older Continuity of Continuously...

95

Next we note that x + (t − sd1 )a(x ) ∈ [p + Rpt (Bn−1 (0, 3r/4)×] − 3δ/4, 3δ/4[)] + Bn (0, d1 ) ⊆ C(p, Rp , r, δ) ∀s ∈ [0, 1] and similarly x + (t − sd1 )a(x ) ∈ C(p, Rp , r, δ)

∀s ∈ [0, 1] .

Then by the memberships t − sd1 , t − sd1 ∈] − t2 − d1 , 0[ ⊆] − (t1 /2) − (r/4), 0[⊆] − (t1 /2) − (t1 /8), 0[⊆] − t1 , 0[

∀s ∈ [0, 1] ,

we have x + (t − sd1 )a(x ) ∈ A+ (t1 ) ,

x + (t − sd1 )a(x ) ∈ A+ (t1 )

∀s ∈ [0, 1] .

We now assume that f ∈ C 1 (Ω) satisfies condition (2.71) and we turn to estimate |f (p ) − f (p )|. We have |f (p ) − f (p )| = |f (x + t a(x )) − f (x + t a(x ))| ≤ |f (x + t a(x )) − f (x + (t − d1 )a(x ))| +|f (x + (t − d1 )a(x )) − f (x + (t − d1 )a(x ))| +|f (x + (t − d1 )a(x )) − f (x + t a(x ))|  1  t    |a(x ) · ∇f (x + sa(x ))| ds + |∇f (Gd1 (s))| |Gd1 (s)| ds ≤ t −d1



0

t

|a(x ) · ∇f (x + sa(x ))| ds

+ t −d

 ≤

1

t t −d1  1

+ 

0

Mt1 ,α (f )|s|α−1 ds

Mt1 ,α (f )|(1 − s)t + st − d1 |α−1 |Gd1 (s)| ds

t

+ t −d1

Since

Mt1 ,α (f )|s|α−1 ds .

(1 − s)t + st − d1 ∈] − t2 − d1 , −d1 [

∀s ∈ [0, 1] ,

we have 



|f (p ) − f (p )| ≤ Mt1 ,α (f )



|t |+d1 |t |

sα−1 ds

(2.73)

96

2 Preliminaries

 +length(Gd1 )dα−1 Mt1 ,α (f ) 1

+ Mt1 ,α (f )



|t |+d1

sα−1 ds

|t |

dα (1 + d1 Lip(a)) ≤ Mt1 ,α (f ) 1 + |p − p |dα−1 1 α √ √ $ %  2 2 2 dα 1 (1 + Lip(γ ))(1 + |t |Lip(a)) + × + p 1 α (1 − ϑ)1/2 (1 − ϑ)1/2 (cf. (2.72)). Next we observe that   α dα 1 ≤ |p − p |

and that |p − p | |p − p | = 1−α 1−α min {|p − p |, r/4} d1 ! if |p − p | ≤ r/4 , |p − p |α   1−α ≤ |p −p | diam(Ω)   α   (r/4)1−α ≤ (r/4)1−α |p − p | if |p − p | ≥ r/4 . Hence, inequality (2.73) implies the validity of inequality (2.70) and the proof is complete.   We are now ready to prove a sufficient condition for the H¨older continuity of a continuously differentiable real valued function defined on an open subset of Rn of class C 1 . Proposition 2.82. Let n ∈ N \ {0, 1}. Let Ω be a bounded open subset of Rn of class C 1 . Let U be an open neighborhood of ∂Ω of class C ∞ . Let a ∈ C ∞ (U , Rn ) satisfy the conditions in (2.66) for some ϑ ∈]0, 1[ and τ ∈]0, +∞[. Let t1 ∈]0, +∞[ be as in Lemma 2.79. Let α ∈]0, 1]. Then there exist B ∈]0, +∞[ and a compact subset H of Ω such that   sup |f | + |f : Ω|α ≤ B max sup |f |, sup |∇f |, Mt1 ,α (f ) Ω

H

H

for all f ∈ C 1 (Ω) such that Mt1 ,α (f ) ≡ sup{|t|1−α |∇f (x + ta(x))| : (x, t) ∈ (∂Ω)×] − t1 , 0[} < +∞ . (2.74) Proof. Since Ω is of class C 1 , for each point x ∈ ∂Ω there exists a coordinate cylinder C(x, Rx , rx , δx ) for Ω around x and if rx + δx is less than the distance between ∂Ω and Rn \ A(t1 ), then we have C(x, Rx , rx , δx ) ⊆ A(t1 ). Thus we now choose r∗ , δ∗ ∈]0, +∞[ such that r∗ + δ∗ < dist(∂Ω, Rn \ A(t1 )) . Since we plan to invoke Lemma 2.81, we also assume that

2.19 A Sufficient Condition for the H¨older Continuity of Continuously...

97

2δ∗ < min{τ /2, t1 } , where τ is as in (2.66). Then, by Lemma 2.63 of the uniform cylinders, there exist r ∈]0, r∗ [, δ ∈]0, δ∗ [, with r < δ, such that if x ∈ ∂Ω, then there exists Rx in On (R) such that C(x, Rp , r, δ) is a coordinate cylinder for Ω around x and the corresponding function γx satisfies the condition ∇γx (0) = 0 and the inequalities |∇γx (η)| ≤ 1/3

∀η ∈ Bn−1 (0, r) ,

(2.75)

sup γx C 1 (Bn−1 (0,r)) < +∞ .

x∈∂Ω

Since r + δ < dist(∂Ω, Rn \ A(t1 )), we have C(x, Rx , r, δ) ∩ Ω ⊆ A+ (t1 ) = A(t1 ) ∩ Ω

∀x ∈ ∂Ω .

Since ∂Ω is compact, there exists a finite family {x(j) }m j=1 of points of ∂Ω such that m  [x(j) + Rxt (j) (Bn−1 (0, r/4)×] − δ/4, δ/4[)] , ∂Ω ⊆ j=1

and we note that the right-hand side is an open neighborhood of ∂Ω. We now set  t1 (1 − ϑ)1/2 , , μ ≡ min r/4, √ j=1,...,m 2 2(Lip(a) + 1) 2   m  n (j) t dist ∂Ω, R \ [x + Rx(j) (Bn−1 (0, r/4)×] − δ/4, δ/4[)] , j=1

and we choose t2 ∈]0, μ[. In particular, we have A+ (t2 ) ⊆ A(t2 ) ⊆ {x ∈ Ω : dist(x, ∂Ω) < μ} m  [x(j) + Rxt (j) (Bn−1 (0, r/4)×] − δ/4, δ/4[)] . ⊆ j=1

Then Lemma 2.81 implies that there exists Bj ∈]0, +∞[ such that |f : [x(j) +Rxt (j) (Bn−1 (0, r/4)×]−δ/4, δ/4[)]∩A+ (t2 )|α ≤ Bj Mt1 ,α (f ) (2.76) for all j ∈ {1, . . . , m} and for all f ∈ C 1 (Ω) such that Mt1 ,α (f ) < +∞. Now let Ω1 be an open subset of class C ∞ of Ω such that Ω \ A+ (t2 ) ⊆ Ω1 ⊆ Ω1 ⊆ Ω (cf. Lemma 2.70). Since Ω1 is of class C 1 , C 1 (Ω1 ) is continuously embedded into C 0,α (Ω1 ) and thus there exists cα (Ω1 ) ∈]0, +∞[ such that

98

2 Preliminaries

!

)

|f : Ω1 |α ≤ cα (Ω1 ) sup sup |f |, sup |∇f | Ω1

(2.77)

Ω1

for all f ∈ C 1 (Ω1 ) (cf. Corollary 2.57 (i), (iii)). Then we take f ∈ C 1 (Ω) such that Mt1 ,α (f ) < +∞ and we turn to estimate |f : Ω|α . To do so, we observe that Ω ⊆ Ω1 ∪

m ' 

[x(j) + Rxt (j) (Bn−1 (0, r/4)×] − δ/4, δ/4[)] ∩ A(t2 )

( . (2.78)

j=1

Let Λ be a Lebesgue number corresponding to the open cover of Ω in the right-hand side of (2.78). We can clearly assume that Λ < δ/12 . If p , p ∈ Ω and |p − p | ≤ Λ, then both p and p belong to at least one of the open sets in the right-hand side of (2.78) and thus inequalities (2.76) and (2.77) imply that     ˜ |f (p )−f (p )| ≤ max BMt1 ,α (f ), cα (Ω1 ) sup |f |, cα (Ω1 ) sup |∇f | |p −p |α , Ω1

Ω1

(2.79) where ˜≡ B

max

j∈{1,...,m}

Bj .

In order to estimate |f (p ) − f (p )| in case |p − p | > Λ, we need to estimate supΩ |f | (cf. Remark 2.20). To do so, we note that x(j) −

t2 a(x(j) ) ∈ A(t2 )+ ⊆ Ω 2

∀j ∈ {1, . . . , m} ,

and that our assumptions t2 < μ < r/4, r < δ imply that x(j) −

t2 a(x(j) ) ∈ Bn (x(j) , r/8) ⊆ x(j) + Rxt (j) (Bn−1 (0, r/4)×] − δ/4, δ/4[) 2

for all j ∈ {1, . . . , m}. By Lemma 2.70 of the Appendix, there exists an open subset Ω2 of class C ∞ of Ω such that   t2 Ω1 ∪ x(j) − a(x(j) ) : j ∈ {1, . . . , m} ⊆ Ω2 ⊆ Ω2 ⊆ Ω . 2 If p ∈ Ω \ Ω2 , then p ∈ Ω \ Ω1 and p ∈ A(t2 )+ ⊆

m  j=1

[x(j) + Rxt (j) (Bn−1 (0, r/4)×] − δ/4, δ/4[)]

2.20 Schauder Spaces on a Compact Manifold and on the Boundary...

99

and accordingly, there exists ˜j ∈ {1, . . . , m} such that ˜

p ∈ A(t2 )+ ∩ [x(j) + Rxt (˜j) (Bn−1 (0, r/4)×] − δ/4, δ/4[)] . ˜

Since both p and x(j) −

t2 (˜ j) 2 a(x )

belong to

˜

A(t2 )+ ∩ [x(j) + Rxt (˜j) (Bn−1 (0, r/4)×] − δ/4, δ/4[)] ˜

and x(j) −

t2 (˜ j) 2 a(x )

belongs to Ω2 , we have t2 t2 ˜ ˜ ˜ a(x(j) ))| + |f (x(j) − a(x(j) ))| 2  2  α ˜ t ,α (f ) p − x(˜j) − t2 a(x(˜j) ) + sup |f | . ≤ BM 1   2 ˜

|f (p)| ≤ |f (p) − f (x(j) −

Ω2

˜

Since both p and x(j) −

t2 (˜ j) 2 a(x )

belong to

˜

x(j) + Rxt (˜j) (Bn−1 (0, r/4)×] − δ/4, δ/4[) that has a diameter less than or equal to 2(r/4) + 2(δ/4) < δ, we have ˜ t ,α (f )δ α + sup |f | . |f (p)| ≤ BM 1 Ω2

If instead p ∈ Ω2 , we certainly have |f (p)| ≤ supΩ2 |f |. Hence, ˜ t ,α (f )δ α + sup |f | sup |f | ≤ BM 1 Ω

Ω2

and |f : Ω|α

    ˜ t ,α (f ), cα (Ω1 ) sup |f |, cα (Ω1 ) sup |∇f | , 2 sup |f | . ≤ max max BM 1 Λα Ω Ω1 Ω1

Then by taking H = Ω2 , we conclude that B as in the statement does exist.

 

2.20 Schauder Spaces on a Compact Manifold and on the Boundary of a Bounded Open Subset of Rn Let n ∈ N \ {0, 1}. Let m ∈ N \ {0} and α ∈ [0, 1]. A subset M of Rn is a differential manifold of dimension s ∈ {1, . . . , n − 1} and of class C m,α embedded in Rn if, for every p ∈ M , there exist an open neighborhood W of p in Rn and a

100

2 Preliminaries

  map ψ ∈ C m,α Bs (0, 1), Rn such that ψ is a homeomorphism of Bs (0, 1) onto W ∩ M , ψ(0) = p, and the Jacobian matrix Dψ has rank s at all points of Bs (0, 1), i.e., if there exists a parametrization ψ of class C m,α for M around p. If ψ is such a parametrization, then Lemma 2.41 implies that the operator dψ(u) maps Rs onto the s-dimensional tangent space Tψ(u) M at ψ(u), for all u ∈ Bs (0, 1). If we further assume that M is compact, then thereexist p(1) ,. . .  , p(r) ∈ M , and parametrizations {ψl }l=1,...,r , with ψl ∈ C m,α Bs (0, 1), Rn , such that ∪rl=1 ψl (Bs (0, 1)) = M . Since C m,α (Bn−1 (0, 1)) is a subspace of C m,α1 (Bn−1 (0, 1)) for all α1 ∈ [0, α], a differential manifold of class C m,α embedded in Rn is also of class C m,α1 for all α1 ∈ [0, α]. If in addition m > 1, then C m,α (Bn−1 (0, 1)) is a subspace of C h,β (Bn−1 (0, 1)) for all h ∈ {1, . . . , m − 1} and β ∈ [0, 1], and thus a differential manifold of class C m,α embedded in Rn is also of class C h,β for all h ∈ {1, . . . , m − 1} and β ∈ [0, 1]. h,β Let m ∈ N\{0}, α ∈ [0, 1], β ∈ [0, α], h ∈ {0, . . . , m}. We denote  by C (M ) the linear space of functions f from M to R such that f ◦ ψl ∈ C h,β Bs (0, 1) for all l ∈ {1, . . . , r}, and we introduce the norm f C h,β (M ) ≡ sup f ◦ ψl C h,β (Bs (0,1))

∀f ∈ C h,β (M ).

l=1,...,r

It is well known that choosing a different finite family of parametrizations as {ψl }l=1,...,r , we would obtain an equivalent norm. Moreover, if h = 0, then C h,β (M ) coincides with the space of β-H¨older continuous functions on the set M and the above norm is equivalent to the one that we have already introduced in Section 2.6. If 1 ≤ h < m, we note that the assumption that M is of class C m,α implies that M is of class C m−1,1 and that accordingly we can define C h,β (M ) as above for all β ∈ [0, 1], and not only for β ∈ [0, α], while instead for h = m we have defined C h,β (M ) only for β ∈ [0, α].  By the properties of C h,α Bs (0, 1) of Proposition 2.54, we deduce the validity of the following.

Theorem 2.83. Let n ∈ N \ {0, 1}, s ∈ {1, . . . , n − 1}, m ∈ N \ {0}, α ∈ [0, 1] (cf. (2.33)). Let M be a compact manifold of dimension s, of class C m,α and embedded in Rn . Let h ∈ {1, . . . , m}. Then the following statements hold.   (i) The space C h,α (M ),  · C h,α (M ) is complete. (ii) C h (M ) is continuously embedded into C h−1,1 (M ).  (iii) If α ∈ [0, α[, then C h,α (M ) is compactly embedded into C h,α (M ). (iv) Let α1 , α2 ∈ [0, α]. Then the pointwise multiplication from C h,α1 (M ) × C h,α2 (M ) to C h,min{α1 ,α2 } (M ) that takes a pair of functions (f, g) to the pointwise product f g is continuous.

2.20 Schauder Spaces on a Compact Manifold and on the Boundary...

101

Under the assumptions of Theorem 2.83, if 1 ≤ h < m, then Ω is of class C h,1 and thus by applying Theorem 2.83 with h instead of m we deduce that statements (iii) and (iv) of Theorem 2.83 as it is hold also by replacing α with some β ∈]0, 1]. We retain the same notation for C 1 manifolds that are embedded in Rn of Section 2.1. Accordingly, we denote by dσ the area element of a manifold, by LM the σ-algebra of measurable subsets of M , and by mM (or simply by ms ) the sdimensional measure of M (cf. e.g., Naumann and Simader [232]) . We plan to show that if Ω is of class C m,α with m ≥ 1 and α ∈ [0, 1], then the functions of C h,α (∂Ω) with h ∈ {0, . . . , m} can be extended to functions of class C h,α in the whole of Rn . Since locally around the boundary points, the boundary of sets of class C m,α can be mapped onto (n − 1)-dimensional balls by means of a parametrization, we first prove the following well-known technical lemma for functions defined on an (n − 1)-dimensional ball. Lemma 2.84. Let n ∈ N \ {0, 1}, h ∈ N, α ∈ [0, 1] (cf. (2.33)). Let ρ ∈]0, +∞[. Then there exists a linear and continuous operator G from {f ∈ C h,α (Bn−1 (0, ρ)) : supp (f ) ⊆ Bn−1 (0, ρ/2)} to {g ∈ C h,α (Bn (0, ρ)) : supp (g) ⊆ Bn (0, 3ρ/4)} such that G[f ](x1 , . . . , xn−1 , 0) = f (x1 , . . . , xn−1 )

∀(x1 , . . . , xn−1 ) ∈ Bn−1 (0, ρ)

for all f ∈ C h,α (Bn−1 (0, ρ)) with supp (f ) ⊆ Bn−1 (0, ρ/2). Proof. Let f  (x1 , . . . , xn−1 , xn ) ≡ f (x1 , . . . , xn−1 )

∀x ∈ Bn (0, ρ)

for all f ∈ C h,α (Bn−1 (0, ρ)). By the definition of the C h,α -norm, we have f  C h,α (Bn (0,ρ)) ≤ f C h,α (Bn−1 (0,ρ)) . By the cut-off function Lemma A.36, there exists a function θ ∈ Cc∞ (Rn ) such that θ = 1 on Bn (0, 5ρ/8) and supp θ ⊆ Bn (0, 3ρ/4). Then we set G[f ] ≡ θf 

∀f ∈ C h,α (Bn−1 (0, ρ)) .

By the continuity of the pointwise product in C h,α (Bn (0, ρ)) and by the above inequality, the operator G is continuous from C h,α (Bn−1 (0, ρ)) to C h,α (Bn (0, ρ)). Moreover, G[f ](x1 , . . . , xn−1 , 0) = f (x1 , . . . , xn−1 )

∀(x1 , . . . , xn−1 ) ∈ Bn−1 (0, ρ) ,

supp G[f ] ⊆ Bn (0, 3ρ/4) for all f ∈ C h,α (Bn−1 (0, ρ)) such that supp (f ) ⊆ Bn−1 (0, ρ/2).

 

102

2 Preliminaries

We are now ready to prove the following. Theorem 2.85. Let n ∈ N \ {0, 1}. Let m ∈ N \ {0}, h ∈ {0, . . . , m}, α ∈ [0, 1] (cf. (2.33)). Let Ω be a bounded open subset of Rn of class C m,α . Then (i) ∂Ω is a manifold of class C m,α and dimension n − 1 embedded in Rn , and  h,α h,α Ω to C (∂Ω) for the restriction map is linear and continuous from C all h = 0, . . . , m. (ii) Let Ω  be a bounded open subset of Rn such that Ω ⊆ Ω  . Then there exists a linear and continuous extension operator FΩ  from C h,α (∂Ω) to C h,α (Ω  ) such that supp FΩ  [f ] is compact and contained in Ω 

FΩ  [f ]|∂Ω = f ,

for all f ∈ C h,α (∂Ω). (iii) There exists a linear and continuous extension operator F of C h,α (∂Ω) to C h,α (Ω) such that F [f ]|∂Ω = f for all f ∈ C h,α (∂Ω).   Proof. Let {φa }a∈{1,...,b} with φa ∈ C m,α Bn (0, 1), Rn be parametrizations of Ω around points of ∂Ω such that ∂Ω ⊆

b 

φa (Bn (0, 1/2))

a=1

(see Definition 2.65). By Remark 2.66, we can assume that φa (Bn (0, 1)) ⊆ Ω  and that φa (Bn (0, 1)) is regular in the sense of Whitney for all a ∈ {1, . . . , b}. Next we set ∀η ∈ Bn−1 (0, 1) φ˜a (η) ≡ φa (η, 0) for all a ∈ {1, . . . , b}. Then {φ˜a }a∈{1,...,b} is a family of parametrizations for ∂Ω b such that ∂Ω ⊆ a=1 φ˜a (Bn−1 (0, 1/2)). We now  turn to prove statement (i). We first point out that if h > 0, a function g of C h,α Ω is defined in Ω and admits a continuous extension to Ω by definition (see Section 2.11). Thus we understand that the restriction to ∂Ω of g of statement (i) is referred to the restriction to ∂Ω of the continuous extension of g to Ω. By Definition 2.65 of set of class C m,α (with the diffeomorphisms), it follows that ∂Ω is a manifold of class C m,α and dimension n − 1 embedded in Rn . By the definition of norm in C h,α (∂Ω), we can take f C h,α (∂Ω) =

sup a∈{1,...,b}

f ◦ φ˜a C h,α (Bn−1 (0,1))

∀f ∈ C h,α (∂Ω) .

    Next we note that φa ∈ C m,α Bn (0, 1), Rn ⊆ C h,1 Bn (0, 1), Rn when h < m. Then Theorem 2.59 on the composition of functions in Schauder spaces implies that there exists c1 > 0 such that g|∂Ω C h,α (∂Ω) =

sup a∈{1,...,b}

g|∂Ω ◦ φ˜a C h,α (Bn−1 (0,1))

2.20 Schauder Spaces on a Compact Manifold and on the Boundary...

≤

sup a∈{1,...,b}

103

g ◦ φa C h,α (B+ (0,1)) ≤ c1 gC h,α (Ω) n

for all g ∈ C h,α (Ω) both in case h < m and h = m. Hence, statement (i) holds true. We now turn to prove statement (ii). By Lemma A.35 on the partition of unity, there exists a finite family {θa }ba=1 of functions of class Cc∞ (Rn ) such that supp θa ⊆ φa (Bn (0, 1/2))

∀a ∈ {1, . . . , b}

b 

and

θa (x) = 1 ∀x ∈ ∂Ω .

a=1

If a ∈ {1, . . . , b}, then by the definition of norm in C h,α (∂Ω), we can take θa f C h,α (∂Ω) =

sup l∈{1,...,b}

(θa f ) ◦ φ˜l C h,α (Bn−1 (0,1))

∀f ∈ C h,α (∂Ω)

and we note that supp (θa f ) ⊆ φ˜a (Bn−1 (0, 1/2))

∀f ∈ C h,α (∂Ω) .

Let G be the operator for the unit ball of Lemma 2.84. Let G be the norm of (−1) G. Since φa (Bn (0, 1)) is regular in the sense of Whitney and φa belongs to m,α h,1 (φa (Bn (0, 1))) ⊆ C (φa (Bn (0, 1))) when h < m, the composition TheC orem 2.59 for functions in a Schauder space implies that there exists c > 0 such that G[(θa f ) ◦ φ˜a ] ◦ φ(−1) C h,α (φa (Bn (0,1))) a

(2.80)

≤ c G[(θa f ) ◦ φ˜a ]C h,α (Bn (0,1)) ≤ c G(θa f ) ◦ φ˜a C h,α (Bn−1 (0,1)) ≤ c Gθa f C h,α (∂Ω)

∀f ∈ C h,α (∂Ω)

for all a ∈ {1, . . . , b} both in case h < m and h = m. Since the pointwise product is continuous in C h,α (∂Ω), there exists c > 0 such that θa f C h,α (∂Ω) ≤ c f C h,α (∂Ω)

∀f ∈ C h,α (∂Ω)

for all a ∈ {1, . . . , b}. We also note that supp G[(θa f ) ◦ φ˜a ] ⊆ Bn (0, 3/4) ⊆ Bn (0, 3/4) ⊆ Bn (0, 1) supp [G[(θa f ) ◦ φ˜a ] ◦ φ(−1) ] ⊆ φa (Bn (0, 3/4)) a

for all f ∈ C h,α (∂Ω) and for all a ∈ {1, . . . , b}. The operator Ea from {g ∈ C h,α (φa (Bn (0, 1))) : supp (g) ⊆ φa (Bn (0, 3/4))} to C h,α (Ω  ) defined by

(2.81)

104

2 Preliminaries

 Ea [g](y) ≡

g(y) if y ∈ φa (Bn (0, 3/4)) , 0 if y ∈ Ω  \ φa (Bn (0, 3/4)) ,

for all g ∈ C h,α (φa (Bn (0, 1))) is linear and continuous for all a ∈ {1, . . . , b}. (−1) Since the support of G[(θa f ) ◦ φ˜a ] ◦ φa is compact and contained in the set  h,α φa (Bn (0, 3/4)) ⊆ Ω for all f ∈ C (∂Ω) and for all a ∈ {1, . . . , b}, then the operator F defined by F [f ] ≡

b 

1 2 Ea G[(θa f ) ◦ φ˜a ] ◦ φ(−1) a

∀f ∈ C h,α (∂Ω)

a=1

satisfies the properties of the statement. To prove statement (iii), it suffices to take an open ball Ω  that contains Ω and to take F equal to the composition of the corresponding operator FΩ  of statement (ii) with the restriction operator from C h,α (Ω  )  h,α Ω .   to C

2.21 Tangential Derivatives Let n ∈ N \ {0, 1}. Let Ω be a bounded open subset of Rn of class C 1 . Let ν ≡ (ν1 , . . . , νn ) ≡ νΩ be the outward unit normal to ∂Ω. The tangential gradient ∇∂Ω f of f ∈ C 1 (∂Ω) is defined as on ∂Ω , (2.82) ∇∂Ω f ≡ ∇f˜ − (ν · ∇f˜)ν where f˜ is an extension of f of class C 1 in an open neighborhood of ∂Ω (see Theorem 2.85). The vector field ∇∂Ω f is independent of the specific choice of the extension f˜ of f . To prove it, it suffices to show that if f˜ vanishes on ∂Ω and p ∈ ∂Ω, then the vector a ≡ ∇f˜(p) − (ν(p) · ∇f˜(p))ν(p)   equals zero. To do so, we note that if p ∈ M and ψ ∈ C 1 Bn−1 (0, 1), Rn is a parametrization for the (n − 1)-dimensional manifold ∂Ω around p, with ψ(0) = p, then Lemma 2.41 implies that dψ(0) maps Rn−1 onto the (n − 1)-dimensional tangent space Tψ(0) (∂Ω) to ∂Ω at ψ(0) and accordingly ν(ψ(0))t Dψ(0) = 0. Since f˜ ◦ ψ = 0 on Bn−1 (0, 1), we have at Dψ(0) = (∇f˜(ψ(0)))t Dψ(0) − (ν(ψ(0)) · ∇f˜(ψ(0)))ν(ψ(0))t Dψ(0)   = ∇ f˜ ◦ ψ (0) − 0 = 0 .

2.21 Tangential Derivatives

105

In particular, a is orthogonal to the image of dψ(0), i.e., to the (n − 1)-dimensional space Tp (∂Ω). Now, by the definition of a we have a · ν(p) = 0, and thus a belongs to Tp (∂Ω). Hence, a = 0 and ∇∂Ω f is independent on the specific extension of f . If l, r ∈ {1, . . . , n}, then Mlr denotes the tangential derivative operator from C 1 (∂Ω) to C 0 (∂Ω) that takes f to Mlr [f ] ≡ νl

∂ f˜ ∂ f˜ − νr ∂xr ∂xl

on ∂Ω ,

(2.83)

where f˜ is any continuously differentiable extension of f to an open neighborhood of ∂Ω. The tangential derivative Mlr is related to the tangential gradient ∇∂Ω by the formulas n  ∂ f˜ − (ν · ∇f˜)νr = Mlr [f ]νl (2.84) (∇∂Ω f )r = ∂xr l=1

and Mlr [f ] = νl (∇∂Ω f )r − νr (∇∂Ω f )l . Since ∇∂Ω f does not depend on the specific choice of the extension f˜ of f , the last equality implies that also Mlr [f ] is independent of the specific choice of f˜. We also need the following consequence of the Divergence Theorem. Lemma 2.86. Let n ∈ N \ {0, 1}. Let Ω be a bounded open subset of Rn of class C 1 . If ϕ, ψ ∈ C 1 (∂Ω), then   Mlr [ϕ]ψ dσ = − ϕMlr [ψ] dσ ∂Ω

∂Ω

for all l, r ∈ {1, . . . , n}. Proof. By Theorem 2.85, there exist Φ, Ψ ∈ Cc1 (Rn ) such that ϕ = Φ|∂Ω and ψ = Ψ|∂Ω . By taking the convolution of Φ and Ψ with a family of mollifiers, we can show that there exist sequences {Φj }j∈N and {Ψj }j∈N in Cc∞ (Rn ) such that lim Φj = Φ

j→∞

and

lim Ψj = Ψ

j→∞

uniformly on the compact subsets of Rn , together with their first order partial derivatives (see Proposition A.33). By the Divergence Theorem, we have   Mlr [ϕ]ψ dσ + ϕMlr [ψ] dσ ∂Ω ∂Ω       ∂Φ ∂Φ ∂Ψ ∂Ψ = − νr − νr νl Ψ dσ + νl Φ dσ ∂xr ∂xl ∂xr ∂xl ∂Ω ∂Ω   ∂(ΦΨ ) ∂(ΦΨ ) ∂(Φj Ψj ) ∂(Φj Ψj ) νl − νr dσ = lim νl − νr dσ = j→∞ ∂x ∂x ∂x ∂xl r l r ∂Ω ∂Ω  ∂ ∂(Φj Ψj ) ∂ ∂(Φj Ψj ) = lim − dx = 0 j→∞ Ω ∂xl ∂xr ∂xr ∂xl

106

2 Preliminaries

 

and thus the proof is complete.

Next we introduce the following auxiliary lemma, whose proof is based on the definition of norm in a Schauder space. Lemma 2.87. Let n ∈ N \ {0, 1}. Let m ∈ N \ {0}, α ∈ [0, 1]. Let Ω be a bounded open connected subset of Rn of class C m,α . Then the following statements hold. (i) Mlr is a linear and continuous operator from C m,α (∂Ω) to C m−1,α (∂Ω) for all l, r ∈ {1, . . . , n}. (ii) A function f ∈ C 1 (∂Ω) belongs to C m,α (∂Ω) if and only if Mlr [f ] belongs to C m−1,α (∂Ω) for all l, r ∈ {1, . . . , n}. (iii) The norm  · C m,α (∂Ω) is equivalent to the norm on C m,α (∂Ω) defined by f C 0 (∂Ω) +

n 

Mlr [f ]C m−1,α (∂Ω)

∀f ∈ C m,α (∂Ω) .

(2.85)

l,r=1

Proof. (i) Since νΩ ∈ C m−1,α (∂Ω, Rn ) and the extension and restriction operators of Theorem 2.85 (i), (ii) are continuous, the continuity of the pointwise product in C m−1,α (∂Ω) of Theorem 2.83 ensures the continuity of Mlr (see the definition (2.83) of Mlr ). (ii) The necessity is an immediate consequence of statement (i). We now prove the sufficiency. Let f ∈ C 1 (∂Ω) be such that Mlr [f ] ∈ C m−1,α (∂Ω) for all l, r ∈ {1, . . . , n}. By the definition of C m,α (∂Ω),   it suffices to show that if ψ is a parametrization of class C m,α Bn−1 (0, 1), Rn for the (n − 1)-dimensional man-

ifold ∂Ω, then f ◦ ψ ∈ C m,α (Bn−1 (0, 1)). Since f ◦ ψ belongs to C 1 (Bn−1 (0, 1)), it suffices to show that the first order partial derivatives of f ◦ ψ belong to the space C m−1,α (Bn−1 (0, 1)). Since the vector ν ◦ ψ(ξ) is normal to the tangent space Tψ(ξ) (∂Ω) = dψ(ξ)[Rn−1 ] to ∂Ω at the point ψ(ξ), we have ν t ◦ ψ(ξ) Dψ(ξ) = 0 . By the extension Theorem 2.85 (ii), there exists f˜ ∈ Cc1 (Rn ) such that f˜|∂Ω = f , and thus we have ∇(f ◦ ψ)(ξ) = ∇(f˜ ◦ ψ)(ξ) = (∇f˜)t ◦ ψ(ξ)Dψ(ξ) (2.86) 1   2 t t = (∇f˜) ◦ ψ(ξ) − (∇f˜) ◦ ψ(ξ) · ν ◦ ψ(ξ) ν ◦ ψ(ξ) Dψ(ξ) = (∇∂Ω f )t ◦ ψ(ξ)Dψ(ξ) = ν t ◦ ψ(ξ) (Mlr [f ] ◦ ψ(ξ))l,r=1,...,n Dψ(ξ)

∀ξ ∈ Bn−1 (0, 1) ,

(cf. (2.82) and (2.84) for the definition of the tangential gradient ∇∂Ω f and for the formula in terms of the tangential derivatives). Since Mlr [f ] belongs to C m−1,α (∂Ω), ν belongs to C m−1,α (∂Ω, Rn ), and ψ is a parametrization of class   C m,α Bn−1 (0, 1), Rn

2.21 Tangential Derivatives

107

  of ∂Ω, then Mlr [f ]◦ψ belongs to the space C m−1,α Bn−1 (0, 1) and ν◦ψ belongs to C m−1,α (Bn−1 (0, 1), Rn ) for all l, r in {1, . . . , n}. Then the components of the function in the right-hand side of (2.86) are sums of products of functions of class C m−1,α (Bn−1 (0, 1)) and thus belong to C m−1,α (Bn−1 (0, 1)). Hence, the first order partial derivatives of f ◦ ψ belong to C m−1,α (Bn−1 (0, 1)) and we conclude that f ◦ ψ belongs to C m,α (Bn−1 (0, 1)). (iii) Let  · †C m,α (∂Ω) be the norm on C m,α (∂Ω) defined by (2.85). By the definition of the norm  · C m,α (∂Ω) and by statement (i), the identity map from (C m,α (∂Ω),  · C m,α (∂Ω) ) to (C m,α (∂Ω),  · †C m,α (∂Ω) ) is continuous. Thus it suffices to show that we can estimate the norm  · C m,α (∂Ω) in terms of a constant multiple of the norm  · †C m,α (∂Ω) . Let {ψa }a∈{1,...,b} , with ψa in   C m,α Bn−1 (0, 1), Rn , be parametrizations of ∂Ω such that b 

ψa (Bn−1 (0, 1)) = ∂Ω

a=1

and f C m,α (∂Ω) =

sup a∈{1,...,b}

f ◦ ψa C m,α (Bn−1 (0,1))

∀f ∈ C m,α (∂Ω) . (2.87)

  By the second inequality in (2.32) for the C m,α Bn−1 (0, 1) -norm and by the definition of the  · †C m,α (∂Ω) norm, we have f ◦ ψa C m,α (Bn−1 (0,1)) ≤ f ◦ ψa C 0 (Bn−1 (0,1)) +

(2.88) n 

∂ξj (f ◦ ψa )C m−1,α (Bn−1 (0,1)) .

j=1

Moreover, f ◦ ψa C 0 (Bn−1 (0,1)) ≤ f C 0 (∂Ω) ≤ f †C m,α (∂Ω) .

(2.89)

Thus it suffices to estimate the norms ∂ξj (f ◦ ψa )C m−1,α (Bn−1 (0,1)) in terms of f †C m,α (∂Ω) for all f ∈ C m,α (∂Ω), a ∈ {1, . . . , b}, and j ∈ {1, . . . , n − 1}. By formula (2.86) with ψ = ψa , we have ∂ξj (f ◦ ψa )C m−1,α (Bn−1 (0,1)) 0 n 0  0 ∂(ψa )r 0 0 0 ≤ . 0νl ◦ ψa Mlr [f ] ◦ ψa ∂ξj 0 m−1,α C (Bn−1 (0,1)) l,r=1

(2.90)

108

2 Preliminaries

  By the continuity of the pointwise product in C m−1,α Bn−1 (0, 1) , there exists c1 > 0 such that 0 n 0  0 0 0νl ◦ ψa Mlr [f ] ◦ ψa ∂(ψa )r 0 (2.91) 0 ∂ξj 0C m−1,α (Bn−1 (0,1)) l,r=1 0 0 n  0 ∂(ψa )r 0 0 0 ≤ c1 νl ◦ ψa Mlr [f ] ◦ ψa C m−1,α (Bn−1 (0,1)) 0 ∂ξj 0C m−1,α (Bn−1 (0,1)) l,r=1 0 0 n  0 ∂(ψa )r 0 0 ≤ c1 νl Mlr [f ]C m−1,α (∂Ω) 0 0 ∂ξj 0 m−1,α C (Bn−1 (0,1)) l,r=1 for all f ∈ C m,α (∂Ω), a ∈ {1, . . . , b}, and j ∈ {1, . . . , n − 1}. By the continuity of the pointwise product in C m−1,α (∂Ω), there exists c2 > 0 such that 0 0 0 ∂(ψa )r 0 0 νl Mlr [f ]C m−1,α (∂Ω) 0 (2.92) 0 ∂ξj 0 m−1,α C (Bn−1 (0,1)) l,r=1 0 0 n  0 ∂(ψa )r 0 0 0 ≤ c1 c2 νl C m−1,α (∂Ω) Mlr [f ]C m−1,α (∂Ω) 0 ∂ξj 0C m−1,α (Bn−1 (0,1))

c1

n 

l,r=1

for all f ∈ C m,α (∂Ω), a ∈ {1, . . . , b}, and j ∈ {1, . . . , n − 1}. Then, by combining inequalities (2.87)–(2.92), we conclude that the identity map from the space (C m,α (∂Ω),  · †C m,α (∂Ω) ) to the space (C m,α (∂Ω),  · C m,α (∂Ω) ) is continuous. Now the proof is complete.  

2.22 Schauder Spaces in Open Subsets of Rn , a Case of a Negative Exponent So far, we have defined and considered Schauder spaces only with a positive exponent m ≥ 0. We now consider a case with negative exponent. Namely, the case with m = −1, which we need later in the book. We note that Schauder spaces with negative exponents have been known for a long time and have been used in the analysis of elliptic and parabolic partial differential equations (cf. Triebel [267], Gilbarg and Trudinger [107], Lunardi and Vespri [191]). Definition 2.88. Let n ∈ N \ {0, 1}. Let α ∈]0, 1]. Let Ω be a bounded open subset of Rn . We denote by C −1,α (Ω) the subspace ⎧ ⎫ n ⎨ ⎬  ∂ f0 + fj : fj ∈ C 0,α (Ω) ∀j ∈ {0, . . . , n} , ⎩ ⎭ ∂xj j=1

2.22 Schauder Spaces in Open Subsets of Rn , a Case of a Negative Exponent

109

of the space of distributions D (Ω) in Ω. According to the above definition, the space C −1,α (Ω) is the image of the linear and continuous map Ξ : (C 0,α (Ω))n+1 → D (Ω) n ∂ that takes an (n + 1)-tuple (f0 , . . . , fn ) to f0 + j=1 ∂x fj . Let π denote the j canonical projection π : (C 0,α (Ω))n+1 → (C 0,α (Ω))n+1 /Ker Ξ of (C 0,α (Ω))n+1 onto the quotient space (C 0,α (Ω))n+1 /Ker Ξ. Let Ξ˜ be the unique linear injective map from (C 0,α (Ω))n+1 /Ker Ξ onto the image C −1,α (Ω) of Ξ such that Ξ = Ξ˜ ◦ π . Then, Ξ˜ is a linear bijection from (C 0,α (Ω))n+1 /Ker Ξ onto C −1,α (Ω). Since (C 0,α (Ω))n+1 is a Banach space and Ker Ξ is a closed subspace of 0,α (C (Ω))n+1 , we know that (C 0,α (Ω))n+1 /Ker Ξ is a Banach space (cf. Theorem 2.1). ˜ i.e., we set We endow C −1,α (Ω) with the norm induced by Ξ, f C −1,α (Ω) ≡ inf

 n

fj C 0,α (Ω) :

(2.93)

j=0

 n  ∂ 0,α fj , fj ∈ C (Ω) ∀j ∈ {0, . . . , n} . f = f0 + ∂xj j=1 By definition of the norm  · C −1,α (Ω) , the linear bijection Ξ˜ is an isometry of the space (C 0,α (Ω))n+1 /Ker Ξ onto the space (C −1,α (Ω),  · C −1,α (Ω) ). Since the quotient (C 0,α (Ω))n+1 /Ker Ξ is a Banach space, it follows that −1,α (Ω),  · C −1,α (Ω) ) is also a Banach space. (C Since Ξ is continuous from (C 0,α (Ω))n+1 to D (Ω), a fundamental property of the quotient topology implies that the map Ξ˜ is continuous from the quotient space (C 0,α (Ω))n+1 /Ker Ξ to D (Ω) (cf. Proposition A.5). Hence, (C −1,α (Ω),  · C −1,α (Ω) ) is continuously embedded into D (Ω). Also, the definition of the norm ·C −1,α (Ω) implies that C 0,α (Ω) is continuously embed∂ is continuous from C 0,α (Ω) ded into C −1,α (Ω) and that the partial derivation ∂x j to C −1,α (Ω) for all j ∈ {1, . . . , n}. The elements of C −1,α (Ω) are not integrable functions, but distributions in Ω. We now define a linear functional IΩ on C −1,α (Ω) which extends the integration in Ω to all elements of C −1,α (Ω). We do so by means of the following proposition. Proposition 2.89. Let n ∈ N \ {0, 1}. Let α ∈]0, 1]. Let Ω be a bounded open Lipschitz subset of Rn . Then the following statements hold.

110

2 Preliminaries

(i) If (f0 , . . . , fn ) ∈ C 0,α (Ω)n+1 and f0 +

n  ∂ fj = 0 ∂x j j=1

in the sense of distributions in Ω, then 



n 

f0 dx + Ω

(νΩ )j fj dσ = 0 .

∂Ω j=1

(ii) There exists one and only one linear and continuous operator IΩ from C −1,α (Ω) to R such that    n f0 dx + (νΩ )j fj dσ (2.94) IΩ [f ] = Ω

for all f = f0 +

n

∂ j=1 ∂xj fj

∂Ω j=1

∈ C −1,α (Ω). Moreover, 

IΩ [f ] =

f dx

∀f ∈ C 0,α (Ω) .

Ω

Proof. (i) Since all components of the vector valued function (f1 , . . . , fn ) and its divergence −f0 belong to C 0,α (Ω), which is continuously embedded into L2 (Ω), there exists a sequence {(fl1 , . . . , fln )}l∈N in C ∞ (Ω, Rn ) such that in L2 (Ω, Rn ) ,

lim (fl1 , . . . , fln ) = (f1 , . . . , fn )

l→∞

n n   ∂ ∂ flj = fj l→∞ ∂xj ∂xj j=1 j=1    n (νΩ )j flj dσ = lim

lim

l→∞

∂Ω j=1

in L2 (Ω) , n 

(νΩ )j fj dσ

∂Ω j=1

(cf., e.g., Tartar [264, p. 101]). By the Divergence Theorem on Lipschitz domains, we have     n n ∂ flj dx = (νΩ )j flj dσ ∀l ∈ N Ω j=1 ∂xj ∂Ω j=1 (cf., e.g., Ziemer [277, Theorem 5.8.2, Remark 5.8.3]). Then, by taking the limit as l tends to infinity, we obtain  −

f0 dx = Ω

    n n ∂ fj dx = (νΩ )j fj dσ Ω j=1 ∂xj ∂Ω j=1

and the proof of statement (i) is complete.

2.22 Schauder Spaces in Open Subsets of Rn , a Case of a Negative Exponent

111

(ii) Let L be the linear operator from C 0,α (Ω)n+1 to R that takes (f0 , . . . , fn ) to the function in the right-hand side of (2.94). By (i), we have Ker Ξ ⊆ Ker L. Since the operator Ξ from C 0,α (Ω)n+1 to C −1,α (Ω) is surjective, the Homomorphism Theorem A.1 for linear maps between vector spaces implies the existence of L = IΩ ◦ Ξ, i.e., such that a unique linear map IΩ from C −1,α (Ω) to R such that n ∂ fj , we have (2.94) holds true. Then we note that if f = f0 + j=1 ∂x j |IΩ [f ]| ≤ mn (Ω)f0 C 0,α (Ω) +

n 

fj C 0,α (Ω) mn−1 (∂Ω) .

j=1

By taking the infimum on all (f0 , . . . , fn ) ∈ C 0,α (Ω)n+1 such that f = f0 +

n  ∂ fj , ∂xj j=1

we obtain |IΩ [f ]| ≤ max{mn (Ω), mn−1 (∂Ω)}f C −1,α (Ω) . Hence, the linear map IΩ is continuous. The last equality of the statement follows   by taking f0 = f , f1 = · · · = fn = 0 in (2.94).

Chapter 3

Preliminaries on Harmonic Functions

Abstract This chapter is devoted to the properties of harmonic functions that we exploit in this book. For most of the proofs we refer to classic monographs and in particular to Evans (Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010), Folland (Introduction to partial differential equations. Princeton University Press, Princeton, NJ, second edition, 1995), and Gilbarg and Trudinger (Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 1983). We present the Maximum Principle, the Liouville Theorem, the interior inequalities for the derivatives of a harmonic function, Weyl’s Lemma, the Lemma of Hopf-Ole˘ınik, some properties of harmonic functions around isolated singularities, the Kelvin transform, and some properties of harmonic functions at infinity. Unless otherwise specified, we understand that throughout this chapter n is a natural number bigger than or equal to two. Namely, n ∈ N, n ≥ 2.

3.1 Basic Properties of Harmonic Functions Definition 3.1. Let Ω be an open subset of Rn . We say that a function u from Ω to R is harmonic in Ω provided that u ∈ C 2 (Ω) and that Δu = 0 in Ω, where Δu ≡

n  ∂2u j=1

∂x2j

.

© Springer Nature Switzerland AG 2021 M. Dalla Riva et al., Singularly Perturbed Boundary Value Problems, https://doi.org/10.1007/978-3-030-76259-9 3

113

114

3 Preliminaries on Harmonic Functions

Example 3.2. (j) If n ≥ 3 and x0 ∈ Rn , then the function |x − x0 |2−n is harmonic in Rn \ {x0 }. (jj) If n = 2 and x0 ∈ R2 , then the function log |x − x0 | is harmonic in R2 \ {x0 }. As we shall see in Weyl’s Lemma 3.14, we can replace the assumption that u ∈ C 2 (Ω) in the definition of harmonic function with the assumption that u is locally integrable in Ω, provided that we ask that equality Δu = 0 is satisfied in the sense of distributions. Definition 3.3. Let Ω be an open subset of Rn . Let u be a function from Ω to R. We say that u (i) satisfies the strong Maximum Principle in Ω if either u(x) < sup u Ω

∀x ∈ Ω ,

or u is constantly equal to supΩ u, i.e., u(x) = sup u Ω

∀x ∈ Ω ,

(ii) satisfies the strong Minimum Principle in Ω if either u(x) > inf u

∀x ∈ Ω ,

u(x) = inf u

∀x ∈ Ω .

Ω

or Ω

As shown by the following statement, harmonic functions satisfy the strong Maximum Principles in an open connected set. For a proof, we refer for example to Folland [102, (2.13), p. 72] or Gilbarg and Trudinger [107, Cor. 3.2, p. 33]). Theorem 3.4. Let Ω be an open connected subset of Rn . If u is a real valued harmonic function in Ω, then u satisfies both the strong Maximum Principle and the strong Minimum Principle in Ω. Then we have the following corollary (cf., e.g., Folland [102, (2.14), p. 72]). Corollary 3.5. Let Ω be a bounded open connected subset of Rn . If u ∈ C 0 (Ω) is a real valued harmonic function in Ω, then max u = max u , Ω

∂Ω

min u = min u , Ω

∂Ω

max |u| = max |u| . Ω

∂Ω

In particular, Corollary 3.5 says that a harmonic function u attains its maximum and minimum on the boundary. Then we can immediately derive the following uniqueness theorem (cf., e.g., Folland [102, (2.15), p. 72]). Theorem 3.6. Let Ω be an open bounded connected subset of Rn . Let u, v ∈ C 0 (Ω). If both u, and v are real valued harmonic functions in Ω and if u = v on ∂Ω, then u = v on Ω.

3.1 Basic Properties of Harmonic Functions

115

Example 3.7. As the following examples show, the previous corollary no longer holds if we omit the assumption that Ω is bounded. (i) Let n ≥ 3. Then the functions 1 and |x|2−n are both harmonic in Rn \ Bn (0, 1) continuous in Rn \ Bn (0, 1) and coincide on ∂Bn (0, 1). (ii) Let n = 2. Then the functions 0 and log |x| are both harmonic R2 \ B2 (0, 1) continuous in R2 \ B2 (0, 1) and coincide on ∂B2 (0, 1). Then we have the following theorem, which says that harmonic functions are infinitely many times differentiable. For a proof we refer for example to Folland [102, (2.11), p. 71]. Theorem 3.8 (of K¨obe). Let Ω be an open subset of Rn . If a real valued function u is harmonic in Ω, then u is of class C ∞ in Ω. We also mention that a uniform limit on compact sets of a sequence of harmonic functions is harmonic. Indeed, the following theorem holds (cf., e.g., Folland [102, (2.12), p. 71]). Theorem 3.9. Let Ω be an open subset of Rn . If {uj }j∈N is a sequence of harmonic functions in Ω and if the sequence {uj }j∈N converges to a function u uniformly on the compact subsets of Ω, then u is harmonic in Ω. We also need to know that an entire bounded harmonic function is constant. Indeed, the following holds (cf., e.g., Folland [102, (2.16), p. 72]). Theorem 3.10 (of Liouville). If u is a real valued harmonic function in Rn and if u is bounded, then u is constant. By the K¨obe Theorem 3.8, a harmonic function is of class C ∞ . However, a lot more holds. Indeed, the following statement holds true (cf., e.g., Gilbarg and Trudinger [107, Theorem 2.10, p. 23]). Theorem 3.11 (of the Interior Inequalities). Let Ω be an open subset of Rn . Let u be a real valued harmonic function in Ω. If Bn (x, r) ⊆ Ω, then  |Dα u(x)| ≤

n|α| r

|α| sup |u|

∀α ∈ Nn \ {0} .

∂Bn (x,r)

Then we have the following corollary. Corollary 3.12. Let Ω be an open subset of Rn with Ω = Rn . Let Ω  be an open subset of Ω such that Ω  is contained in Ω and is compact. Let d be the distance between Ω  and ∂Ω. If u is a harmonic function in Ω, then  sup |D u| ≤ α

Ω

n|α| d

|α| sup |u| Ω

∀α ∈ Nn \ {0} .

Here we understand that the right-hand side equals +∞ if supΩ |u| = +∞.

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3 Preliminaries on Harmonic Functions

By Stirling’s inequality, we have |α||α| ≤ e|α| |α|!

∀α ∈ Nn \ {0} .

Thus, the characterization Theorem 2.17 of real analyic functions and Corollary 3.12 imply the validity of the following theorem. Theorem 3.13. Let Ω be an open subset of Rn . If u is a real valued harmonic function in Ω, then u is real analytic. We also need the following well known result of Weyl [275]. Lemma 3.14 (of Weyl). Let Ω be an open subset of Rn . If u is a locally integrable function from Ω to R and if  uΔϕ dx = 0 ∀ϕ ∈ D(Ω) , Ω

i.e., if Δu = 0 in the sense of distributions in Ω, then u equals almost everywhere a harmonic function in Ω. Then we have the following refined form of the Hopf-Ole˘ınik Lemma. The HopfOle˘ınik Lemma in the form we present here is due to Kamynin and Himˇcenko [132], [133, p. 84 of the English translation] (see also Lieberman [188, Theorem 4.1] for a simpler proof; for a historical account and for further developments see Alvarado et al. [5, p. 321]). Lemma 3.15 (of Hopf-Ole˘ınik). Let α ∈]0, 1]. Let Ω be an open subset of Rn of class C 1,α . Let x0 ∈ ∂Ω. Let u ∈ C 0 (Ω) be a real valued harmonic function in Ω. Let u(x) < u(x0 ) for all x ∈ Ω. If the limit ∂u u(x0 + tνΩ (x0 )) − u(x0 ) (x0 ) ≡ lim ∂νΩ t t→0− exists in R, then

∂u ∂νΩ (x0 )

> 0.

3.2 A Fundamental Solution for the Laplace Operator We say that a function E ∈ L1loc (Rn ) is a fundamental solution for Δ provided that  EΔϕ dx = ϕ(0) ∀ϕ ∈ Cc∞ (Rn ) . Rn

In terms of the theory of distributions, it means that ΔE = δ0 ,

3.3 Isolated Singularities of Harmonic Functions

117

where δ0 is the Dirac distribution with mass at 0. We now introduce a specific fundamental solution of Δ, which is commonly addressed to as ‘the fundamental solution of the Laplace operator’. Let Υn be the function from ]0, +∞[ to R defined by  Υn (r) ≡

1 s2

log r

1 2−n (2−n)sn r

∀r ∈]0, +∞[, ∀r ∈]0, +∞[,

if n = 2 , if n > 2 ,

(3.1)

where sn denotes the (n − 1)-dimensional measure of ∂Bn (0, 1). We denote by Sn the function from Rn \ {0} to R defined by Sn (x) ≡ Υn (|x|)

∀x ∈ Rn \ {0} .

(3.2)

It is well known that Sn is a fundamental solution of Δ. Actually, one can prove that the set of all radial fundamental solutions of Δ is delivered by the set {Sn + c : c ∈ R} . Clearly, ∂ 1 xj Sn (x) = ∀j ∈ {1, . . . , n} , ∂xj sn |x|n 1 1 x 1 , |∇Sn (x)| = ∇Sn (x) = sn |x|n sn |x|n−1

(3.3)

for all x ∈ Rn \ {0}. We note that contrary to Sn , the gradient ∇Sn has the same form for all n ≥ 2. A simple computation shows that     2   1  ∂ Sn 1 1 δij nxi xj     ∀x ∈ Rn \ {0} , (3.4)  ∂xi ∂xj (x) =  sn |x|n − |x|n+2  ≤ ωn |x|n for all i, j ∈ {1, . . . , n}, where ωn denotes the n-dimensional measure of Bn (0, 1).

3.3 Isolated Singularities of Harmonic Functions Definition 3.16. Let Ω be an open subset of Rn . Let x0 ∈ Ω. Let u be a harmonic function from Ω\{x0 } to R. We say that x0 is a removable singularity for u provided that there exists a harmonic function v in Ω such that u(x) = v(x)

∀x ∈ Ω \ {x0 } .

We now introduce a criterium to establish whether a singularity is removable for a harmonic function. For a proof, we refer for example to Folland [102, (2.69), p. 111].

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3 Preliminaries on Harmonic Functions

Theorem 3.17. Let Ω be an open subset of Rn . Let x0 ∈ Ω. Let u be a harmonic function from Ω \ {x0 } to R. Then x0 is removable for u if and only if u(x) = o(Sn (x − x0 ))

as x → x0

i.e., if and only if lim

x→x0

u(x) = 0. Sn (x − x0 )

Remark 3.18. By exploiting the previous theorem, one can easily show that u has a removable singularity at x0 if and only if there exists at least one punctured neighborhood of x0 on which u is bounded. (A punctured neighborhood of a point is by definition a neighborhood of the point minus the point itself.)

3.4 Behavior at Infinity of Harmonic Functions In order to analyze the behavior at infinity of a harmonic function, we need to introduce the inversion with respect to a sphere and the Kelvin transform. Definition 3.19. Let r ∈]0, +∞[. We define the inversion map with respect to the sphere ∂Bn (0, r) to be the map ( · )∗ from Rn \ {0} to itself defined by y∗ ≡ y

r2 |y|2

∀y ∈ Rn \ {0} .

As one can easily verify, the inversion with respect to the sphere ∂Bn (0, r) is a diffeomorphism of class C ∞ from Rn \ {0} onto Rn \ {0} of involutory character, i.e., y ∗∗ = y for all y ∈ Rn \ {0}. We are now ready to introduce the Kelvin transform. Definition 3.20. Let A be an open subset of Rn \ {0}. Let (·)∗ be the inversion map with respect to the unit sphere ∂Bn (0, 1). Let   y : y ∈ A . A∗ ≡ |y|2 Let u be a function from A to R. Then we define the Kelvin transform of u to be the function u ˜ from A∗ to R delivered by   y 2−n u ˜(y) ≡ |y| u ∀y ∈ A∗ . |y|2 By the chain rule, one can verify that a function u from A to R is harmonic if and only if its Kelvin transform u ˜ is harmonic in A∗ . We also note that there exists ρ ∈]0, +∞[ such that Rn \ Bn (0, ρ) ⊆ A if and only if Bn (0, ρ−1 ) \ {0} ⊆ A∗ . In particular, A ∪ {∞} is an open neighborhood

3.4 Behavior at Infinity of Harmonic Functions

119

of ∞ if and only if A∗ ∪ {0} is an open neighborhood of 0. We are now ready to introduce the following. Definition 3.21. Let Ω be an open subset of Rn such that there exists a compact subset K of Rn with Ω ⊇ Rn \ K. We say that a function u from Ω to R is harmonic at infinity provided that there exists ρ > 0 such that Ω ⊇ Rn \ Bn (0, ρ), u is harmonic in Rn \ Bn (0, ρ), and the Kelvin transform of u|Rn \Bn (0,ρ) has a removable singularity at 0. Then we have the following characterization of functions which are harmonic at infinity. For a proof we refer for example to Folland [102, (2.74), p. 114]. Theorem 3.22. Let Ω be an open subset of Rn such that there exists a compact subset K of Rn with Ω ⊇ Rn \ K. Let u be a harmonic function from Ω to R. Then the following statements are equivalent. (i) u is harmonic at infinity. (ii) u(x) = O(|x|2−n ) as x tends to ∞, i.e., u(x)|x|n−2 is bounded in at least a punctured neighborhood of infinity. (iii) If n ≥ 3, then limx→∞ u(x) = 0. If n = 2, then u(x) = o(log |x|) as x tends u(x) to ∞, i.e., limx→∞ log |x| = 0. Remark 3.23. A constant function c ∈ R \ {0} in Rn is harmonic at infinity if and only if n = 2. Corollary 3.24. Let Ω be an open subset of Rn such that there exists ρ ∈]0, +∞[ with Ω ⊇ Rn \ Bn (0, ρ). If u is harmonic in Ω and is harmonic at infinity, then the following statements hold. (i) If n ≥ 3 then limx→∞ u(x) = 0.  (ii) If n = 2 and r > ρ, then limx→∞ u(x) = −∂Bn (0,r) u dσ. Proof. Statement (i) is an immediate consequence of the previous statement. We now consider statement (ii). We know that the Kelvin transform u ˜(y) ≡ u( |y|y 2 ) is harmonic in y ∈ B2 (0, ρ−1 ) \ {0} and has a removable singularity at 0. Let v denote an extension of u ˜ in B2 (0, ρ−1 ). Then by the Spherical Mean Value Property for harmonic functions, we have    y u dσy v(0) = − |y|2 ∂B2 (0,1/r) (cf., e.g., Folland [102, Theorem 2.8, p. 69]). Then we can change the variable in the integral by setting x = |y|y 2 = r2 y and we obtain v(0) =

1 r−1 2π



u(x)r−2 dσx = ∂B2 (0,r)

1 2πr

 u dσ . ∂B2 (0,r)

Since v(0) = limy→0 u( |y|y 2 ) = limx→∞ u(x), the proof is complete.

 

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3 Preliminaries on Harmonic Functions

Let Ω be an open subset of Rn . Let u ∈ C 1 (Ω). Then we set ∂r u(x) = ∇u(x) · Clearly, ∂r u(x) = lim

x |x|

∀x ∈ Ω \ {0} .

x u(x + h |x| ) − u(x)

h

h→0

x for all x ∈ and thus ∂r u(x) coincides with the derivative of u in the direction |x| Ω \ {0} (the radial derivative of u). Then we have the following result on the decay of the radial derivative of a harmonic function at infinity. For a proof, we refer for example to Folland [102, (2.75), p. 114].

Proposition 3.25. Let Ω be an open subset of Rn such that there exists ρ > 0 with Ω ⊇ Rn \ Bn (0, ρ). Let u ∈ C 2 (Ω) be harmonic at infinity. (i) If n ≥ 3, then ∂r u(x) = O(|x|1−n ) as x tends to ∞, i.e., ∂r u(x)|x|n−1 is bounded in at least a punctured neighborhood of infinity. (ii) If n = 2, then ∂r u(x) = O(|x|−2 ) as x tends to ∞, i.e., ∂r u(x)|x|2 is bounded in at least a punctured neighborhood of infinity. The above properties at infinity of harmonic functions imply the validity of the following technical proposition that we exploit later in the book. Proposition 3.26. Let Ω be an open subset of Rn such that there exists ρ > 0 with Ω ⊇ Rn \ Bn (0, ρ). Let u, v be real valued, harmonic in Ω and harmonic at infinity. Then we have  ∂ u v dσ = 0 . (3.5) lim s→+∞ ∂B (0,s) ∂νB (0,s) n n Proof. Since

∂v ∂νBn (0,s) (y)

=

y |y|

 u ∂Bn (0,s)

· ∇v(y) = ∂r v(y) for all y ∈ ∂Bn (0, s), we have ∂v

∂νBn (0,s)

 dσ =

u∂r v dσ , ∂Bn (0,s)

for all s ∈]ρ, +∞[. Since u and v are harmonic at infinity, Theorem 3.22 and Proposition 3.25 ensure the existence of C > 0 and ρ1 > ρ such that |u(x)| ≤ C|x|2−n

if x ∈ Rn \ Bn (0, ρ1 )

if n ≥ 2 ,

and such that |∂r v(x)| ≤ C|x|1−n if x ∈ Rn \ Bn (0, ρ1 ) |∂r v(x)| ≤ C|x|−2 if x ∈ Rn \ Bn (0, ρ1 )

and n ≥ 3 , and n = 2 .

Then we have       u∂r v dσ  ≤ sn sn−1 C 2 s2−n s1−n = sn C 2 s2−n   ∂Bn (0,s) 

∀s ∈]ρ1 , +∞[

3.4 Behavior at Infinity of Harmonic Functions

for n ≥ 3 and       u∂r v dσ  ≤ 2πsC 2 s−2 = 2πC 2 s−1   ∂Bn (0,s) 

121

∀s ∈]ρ1 , +∞[

for n = 2. Thus the validity of the limiting relation (3.5) follows.

 

Chapter 4

Green Identities and Layer Potentials

Abstract This chapter is devoted to the Green Identities and to the layer potentials corresponding to the fundamental solution of the Laplace operator, cf. (3.2). We first consider the Green Identities for bounded domains and then the case of exterior domains for functions which are harmonic at infinity. Next, we introduce the single and double layer potentials and the corresponding mapping properties, especially in spaces of H¨older continuous functions and in Schauder spaces. Although such properties can be found in the classic monographs of G¨unter (Potential theory and its applications to basic problems of mathematical physics. Translated from the Russian by John R. Schulenberger. Frederick Ungar Publishing, New York, 1967) and Kupradze et al. (Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, volume 25 of North-Holland Series in Applied Mathematics and Mechanics. North-Holland Publishing, Amsterdam-New York, Russian edition, 1979. Edited by V. D. Kupradze.), here we prove the corresponding statements with optimal H¨older exponents. To do so, we exploit a result of Miranda (Atti Accad Naz Lincei Mem Cl Sci Fis Mat Natur Sez I (8), 7:303–336, 1965) on the H¨older continuity of singular integrals of convolution type, for which we provide a complete proof. Unless otherwise specified, we understand that throughout this chapter n is a natural number bigger than or equal to two. Namely, n ∈ N, n ≥ 2.

4.1 Green Identities for Bounded Domains The Green Identities play an important role in the representation of the solutions of boundary value problems. Before introducing the Green Identities, we show a variant of the Divergence Theorem. The difference between the statement here below and the standard formulation which can be found in most textbooks is that, instead © Springer Nature Switzerland AG 2021 M. Dalla Riva et al., Singularly Perturbed Boundary Value Problems, https://doi.org/10.1007/978-3-030-76259-9 4

123

124

4 Green Identities and Layer Potentials

of having a vector field that is C 1 on the closure of a certain open set, we deal with a vector field which is C 1 in the interior and only continuous on the closure. The proof is performed by adapting an argument of Dautray and Lions [86, pp. 226–229] and is here included for the sake of completeness. For a more general statement on open sets with Ahlfors regular boundary, we refer to Hofmann et al. [125, Prop. 2.13]. Theorem 4.1 (of the Divergence). Let Ω be a bounded open Lipschitz subset of Rn . Let v ∈ C 0 (Ω, Rn ) be a continuous vector field on Ω such that v|Ω ∈ C 1 (Ω, Rn ) and div v ∈ L1 (Ω). Then we have   div v dx = v · νΩ dσ . (4.1) Ω

∂Ω

Proof. Since Ω is compact, we can exploit Lemma A.35 on the partition of unity and write v as a finite sum of functions which have compact support in Ω and of functions which have support in a coordinate cylinder. Now, the terms which have compact support in Ω can be extended to functions of class C 1 in Rn and thus satisfy equality (4.1) by the classical Divergence Theorem in bounded open Lipschitz sets (cf., e.g., Ziemer [277, Theorem 5.8.2, Remark. 5.8.3]). Thus it suffices to prove equality (4.1) in the case in which the support of v is contained in a coordinate cylinder. Namely, we can assume that there exist p ∈ ∂Ω, R ∈ On (R), and r, δ ∈ ]0, +∞[ such that supp v ⊆ C(p, R, r, δ) ∩ Ω , where C(p, R, r, δ) is a coordinate cylinder for Ω around p. Then we have   div v dx = div v dx Ω

and



C(p,R,r,δ)∩Ω

 v · νΩ dσ = ∂Ω

v · νΩ dσ , C(p,R,r,δ)∩∂Ω

and thus it suffices to prove that   div v dx = C(p,R,r,δ)∩Ω

v · νΩ dσ .

(4.2)

C(p,R,r,δ)∩∂Ω

Without loss of generality we can assume that p is the point at the origin and that R is the identity matrix. Then we denote by γ ∈ C 0,1 (Bn−1 (0, r), ] − δ, δ[) the function that represents ∂Ω in C(p, R, r, δ). We have C(p, R, r, δ) ∩ Ω = {(η, y) ∈ Bn−1 (0, r)×] − δ, δ[ : y < γ(η)} . and, for all  ∈ [0, δ/2[, we set A = {(η, y) ∈ Bn−1 (0, r)×] − δ, δ[ : y < γ(η) − } .

4.1 Green Identities for Bounded Domains

125

We can assume that δ and r have been chosen small enough so that A ⊆ Ω for all  ∈]0, δ/2[. Thus v is of class C 1 on the neighborhood Ω of A . Moreover, the support of the restriction v|A is contained in A ∪ ∂  A , where ∂  A is the portion of the boundary of A where y = γ(η) −  and η ∈ Bn−1 (0, r). Namely, ∂  A = {(η, γ(η) − ) : η ∈ Bn−1 (0, r)} . By the Divergence Theorem on bounded open Lipschitz sets for C 1 functions on a neighborhood of the Lipschitz domain A it follows that   div v dx = v · νA dσ ∀ ∈]0, δ/2[ , (4.3) ∂  A

A

(cf., e.g., Ziemer [277, Theorem. 5.8.2, Remark. 5.8.3]). We now take the limit as  → 0+ of the integrals in the left and right-hand sides of (4.3). Since A ⊆ A0 = C(p, R, r, δ) ∩ Ω

∀ ∈ [0, δ/2[

and div v ∈ L1 (Ω), the Dominated Convergence Theorem implies that   lim div v dx = div v dx . →0+

A

(4.4)

C(p,R,r,δ)∩Ω

Moreover, we have  νA (η, γ(η) − ) =

−∇γ(η) 1

 #

1 1 + |∇γ(η)|2

= νΩ (η, γ(η))

for almost all η ∈ Bn−1 (0, r). Then we compute     −∇γ(η) v · νA dσ = v(η, γ(η) − ) · dη . 1 ∂  A Bn−1 (0,r) Since v is (uniformly) continuous on Ω, it follows that     −∇γ(η) lim v · νA dσ = lim v(η, γ(η) − ) · dη 1 →0+ ∂  A →0+ Bn−1 (0,r)     −∇γ(η) = v(η, γ(η)) · v · νΩ dσ . dη = 1 Bn−1 (0,r) C(p,R,r,δ)∩∂Ω (4.5) Now the validity of (4.2) follows by (4.3), (4.4), and (4.5) and the theorem is proven.  

126

4 Green Identities and Layer Potentials

As a consequence of Theorem 4.1 we deduce the validity of the following. Theorem 4.2 (First Green Identity). Let Ω be a bounded open Lipschitz subset of Rn . If u, v ∈ C 1 (Ω), v ∈ C 2 (Ω), and uΔv ∈ L1 (Ω), then  u ∂Ω

∂v dσ = ∂νΩ

   n ∂u ∂v dx + uΔv dx . Ω j=1 ∂xj ∂xj Ω

The above first Green Identity holds in particular if u, v ∈ C 2 (Ω). As an immediate corollary, we have the following. Theorem 4.3 (Second Green Identity). Let Ω be a bounded open Lipschitz subset of Rn . If u, v ∈ C 1 (Ω) ∩ C 2 (Ω) and vΔu, uΔv ∈ L1 (Ω), then the following identity holds   ∂u ∂v (Δu) v − u (Δv) dx = v−u dσ . (4.6) ∂ν ∂ν Ω Ω Ω ∂Ω The above second Green Identity holds in particular if u, v ∈ C 2 (Ω). Remark 4.4. We note that the first Green Identity of Theorem 4.2 and the second Green Identity of Theorem 4.3 hold also in case u and v are complex-valued. Next we prove the following third Green Identity, which enables us to write a (sufficiently regular) function on Ω in terms of an integral operator applied to Δu ∂u on ∂Ω. in Ω and of integral operators applied to u|∂Ω and ∂ν Ω Theorem 4.5 (Third Green Identity). Let Ω be a bounded open Lipschitz subset of Rn . If u ∈ C 1 (Ω)∩C 2 (Ω) and Δu is integrable in Ω, then the following formula holds.  u(x) = Δu(y)Sn (x − y) dy (4.7)  Ω ∂ ∂u + u(y) (Sn (x − y)) − (y)Sn (x − y) dσy ∀x ∈ Ω , ∂ν ∂ν Ωy Ω ∂Ω where

∂ (Sn (x − y)) ≡ −νΩ (y) · ∇Sn (x − y) ∂νΩy

(4.8)

for all x ∈ Rn and y ∈ ∂Ω such that y = x and νΩ (y) exists. Proof. Let 0 > 0 be such that Bn (x, 0 ) ⊆ Ω. We note that if  ∈]0, 0 ], then Ω ≡ Ω \ Bn (x, ) is a bounded open Lipschitz subset of Rn . Since x ∈ / Ω , we have Sn (x − ·) ∈ C ∞ (Ω ). Moreover, u ∈ C 1 (Ω ). Since Sn (x − ·) is continuous and bounded in Ω and Δu is integrable in Ω, then Δu(y)Sn (x − y) is integrable in y ∈ Ω . Since Sn (x − y) is harmonic in y ∈ Ω , then u(y)Δy Sn (x − y) = 0 for all y ∈ Ω . Hence, we can apply the second Green

4.1 Green Identities for Bounded Domains

127

Identity to the functions u(y) and v(y) = Sn (x − y). Next we denote by ν the outward unit normal to ∂Ω . Then we have ∂Ω = ∂Ω ∪ ∂Bn (x, ) and  νΩ (y) if y ∈ ∂Ω and if νΩ (y) exists, ν (y) ≡ x−y |x−y| if y ∈ ∂Bn (x, ) . Since Δv = 0 in Ω , the second Green Identity implies that   ∂ ∂u −Sn (x − y)Δu(y) dy = u(y) (Sn (x − y)) − (y)Sn (x − y) dσy ∂νy ∂ν Ω ∂Ω and thus

 −

u(y) ∂Bn (x,)



∂ ∂u (Sn (x − y)) − (y)Sn (x − y) dσy ∂νy ∂ν

Sn (x − y)Δu(y) dy

= Ω



+

u(y) ∂Ω

∂ ∂u (Sn (x − y)) − (y)Sn (x − y) dσy . ∂νΩy ∂νΩ

Thus it suffices to show that (i)



 Sn (x − y)Δu(y) dy =

lim

→0

Ω

Sn (x − y)Δu(y) dy , Ω

(ii)  u(y)

lim

→0

∂Bn (x,)

∂ ∂u Sn (x − y) − (y)Sn (x − y) dσy = −u(x) . ∂νy ∂ν

We first consider (i). We note that   Sn (x − y)Δu(y) dy = χΩ (y)Sn (x − y)Δu(y) dy , Ω

Ω

and that |χΩ (y)Sn (x − y)Δu(y)| ≤ |χΩ (y)Sn (x − y)Δu(y)| , lim χΩ (y)Sn (x − y)Δu(y) = χΩ (y)Sn (x − y)Δu(y) →0

for all y ∈ Ω \ {x}. Since Δu ∈ C 0 (Bn (x, 0 )) and Sn (x − ·) ∈ L1 (Bn (x, 0 )), the function Sn (x − y)Δu(y) is integrable in y ∈ Bn (x, 0 ). On the other hand, we have already observed that the same function is integrable in Ω \ Bn (x, 0 ), and accordingly χΩ (y)Sn (x − y)Δu(y) ∈ L1 (Ω). Then the Dominated Convergence Theorem implies that

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4 Green Identities and Layer Potentials



 χΩ (y)Sn (x − y)Δu(y) dy =

lim

→0

Ω

χΩ (y)Sn (x − y)Δu(y) dy . Ω

Hence, (i) holds true. Next we turn to the proof of (ii). It clearly suffices to prove that  (j) lim→0 ∂Bn (x,) u(y) ∂ν∂ (Sn (x − y)) dσy = −u(x), y  ∂u (y)S (jj) lim→0 ∂Bn (x,) ∂ν n (x − y) dσy = 0. We first consider (j). Clearly,  ∂ u(y) (Sn (x − y)) dσy ∂νy ∂Bn (x,)  x − y dσy x−y · =− u(y) |x − y| |x − y|n sn ∂Bn (x,)   1 = − n−1 u(y)dσy = −− u(y)dσy  sn ∂Bn (x,) ∂Bn (x,) Since u is continuous at x, we have  lim −

→0 ∂B (x,) n

∀ ∈]0, 0 [ .

u(y)dσy = u(x)

and thus (j) holds. We now prove (jj). Clearly,     ∂u   (y)Sn (x − y) dσy  ≤ n−1 sn |Υn ()| sup |∇u|   ∂Bn (x,) ∂ν  Bn (x,0 )

∀ ∈]0, 0 [

(see (3.1) for the definition of Υn ).Then, by taking the limit as  tends to zero, we conclude that (jj) holds true and thus the proof is complete.   The above third Green Identity holds in particular if u ∈ C 2 (Ω). We also have the following corollary. Corollary 4.6. Let Ω be a bounded open Lipschitz subset of Rn . If u ∈ C 1 (Ω) is harmonic in Ω, then   ∂ ∂u u(x) if x ∈ Ω , u(y) (Sn (x − y)) − (y)Sn (x − y) dσy = 0 if x ∈ Rn \ Ω . ∂ν ∂ν Ωy Ω ∂Ω In particular,  ∂Ω

∂ (Sn (x − y)) dσy = ∂νΩy



1 if x ∈ Ω , 0 if x ∈ Rn \ Ω

(see also (4.8)). Proof. If x ∈ Ω, the statement is an immediate consequence of the third Green Identity. If x ∈ Rn \ Ω, then both the functions u(y) and v(y) ≡ Sn (x − y) are of

4.2 Green Identities for Harmonic Functions on Exterior Domains

129

class C 1 (Ω) ∩ C 2 (Ω) and harmonic in Ω. Then the statement follows by the second Green Identity. The second part of the statement follows by setting u = 1.  

4.2 Green Identities for Harmonic Functions on Exterior Domains In this section we introduce some Green Identities for functions in the exterior Ω − of an open bounded subset Ω of Rn . In particular, we will deal with functions that are harmonic and harmonic at infinity. We begin with the following Corollary 4.7, where we present a counterpart in Ω − of the first Green Identity. Corollary 4.7 (First Green Identity in Exterior Domains). Let Ω be a bounded open Lipschitz subset of Rn . Let u, v ∈ C 1 (Ω − ) be harmonic in Ω − and harmonic at infinity. Then we have  lim

R→+∞

 n  ∂u ∂v ∂v dx = − u dσ . Ω − ∩Bn (0,R) j=1 ∂xj ∂xj ∂Ω ∂νΩ

In particular, |∇u|2 is integrable in Ω − and   |∇u|2 dx = − Ω−

u ∂Ω

∂u dσ . ∂νΩ

(4.9)

(4.10)

− Proof. Let R ∈]0, +∞[ be such that Ω ⊆ Bn (0, R). Let ΩR ≡ Ω − ∩ Bn (0, R). − n Clearly, ΩR is a bounded open Lipschitz subset of R and − ∂ΩR = ∂Ω ∪ ∂Bn (0, R) ,

νΩ − = −νΩ R

∂Ω ∩ ∂Bn (0, R) = ∅ , y a.e. on ∂Ω = ∂Ω − , νΩ − (y) = R |y|

∀y ∈ ∂Bn (0, R) .

− − ) ∩ C 2 (ΩR ). Then the first Green Identity implies that Moreover, u, v ∈ C 1 (ΩR



 n  ∂u ∂v ∂v dx = u dσ − − − ∂x ∂x ∂ν j j ΩR j=1 ∂ΩR ΩR   ∂v ∂v =− u dσ + u dσ . ∂ν ∂ν Ω Bn (0,R) ∂Ω ∂Bn (0,R)

Since both u and v are harmonic at infinity, the limit as R tends to infinity of the last integral in the right-hand side exists and equals zero (cf. Proposition 3.26). Then the limit in the statement exists finite, and formula (4.9) holds. If v = u, then the Monotone Convergence Theorem implies that

130

4 Green Identities and Layer Potentials



 |∇u| dx = lim



2

Ω−

R→∞

χΩ − |∇u| dx = lim 2

Ω−

R

R→∞

− ΩR

|∇u|2 dx  

and formula (4.9) for v = u implies the validity of formula (4.10).

By the previous corollary, we deduce immediately the following form of the second Green Identity for harmonic functions in exterior domains. Corollary 4.8 (Second Green Identity in Exterior Domains). Let Ω be a bounded open Lipschitz subset of Rn . Let u, v ∈ C 1 (Ω − ) be harmonic in Ω − and harmonic at infinity. Then we have  ∂v ∂u u −v dσ = 0 . (4.11) ∂νΩ ∂Ω ∂νΩ Remark 4.9. We note that the first Green Identity of (4.9) and the second Green Identity of (4.11) hold also in case u and v are complex-valued. Then by v=  taking ∂u u in (4.9) one can deduce that the left-hand side of (4.10) equals − ∂Ω u ∂ν dσ Ω when u is complex-valued. We now consider the third Green Identity for harmonic functions in Ω − . Theorem 4.10 (Third Green Identity in Exterior Domains). Let Ω be a bounded open Lipschitz subset of Rn . Let u ∈ C 1 (Ω − ) be harmonic in Ω − and harmonic at infinity. Let u∞ be the limiting value of u at infinity. Then we have  ∂ ∂u u(x) = − u(y) (Sn (x − y))− (y)Sn (x−y) dσy +u∞ ∀x ∈ Ω − . ∂νΩy ∂νΩ ∂Ω If x ∈ (Ω − )j for some j ∈ {1, . . . , κ − }, then we also have  ∂ ∂u u(x) = − u(y) (Sn (x − y)) − (y)Sn (x − y) dσy ∂νΩy ∂νΩ ∂(Ω − )j (see (4.8)). In addition,  ∂ ∂u 0=− u(y) (Sn (x − y)) − (y)Sn (x − y) dσy + u∞ ∂νΩy ∂νΩ ∂Ω

∀x ∈ Ω .

Proof. Let x ∈ Ω − . Let R0 ∈]0, +∞[ be such that Ω ∪ {x} ⊆ Bn (0, R0 ). Let − − ≡ Ω − ∩ Bn (0, R) for all R ≥ R0 . Clearly, ΩR is a bounded open subset of ΩR class C 1 and − ∂ΩR = ∂Ω ∪ ∂Bn (0, R) ,

νΩ − = −νΩ R

(4.12) −

a.e. on ∂Ω = ∂Ω ,

y νBn (0,R) (y) = |y|

∀y ∈ ∂Bn (0, R) ,

for all R ≥ R0 . By assumption, u ∈ C 1 (Ω − ∩ Bn (0, R)) ∩ C 2 (Ω − ∩ Bn (0, R)). − Then we can apply the third Green Identity in the bounded open Lipschitz set ΩR

4.2 Green Identities for Harmonic Functions on Exterior Domains

and obtain

131



∂ ∂u (Sn (x − y)) − (y)Sn (x − y) dσy ∂νΩy ∂νΩ ∂Ω ∂ ∂u u(y) (Sn (x − y)) − (y)Sn (x − y) dσy ∂νBn (0,R)y ∂νBn (0,R) ∂Bn (0,R)

u(x) = −  +

u(y)

for all R ≥ R0 . Thus it suffices to prove that  (i) limR→+∞ ∂Bn (0,R) u(y) ∂νB ∂(0,R) (Sn (x − y)) dσy = u∞ , n y  (ii) limR→+∞ ∂Bn (0,R) ∂νB∂u(0,R) (y)Sn (x − y) dσy = 0. n

By the decay properties of harmonic functions at infinity of Proposition 3.25, we know that possibly choosing a larger R0 , there exists C > 0 such that |∂r u(y)| ≤ C|y|1−n |∂r u(y)| ≤ C|y|−2

if |y| ≥ R0 , n ≥ 3 , if |y| ≥ R0 , n = 2 .

Then we note that       y  ∂    (Sn (x − y)) =  ∇Sn (x − y)  ∂ν |y| Bn (0,R)y 1 1 1 1 ≤ ≤ , n−1 sn |x − y| sn (R − R0 )n−1 1 1 |Sn (x − y)| ≤ (n − 2)sn |x − y|n−2 1 1 ≤ if n ≥ 3 , (n − 2)sn (R − R0 )n−2 1 1 |S2 (x − y)| ≤ |log |x − y|| ≤ log(R + R0 ) 2π 2π

(4.13)

(4.14)

(4.15)

(4.16)

for all y ∈ ∂Bn (0, R) and R > R0 + 1. Then we are ready to prove (i). By the third Green Identity applied to the constant function 1, we know that  ∂ (Sn (x − y)) dσy . 1= ∂ν Bn (0,R)y ∂Bn (0,R) Then we have  u(y) ∂Bn (0,R)



∂ ∂νBn (0,R)y

(Sn (x − y)) dσy − u∞

(u(y) − u∞ )

= ∂Bn (0,R)

and it suffices to prove that

∂ ∂νBn (0,R)y

(Sn (x − y)) dσy ,

132

4 Green Identities and Layer Potentials

 (u(y) − u∞ )

lim

R→+∞

∂Bn (0,R)

∂ ∂νBn (0,R)y

(Sn (x − y)) dσy = 0 .

To do so, we note that the above inequality (4.14) implies that     ∂   (u(y) − u∞ ) (Sn (x − y)) dσy    ∂Bn (0,R)  ∂νBn (0,R)y ≤

sn Rn−1 sn (R − R0 )n−1

sup y∈∂Bn (0,R)

|u(y) − u∞ |

∀R ≥ R0 + 1 .

On the other hand the limiting relation limx→∞ u(x) = u∞ implies that lim

sup

R→+∞ y∈∂Bn (0,R)

|u(y) − u∞ | = 0 .

Hence, statement (i) follows. We now consider statement (ii). If n ≥ 3, the above inequalities (4.13) and (4.15) imply that     ∂u   (y)Sn (x − y) dσy    ∂Bn (0,R) ∂νBn (0,R)      sn Rn−1 ∂u   ≤ sup (y)   n−2 (n − 2)sn (R − R0 ) y∈∂Bn (0,R) ∂νBn (0,R) ≤

sn Rn−1 C (n − 2)sn (R − R0 )n−2 Rn−1

∀R ≥ R0 + 1 .

If instead n = 2, , the above inequalities (4.13) and (4.16) imply that     ∂u   (y)Sn (x − y) dσy    ∂Bn (0,R) ∂νBn (0,R)      2πR ∂u  ≤ log(R + R0 ) sup (y)  2π y∈∂Bn (0,R) ∂νBn (0,R) ≤ CR−1 log(R + R0 )

∀R > R0 + 1 .

Accordingly, (ii) holds both if n ≥ 3 and if n = 2. In case n ≥ 3, one could prove statement (ii) also by observing that that both u(·) and Sn (x − ·) are harmonic at infinity and by invoking Proposition 3.26. To prove the equality in case x ∈ (Ω − )j with j ≥ 1, it suffices to apply the third Green Identity to (Ω − )j and to note that ν(Ω − )j (x) = −νΩ (x) for all x ∈ ∂(Ω − )j . We now prove the last part of the statement. Let x ∈ Ω. Let R0 ∈]0, +∞[ be − ≡ Ω − ∩ Bn (0, R) for all R ≥ R0 . Since u(·) such that Ω ⊆ Bn (0, R0 ). Let ΩR − − and Sn (x − ·) are harmonic in ΩR and of class C 1 (ΩR ), the second Green Identity − in ΩR and formulas (4.12) for νΩ − imply that R

4.3 Preliminaries on Singular Integrals and Layer Potentials

 0=

− ∂ΩR



u(y)

133

∂ ∂u (Sn (x − y)) − (y)Sn (x − y) dσy ∂νΩ − ∂νΩ − Ry

R

∂ ∂u =− u(y) (Sn (x − y)) − (y)Sn (x − y) dσy ∂νΩy ∂νΩ ∂Ω  ∂ ∂u + u(y) (Sn (x − y)) − (y)Sn (x − y) dσy ∂νBn (0,R)y ∂νBn (0,R) ∂Bn (0,R) for all R ≥ R0 . Arguing as above, we can prove the limiting relations (i) and (ii) for x ∈ Ω and deduce the validity of the last formula of the statement.   In conclusion of this section, we note that the third Green Identities in Ω and in Ω − show that we can write a large class of functions u in terms of integrals of the form   ∂ φ(y)Sn (x − y) dσy , ψ(y) (Sn (x − y)) dσy , ∂ν Ωy ∂Ω ∂Ω which are known as single (or simple) layer potential of moment (or density) φ and double layer potential of moment (or density) ψ, respectively, and in terms of integrals of the form  Sn (x − y)f (y) dy , Ω

which are known as volume potentials with density f . In the next sections, we will analyze some properties of the single and double layer potentials. In Chapter 6, we will exploit these results to prove existence theorems for the solutions of boundary value problems for the Laplace equation. Later on, in Chapter 7, we will consider volume potentials and prove existence results for the solutions of boundary value problems for the Poisson equation.

4.3 Preliminaries on Singular Integrals and Layer Potentials We first note that both the single and the double layer potentials can be written in terms of integrals of the form  k(x − y)μ(y) dσy , ∂Ω

where k is a continuous function in Rn \ {0}. Indeed, for the single layer potential it suffices to take k = Sn while we can write the double layer potential as 

 ∂ Sn (x − y) dσy = − ∂νΩy n

ω(y) ∂Ω

l=1

 ∂Ω

∂ Sn (x − y)(νΩ )l (y)ω(y) dσy ∂xl

134

4 Green Identities and Layer Potentials

and each integral of the sum in the right-hand side can be written in the above form ∂ Sn and μ = (νΩ )l ω. Thus our goal is to analyze the properties of with k = ∂x l integrals of the form  k(x − y)μ(y) dσy ∀x ∈ Rn , K[k, μ](x) ≡ ∂Ω

which are called single layer potentials with support in ∂Ω, (convolution) kernel k, and density, or moment, μ, and we will focus on the case where k equals the fundamental solution Sn or a first order partial derivative of Sn . If k equals Sn and if x ∈ ∂Ω, then k(x − y) is integrable in y ∈ ∂Ω. Instead, if k equals one of the first order partial derivatives of Sn , then k(x − y) is not integrable in y ∈ ∂Ω and the above integral may exist only in the sense of the principal value, i.e., as the limit   p.v. k(x − y)μ(y) dσy ≡ lim k(x − y)μ(y) dσy . →0

∂Ω

∂Ω\Bn (x,)

Since the first order partial derivatives of Sn are positively homogeneous of degree −(n − 1), we are interested in the properties of the above principal value in case where k is positively homogeneous of degree −(n − 1). We now briefly summarize some properties of the positively homogeneous functions and we introduce some notation. Definition 4.11. Let n ∈ N \ {0}, h ∈ R. We say that a function f from Rn \ {0} to R is positively homogeneous of degree h provided that f (tx) = th f (x)

∀(t, x) ∈]0, +∞[×(Rn \ {0}) .

By setting t = |x|, we have f (x) = f (|x|

x x ) = |x|h f ( ) |x| |x|

∀x ∈ Rn \ {0} ,

an equality which shows that a positively homogeneous function of degree h is entirely determined by its values on the unit sphere ∂Bn (0, 1). Proposition 4.12. Let n ∈ N \ {0}, h ∈ R. Let f ∈ C 0 (Rn \ {0}) be positively homogeneous of degree h. Then the following statements hold.   (i) |f (x)| ≤ sup∂Bn (0,1) |f | |x|h for all x ∈ Rn \ {0}. (ii) If m ∈ N \ {0} and if we further assume that f ∈ C m (Rn \ {0}), then Dβ f is positively homogeneous of degree h − |β| for all β ∈ Nn such that |β| ≤ m. The proof is an advanced Calculus exercise and it is left to the reader. If m ∈ N and α ∈ [0, 1], then we set Km,α

(4.17)

4.3 Preliminaries on Singular Integrals and Layer Potentials

135

m,α ≡ {k ∈ Cloc (Rn \ {0}) : k is positively homogeneous of degree − (n − 1)}

and

0 0 kKm,α ≡ 0k|∂Bn (0,1) 0C m,α (∂B

n (0,1))

∀k ∈ Km,α

(4.18)

m,α n (see Section 2.11 for the  definition of Cloc (R \ {0})). Then one can easily verify that Km,α ,  · Km,α is a Banach space. We will also consider the subspace

Km,α;o ≡ {k ∈ Km,α : k is odd}

(4.19)

of Km,α . Since positively homogeneous functions are uniquely determined by their values on the unit sphere, we can prove the following lemma. Lemma 4.13. Let α ∈ [0, 1], j ∈ {1, . . . , n}. Then the linear operator from K1,α to ∂k is linear and continuous . C 0,α (∂Bn (0, 1)) that takes k to ∂x j |∂Bn (0,1)

Proof. Let An (1/2, 1) ≡ Bn (0, 1) \ Bn (0, 1/2). By Theorem 2.85 (i), the restriction operator R from the subspace  1,α 1,α C−(n−1) (An (1/2, 1)) ≡ f|An (1/2,1) : f ∈ Cloc (Rn \ {0}) ,  f is positively homogeneous of degree − (n − 1) of C 1,α (An (1/2, 1)) to C 1,α (∂Bn (0, 1)) is linear and continuous. Since positively homogeneous functions are uniquely determined by their values on the unit sphere, the restriction operator R is a bijection. We now prove that 1,α (An (1/2, 1)) is closed in the space C 1,α (An (1/2, 1)). Let {gj }j∈N be a seC−(n−1) 1,α quence of C−(n−1) (An (1/2, 1)) that converges to g in C 1,α (An (1/2, 1)). We need 1,α to show that g belongs to C−(n−1) (An (1/2, 1)). To do so, it suffices note that g is the

restriction to An (1/2, 1) of the positively homogeneous function |x|−(n−1) g(x/|x|) 1,α (Rn \ {0}). Since the comand to prove that |x|−(n−1) g(x/|x|) is of class Cloc 1,α position of a function of C (An (1/2, 1)) and of the function x/|x| is of class 1,α (Rn \{0}) and the pointwise product of functions in Schauder spaces is continCloc 1,α (Rn \ {0}) (cf. uous, we deduce that the function |x|−(n−1) g(x/|x|) belongs to Cloc 1,α Proposition 2.54, Theorem 2.59). Hence, C−(n−1) (An (1/2, 1)) is a Banach space. Then the Open Mapping Theorem 2.2 implies that R is a homeomorphism. Then the definition of norm in K1,α implies that the map from K1,α to the 1,α (An (1/2, 1)) that takes k to k|An (1/2,1) is linear and continuous. space C−(n−1) Since

∂ ∂xj

1,α is linear and continuous from C−(n−1) (An (1/2, 1)) to C 0,α (An (1/2, 1))

and the restriction operator is linear and continuous from C 0,α (An (1/2, 1)) to ∂ is linear and C 0,α (∂Bn (0, 1)) (cf. Theorem 2.85 (i)), we conclude that ∂x j continuous from K1,α to C 0,α (∂Bn (0, 1)).

|∂Bn (0,1)

 

136

4 Green Identities and Layer Potentials

Then we introduce the following elementary H¨older continuity lemma for positively homogeneous functions, that we prove by an argument of Cialdea [52, VIII, p. 47] who has considered case α = 1. 0,α (Rn \ {0}) be a Lemma 4.14. Let n ∈ N \ {0}, h ∈ N. Let α ∈]0, 1]. Let k ∈ Cloc positively homogeneous function of degree −h. Then

|k(x) − k(y)| ≤ (2α + 2h) max{ sup |k|, |k : ∂Bn (0, 1)|α }|x − y|α (min{|x|, |y|})−h−α ∂Bn (0,1)

for all x, y ∈ Rn \ {0}. Proof. Let c ≡ max{sup∂Bn (0,1) |k|, |k : ∂Bn (0, 1)|α }. We first consider case h > 0. By the triangular inequality, we have         x y −h −h   |k(x) − k(y)| ≤ k |x| − k |y|  |x| |y|           −h   x y   −h y   |x| − |y|−h  |x| + k ≤ k −k   |x| |y| |y| α      x y  |x|−h + c |x|−h − |y|−h  ≤ c  −  |x| |y| α      −1  h−1   x|y| − y|x|  −h  −1  ≤c  |x| |x| + c − |y| |x|−l |y|l−(h−1)  |x||y| l=0 α    (x − y)|y| + y(|y| − |x|)  |x|−h ≤ c   |x||y| +c |x − y|α (|x| + |y|)1−α

h−1 

|x|−l−1 |y|l−h

l=0

for all x, y ∈ Rn \ {0}. By inequality (2.3) with p = 1 − α, q = 1, we have (|x| + |y|)1−α ≤ |x|1−α + |y|1−α

∀x, y ∈ Rn \ {0}

and thus |k(x) − k(y)| ≤ 2α c |x − y|α |x|−α−h h−1  h−1    α −l−α l−h −l−1 l−h+1−α +c |x − y| |x| |y| + |x| |y| l=0 α 

≤ 2 c |x − y| (min{|x|, |y|}) α

−(h+α)

l=0 

+ c |x − y|α 2h(min{|x|, |y|})−(h+α)

for all x, y ∈ Rn \ {0}. Indeed, −α − h and all the exponents which appear in the two above summation signs are negative and accordingly |x|−α−h ≤ (min{|x|, |y|})−(h+α) ,

|x|−l−α |y|l−h ≤ (min{|x|, |y|})−(h+α) ,

4.3 Preliminaries on Singular Integrals and Layer Potentials

137

|x|−l−1 |y|l−h+1−α ≤ (min{|x|, |y|})−(h+α) for all x, y ∈ Rn \ {0} and l ∈ {1, . . . , h − 1}. Hence, the proof of case h > 0 is complete. If h = 0, then c |x|−h − |y|−h  = 0 and the above inequalities hold with h = 0 in the right-hand side and without the terms which contain the summation sign.   Then we introduce the following two preliminary lemmas, which we prove by an argument of Agmon et al. [2, Lemma 3.1, p. 644]. We use the following notation, (η, t) denotes a vector of Rn with η ∈ Rn−1 and t ∈ R.

(4.20)

Lemma 4.15. Let k ∈ C 1 (Rn \{0}) be a positively homogeneous function of degree −(n − 1). If  k(η, 0) dση = 0 ,

(4.21)

∂Bn−1 (0,1)

then

∂ ∂t k(η, t)

Proof. Since

is integrable in the variable η ∈ Rn−1 for all t ∈ R \ {0} and  ∂ k(η, t) dη = 0 ∀t ∈ R \ {0} . (4.22) Rn−1 ∂t

∂ ∂t k(η, t)

is positively homogeneous of degree −(n − 1) − 1 < −(n − 1) ,

∂ the continuous function ∂t k(·, t) is integrable in Rn−1 for all t ∈ R\{0} (cf. Proposition 4.12). Next we note that

k(η, t) = k1 (η, t) + k2 (η, t)

∀(η, t) ∈ Rn \ {0} ,

where k1 (η, t) ≡

η , 0) k( |η|

(|η|2 + t2 )

(n−1) 2

,

k2 (η, t) ≡

η η t , |(η,t)| ) − k( |η| , 0) k( |(η,t)|

(|η|2 + t2 )

(n−1) 2

,

for all (η, t) ∈ Rn \ {0}. Since k ∈ C 1 (Rn \ {0}) and k is Lipschitz continuous on ∂Bn (0, 1), then we have |k2 (η, t)|

(4.23)     η |t| η  1 + ≤ |k : ∂Bn (0, 1)|1  − |(η, t)| |η|  |(η, t)| |(η, t)|n−1     |t| |η|  |(η, t)| − |η|  ≤ |k : ∂Bn (0, 1)|1  + |(η, t)|n |(η, t)|n  |η|   |t| 2|t| |t| ≤ |k : ∂Bn (0, 1)|1 + = |k : ∂Bn (0, 1)|1 |(η, t)|n |(η, t)|n |(η, t)|n

138

4 Green Identities and Layer Potentials

and the right-hand side is integrable in η ∈ Rn−1 for all t ∈ R \ {0}. Moreover, by setting η = ξt, we have  k2 (η, t) dη Rn−1



(t)ξ sgn (t) sgn (t)ξ k( sgn |(ξ,1)| , |(ξ,1)| ) − k( |ξ| , 0)

=

(|ξ|2 + 1)

Rn−1

(n−1) 2

dξ

∀t ∈ R \ {0}

and accordingly ∂ ∂t

 Rn−1

k2 (η, t) dη = 0

Moreover, k2 (η, t) = k(η, t) − k(

∀t ∈ R \ {0} . η |(η, t)|, 0) |η|

and thus       ∂    k2 (η, t) =  ∂ k(η, t) − k( η |(η, t)|, 0)    ∂t   ∂t |η|     n−1  ∂k η  ∂k ηj t  =  (η, t) − ( |(η, t)|, 0) ∂ηj |η| |η| |(η, t)|   ∂t j=1 ⎛ ⎞   n−1    ∂k    ∂k  ⎠ |(η, t)|−n ≤ ⎝ sup   + sup   ∂t ∂η j ∂Bn (0,1) j=1 ∂Bn (0,1)

(4.24)

∀η ∈ Rn−1 ,

for all t ∈ R \ {0}. Then (4.23) and (4.24) and the classical differentiability theorem for integrals depending on a parameter imply that   ∂ ∂ 0= k2 (η, t) dη k2 (η, t) dη = ∀t ∈ R \ {0} . (4.25) ∂t Rn−1 ∂t n−1 R We now turn to consider k1 . By integrating on the spheres and by assumption (4.21), we have  ∂ k1 (η, t) dη ∂t n−1 R  η k( |η| , 0) = −(n − 1)t dη n+1 Rn−1 |(η, t)|  +∞  η k( |η| , 0) dση dρ = −(n − 1)t n+1 0 ∂Bn−1 (0,ρ) |(η, t)|   +∞ ξ k( |ξ| , 0) n−2 ρ = −(n − 1)t n+1 dσξ dρ 0 ∂Bn−1 (0,1) (ρ2 + t2 ) 2

4.3 Preliminaries on Singular Integrals and Layer Potentials



+∞

= −(n − 1)t

ρn−2

139

 dρ

k(ξ, 0) dσξ = 0 n+1 (ρ2 + t2 ) 2 ∂Bn−1 (0,1)    ∂ ∂ ∂ and thus Rn−1 ∂t k(η, t) dη = Rn−1 ∂t k1 (η, t) dη + Rn−1 ∂t k2 (η, t) dη = 0 for all t ∈ R \ {0}. Thus the proof is complete.   0

Lemma 4.16. Let k ∈ C 1 (Rn \{0}) be positively homogeneous of degree −(n−1). If j ∈ {1, . . . , n − 1} and t ∈ R \ {0}, then ∂η∂ j k(·, t) is integrable in Rn−1 and  Rn−1

∂ k(η, t) dη = 0 ∂ηj

(4.26)

(cf. (4.20)). Proof. Since k is positively homogeneous of degree −(n − 1), then

∂k ∂ηj is n−1

positively

∈R for all t in homogeneous of degree −n and thus R \ {0} (cf. Proposition 4.12). Then the Dominated Convergence Theorem together with the Divergence Theorem 4.1 imply that   ∂ ∂ k(η, t) dη = lim k(η, t) dη ρ→+∞ ∂η ∂η n−1 j j R Bn−1 (0,ρ)   ηj ξj = lim k(η, t) dση = lim k(ρξ, t) ρn−2 dσξ ρ→+∞ ∂B ρ→+∞ |η| |ξ| (0,ρ) ∂Bn−1 (0,1)  n−1 = lim k(ξ, t/ρ)ξj ρn−2−(n−1) dσξ = 0 , ∂ ∂ηj k(η, t) is integrable in η

ρ→+∞

∂Bn−1 (0,1)

for all t ∈ R \ {0}. Indeed, limρ→+∞ ρ−1 = 0 and the continuity of k in Rn \ {0} implies that the limit   lim k(ξ, t/ρ)ξj dσξ = k(ξ, 0)ξj dσξ , ρ→+∞

∂Bn−1 (0,1)

∂Bn−1 (0,1)

exists finite for all t ∈ R \ {0}. Hence, the proof is complete.

 

We are now ready to prove the main Theorem 4.17 of this section. The result of the theorem is due to Miranda and has been presented in [212]. Our proof is similar to that of Miranda, but we are able to shorten some parts of it by exploiting the sufficient condition for H¨older continuity of our Proposition 2.82 and we can improve the regularity assumptions on the kernel k which is here of cass C 1,1 and was of class C 2 in the paper of Miranda. A reader interested in understanding all details of the proof may also read Sections 2.13, 2.18, and 2.19 of the Preliminaries (see also Section A.6 in the Appendix). Theorem 4.17. Let α ∈]0, 1[. Let Ω be a bounded open subset of Rn of class C 1,α . Then the following statements hold.

140

4 Green Identities and Layer Potentials

(i) For each (k, μ) ∈ K1,1;o × C 0,α (∂Ω), the map  k(x − y)μ(y) dσy K[k, μ]|Ω (x) =

∀x ∈ Ω

∂Ω

can be extended to a unique α-H¨older continuous function K[k, μ]+ on Ω. Moreover, the map from K1,1;o × C 0,α (∂Ω) to C 0,α (Ω) that takes (k, μ) to K[k, μ]+ is bilinear and continuous. (ii) Let r ∈]0, +∞[ be such that Ω ⊆ Bn (0, r). Then for each (k, μ) ∈ K1,1;o × C 0,α (∂Ω) the map  K[k, μ]|Rn \Ω (x) = k(x − y)μ(y) dσy ∀x ∈ Rn \ Ω , ∂Ω

can be extended to a unique continuous function K[k, μ]− on Rn \ Ω such is α-H¨older continuous. Moreover, the that the restriction K[k, μ]− |Bn (0,r)\Ω

map from K1,1;o × C 0,α (∂Ω) to C 0,α (Bn (0, r) \ Ω) that takes (k, μ) to is bilinear and continuous. K[k, μ]− |Bn (0,r)\Ω

Proof. We first prove statement (i). The classical theorem of continuity for integrals  depending on a parameter implies that the integral ∂Ω k(x − y)μ(y) dσy is continuous in x ∈ Rn \ ∂Ω. Since H¨older continuous functions are uniformly continuous and uniformly continuous functions in Ω can be uniquely extended to the closure Ω, it suffices to show that there exists C > 0 such that sup |K[k, μ](x)| + |K[k, μ] : Ω|α ≤ CkK1,1 μC 0,α (∂Ω)

x∈Ω

for all (k, μ) ∈ K1,1;o × C 0,α (∂Ω). Let ϑ ∈]0, 1[. By Lemma 2.78 there exist an open neighborhood U of class C ∞ of ∂Ω and a ∈ C ∞ (U , Rn ) such that conditions (2.66) hold true for some τ ∈]0, +∞[. Then, by the sufficient condition for H¨older continuity of Proposition 2.82, it suffices to show that (j) If H is a compact subset of Ω, then there exists CH > 0 such that sup |K[k, μ](x)| + sup |∇K[k, μ]| ≤ CH kK1,1 μC 0,α (∂Ω)

x∈H

x∈H

for all (k, μ) ∈ K1,1;o × C 0,α (∂Ω). (jj) If t1 ∈]0, +∞[ is as in Lemma 2.79, then there exist t ∈]0, t1 [ and C2 > 0 such that (4.27) Mt ,α (K[k, μ]|Ω ) ≤ C2 kK1,1 μC 0,α (∂Ω) for all (k, μ) ∈ K1,1;o × C 0,α (∂Ω) (see (2.74) for the definition of Mt ,α (·)). Now, the validity of (j) follows by the classical theorem of continuity for integrals depending on a parameter. Indeed, the distance from H to ∂Ω is positive. (See also Lemma 4.13). Thus it remains to prove statement (jj). In order to estimate ∇K[k, μ](x + ta(x)) for x ∈ ∂Ω and t ∈] − t1 , 0[ we fix j ∈ {1, . . . , n} and we

4.3 Preliminaries on Singular Integrals and Layer Potentials

estimate

141

 ∂xj k(x + ta(x) − y)μ(y) dσy

∂xj K[k, μ](x + ta(x)) = ∂Ω

when (t, x) ∈] − t1 , 0[×∂Ω for a perhaps smaller t1 and when (k, μ) ∈ K1,1;o × C 0,α (∂Ω). By Lemma 2.80 and by the properties of a in (2.66), we have |x − y + ta(x)| ≥ (1 − ϑ)1/2 (|x − y|2 + |t|2 )1/2

(4.28)

for all x, y ∈ ∂Ω such that |x − y| < τ and for all t ∈] − t1 , 0[. Next we note that if μ ∈ C 0,α (∂Ω), then  ∂xj k(x + ta(x) − y)μ(y) dσy (4.29) ∂Ω  ∂xj k(x + ta(x) − y)(μ(y) − μ(x)) dσy = ∂Ω  +μ(x) ∂xj k(x + ta(x) − y) dσy ∀x ∈ ∂Ω ∂Ω

for all t ∈]−t1 , 0[. In order to make the proof more readable, we now find convenient to split it into two parts, in which we consider separately the first and the second integral in the right-hand side of (4.29) and into a series of intermediate steps. Part 1. Here we estimate the first integral in the right-hand side of (4.29). To do so, we split the domain of integration ∂Ω into the two parts (∂Ω) \ Bn (x, τ ) ,

(∂Ω) ∩ Bn (x, τ ) .

Since ∂xj k is positively homogeneous of degree −n, we have       (4.30) ∂xj k(x + ta(x) − y)(μ(y) − μ(x)) dσy    (∂Ω)\Bn (x,τ )   |x − y|α dσy ≤ ∂xj kC 0 (∂Bn (0,1)) |μ : ∂Ω|α n (∂Ω)\Bn (x,τ ) |x + ta(x) − y|  diam(∂Ω)α dσy ≤ ∂xj kC 0 (∂Bn (0,1)) |μ : ∂Ω|α (τ − |t|)n ∂Ω ≤ ∂xj kC 0 (∂Bn (0,1)) |μ : ∂Ω|α mn−1 (∂Ω)diam(∂Ω)α (τ /2)−n for all t ∈] − min{t1 , τ /2}, 0[. • We now turn to consider the integral on (∂Ω) ∩ Bn (x, τ ). By inequality (4.28), we have       (4.31) ∂xj k(x + ta(x) − y)(μ(y) − μ(x)) dσy     (∂Ω)∩Bn (x,τ )

142

4 Green Identities and Layer Potentials

 ≤ ∂xj kC 0 (∂Bn (0,1)) |μ : ∂Ω|α (∂Ω)∩Bn (x,τ )

≤

∂xj kC 0 (∂Bn (0,1)) |μ : ∂Ω|α (1 − ϑ)n/2



(∂Ω)∩Bn (x,τ )

|x − y|α dσy |x + ta(x) − y|n |x − y|α dσy . (|x − y|2 + t2 )n/2

• We now estimate the last integral in the right-hand side of (4.31) by considering separately the case where |x − y| ≥ |t| and the case where |x − y| < |t| for t ∈] − min{t1 , τ /2}, 0[. Then we have  |x − y|α dσy 2 2 n/2 (∂Ω)∩Bn (x,τ ) (|x − y| + t )  ≤

{y∈(∂Ω)∩Bn (x,τ ): |x−y|≥|t|}



(4.32) |x − y|α dσy (|x − y|2 + t2 )n/2

+ {y∈(∂Ω)∩Bn (x,τ ): |x−y| 0 such that  dη  ≤ # whenever m∂Ω (E) ≤ δl1 . (4.65) λ ← |ξ − η| 2 1 +  |∇γl | 2∞ ψ (l) (E) x

Then if we choose

0 < δl ≤ min δl1 , [min{rl /4, τl /4}]λ /2 , inequalities (4.61)–(4.63), (4.65) imply that (4.60) holds for all x in Ql . We now turn to prove the last inequality of the statement. By inequality (4.59) 1 with E = ∂Ω, we have supx∈W ∂Ω |x−y| λ dσy < +∞. By the inclusion Bn−1 (ξ, rl ) ⊆ Bn−1 (0, 2rl ) , which holds for all ξ ∈ Bn−1 (0, rl /8), we have    dη dη dη ≤ = λ λ λ ← |ξ − η| |ξ − η| |η| ψ (l) (∂Ω) Bn−1 (0,rl ) Bn−1 (ξ,rl ) x   dη dη ≤ ≤ < +∞ λ λ Bn−1 (0,2rl ) |η| Bn−1 (0,2 maxl∈{1,...,m} rl ) |η| Then inequalities (4.61)–(4.63) with for all ξ ∈ Bn−1 (0, rl /8) and l ∈ {1,  . . . , m}. 1 E = ∂Ω imply that supx∈m λ dσy < +∞ and accordingly the last Q l ∂Ω |x−y| l=1 inequality of the statement holds true.   We are now ready to prove the following.

4.4 The Single Layer Potential

155

Theorem 4.22. Let Ω be a bounded open Lipschitz subset of Rn . If φ ∈ C 0 (∂Ω), then the following statement holds. (i) vΩ [φ] is harmonic in Rn \ ∂Ω. (ii) vΩ [φ] is continuous in Rn . Proof. Since Dxγ Sn (x−y)φ(y) is continuous in (Rn \∂Ω)×∂Ω for all γ ∈ Nn , then the classical differentiation theorem for integrals depending on a parameter implies that vΩ [φ] ∈ C ∞ (Rn \ ∂Ω) and that  γ D vΩ [φ](x) = Dxγ Sn (x − y)φ(y) dσy ∀x ∈ Rn \ ∂Ω ∂Ω

for all γ ∈ Nn . Since Δx Sn (x − y) = 0 for all x ∈ Rn \ ∂Ω and y ∈ ∂Ω, we conclude that vΩ [φ] is harmonic in Rn \ ∂Ω. In particular, vΩ [φ] is continuous at all points of Rn \ ∂Ω. ˜ ∈ ∂Ω. We want We now prove that vΩ [φ] is continuous at all points of ∂Ω. Let x to verify that   lim φ(y)Sn (x − y) dσy = φ(y)Sn (˜ x − y) dσy . (4.66) x→˜ x

∂Ω

∂Ω

By the continuity of Sn in R \ {0}, we have n

lim φ(y)Sn (x − y) = φ(y)Sn (˜ x − y)

x→˜ x

∀y ∈ (∂Ω) \ {˜ x} .

Now let r > 0 be such that Ω ⊆ Bn (0, r). In order to invoke the Vitali Convergence Theorem A.11, we show that, for each  > 0, there exists δ > 0 such that  |φ(y)Sn (x − y)| dσy ≤  for all E ∈ L∂Ω , m∂Ω (E) ≤ δ , x ∈ Bn (0, r) . E

If n ≥ 3, then we have  |φ(y)Sn (x − y)| dσy ≤ E

φ∞ (n − 2)sn

 E

1 dσy |x − y|n−2

∀x ∈ Rn

and Lemma 4.21 implies that there exists δ > 0 such that  1  dσy ≤ for all E ∈ L∂Ω , m∂Ω (E) ≤ δ , x ∈ Rn . n−2 φ∞ 1 + (n−2)s E |x − y| n If n = 2, then we have   φ∞ |φ(y)Sn (x − y)| dσy ≤ |log |x − y|| dσy 2π E E  φ∞ ( sup ρ1/2 | log ρ||) ≤ |x − y|−1/2 dσy 2π ρ∈]0,2r] E

∀x ∈ Bn (0, r)

156

4 Green Identities and Layer Potentials

and Lemma 4.21 implies that there exists δ > 0 such that   |x − y|−1/2 dσy ≤ φ∞ 1 + 2π (supρ∈]0,2r] ρ1/2 | log ρ||) E for all E ∈ L∂Ω , m∂Ω (E) ≤ δ , x ∈ Bn (0, r) . Then the Vitali Convergence Theorem A.11 implies that, for each sequence {xj }j∈N in Bn (0, r) \ {˜ x} which converges to x ˜, we have   lim φ(y)Sn (xj − y) dσy = φ(y)Sn (˜ x − y) dσy . j→∞

∂Ω

∂Ω

 

Therefore, the limiting relation in (4.66) holds true. Under the assumptions of the previous theorem, we set + vΩ [φ] ≡ vΩ [φ]|Ω

− vΩ [φ] ≡ vΩ [φ]|Ω − .

− Then we have the following theorem, which clarifies the behavior of vΩ [φ] at infinity.

Theorem 4.23. Let Ω be a bounded open Lipschitz subset of Rn . Let φ ∈ C 0 (∂Ω). − − (i) If n ≥ 3, then limx→∞ vΩ [φ](x) = 0 and accordingly vΩ [φ] is harmonic at infinity.  − [φ](x) = 0 and accordingly (ii) Let n = 2. If ∂Ω φ dσ = 0, then limx→∞ vΩ  − − vΩ [φ] is harmonic at infinity. If, instead, ∂Ω φ dσ = 0, then vΩ [φ] is not harmonic at infinity.

Proof. Let r0 > 0 be such that Ω ⊆ Bn (0, r0 ). Clearly, |x − y| ≥ ||x| − |y|| ≥ |x| − r0 > 0

∀y ∈ ∂Ω, x ∈ Rn \ Bn (0, r0 ) .

If n ≥ 3, we have  − |vΩ [φ](x)| ≤ |φ(y)||Sn (x − y)| dσy ∂Ω  1 |φ(y)| ≤ dσy (n − 2)sn ∂Ω |x − y|n−2  1 1 ≤ |φ(y)| dσy (n − 2)sn ||x| − r0 |n−2 ∂Ω

(4.67)

∀x ∈ Rn \ Bn (0, r0 ) .

− − Thus limx→∞ vΩ [φ](x) = 0 and we conclude that vΩ [φ] is harmonic at infinity. If instead n = 2, then we note that  − vΩ [φ](x) = φ(y)S2 (x − y) dσy (4.68) ∂Ω      |x − y| 1 1 −1 dσy + (log |x|) = φ(y) log 1 + φ(y) dσy 2π ∂Ω |x| 2π ∂Ω

4.4 The Single Layer Potential

157

for all x ∈ Ω − \ {0}. We now show that     |x − y| −1 dσy = 0 . φ(y) log 1 + lim x→∞ ∂Ω |x|

(4.69)

Since limρ→0 ρ−1 log(1 + ρ) = 1, there exists δ ∈]0, 1[ such that |log(1 + ρ)| ≤

3 |ρ| 2

∀ρ ∈ [−δ, +δ] .

Moreover, we have    |x − y|  ||x − y| − |x|| |y| r0   ≤ ≤  |x| − 1 = |x| |x| |x|

∀y ∈ ∂Ω, x ∈ R2 \ B2 (0, r0 ) .

Then we choose r1 > r0 such that r0 /r1 < δ and we obtain    |x − y|  r0 r0    |x| − 1 ≤ |x| ≤ r1 < δ ,           log 1 + |x − y| − 1  ≤ 3  |x − y| − 1 ≤ 3 r0    2 |x|  |x| 2 |x| for all y ∈ ∂Ω, x ∈ R2 \ B2 (0, r1 ). Hence         3 r0 |x − y|   −1 dσy  ≤ φ(y) log 1 + |φ(y)| dσy  |x| 2 |x| ∂Ω ∂Ω for all x ∈ R2 \ B2 (0, r1 ). By taking the limit as x tends to infinity, we deduce that the left-hand  side has limit 0 as x tends to infinity. Hence, (4.69) holds. Now if ∂Ω φ dσ = 0, then (4.68) and (4.69) imply that lim v − [φ](x) x→∞ Ω

=0

− − and accordingly vΩ [φ] is harmonic at infinity. Conversely, if vΩ [φ] is harmonic at in− in a punctured neighborhood of infinity. Hence, (4.68) finity, then vΩ [φ] is bounded    and (4.69) imply that ∂Ω φ dσ = 0.

In the next theorem, we introduce a known formula for the first order partial derivatives of a single layer potential. Theorem 4.24. Let α ∈]0, 1[. Let Ω be a bounded open subset of Rn of class C 1,α . ˜ ∈ ∂Ω. Then the following two statements hold. Let φ ∈ C 0,α (∂Ω). Let x (i) Let j ∈ {1, . . . , n}. Then the principal value   ∂Sn ∂Sn p.v. (˜ x − y)φ(y) dσy ≡ lim (˜ x − y)φ(y) dσy →0 ∂x j ∂Ω ∂Ω\Bn (˜ x,) ∂xj

158

4 Green Identities and Layer Potentials

exists finite and we have the following jump formula for the first order partial derivatives of a single layer potential  ∂ ∂Sn 1 x)(νΩ )j (˜ lim vΩ [φ](˜ x+tνΩ (˜ x)) = ∓ φ(˜ x)+p.v. (˜ x−y)φ(y) dσy . 2 t→0∓ ∂xj ∂Ω ∂xj (ii) The function νΩ (˜ x) · ∇Sn (˜ x − y)φ(y) is integrable in y ∈ ∂Ω and we have the following jump formula for the normal derivative of a single layer potential  ∂ ± 1 x) + vΩ [φ](˜ x) = ∓ φ(˜ νΩ (˜ x) · ∇Sn (˜ x − y)φ(y) dσy . ∂νΩ 2 ∂Ω Proof. For a proof of (i), we refer to Miranda [212, 2.II p. 314] or to Cialdea [52, Theorem XI, p. 52], Hackbusch [117, Corollary 8.1.14, p. 280]. By Lemma 2.76 x) · ∇Sn (˜ x − y)φ(y) is integrable in y ∈ ∂Ω. By the (iii), the function νΩ (˜ L’Hˆopital’s Rule, we have ± v ± [φ](˜ x + tνΩ (˜ x)) − vΩ [φ](˜ x) ∂ ± vΩ [φ](˜ x) = lim∓ Ω ∂νΩ t t→0 ± [φ](˜ x + tνΩ (˜ x)) · νΩ (˜ x) . = lim∓ ∇vΩ t→0

Then statement (ii) follows by statement (i).

 

Then we have the following classical Schauder regularity result for the restriction of the single layer potential to Ω + ≡ Ω and to Ω − . Theorem 4.25. Let α ∈]0, 1[. Let Ω be a bounded open subset of Rn of class C 1,α . + + (i) If φ ∈ C 0,α (∂Ω), then vΩ [φ] belongs to C 1,α (Ω). Moreover, the map vΩ [·] is linear and continuous from C 0,α (∂Ω) to C 1,α (Ω). 1,α − [φ] belongs Cloc (Ω − ). Moreover, if r ∈]0, +∞[ (ii) If φ ∈ C 0,α (∂Ω), then vΩ − and Ω ⊆ Bn (0, r), then the map vΩ [·]|Bn (0,r)\Ω is linear and continuous from

C 0,α (∂Ω) to C 1,α (Bn (0, r) \ Ω). + Proof. (i) We already know that vΩ [φ] belongs to C 0 (Ω) ∩ C ∞ (Ω). Moreover, if we set μn ≡ max{n − 2, 1/2}, then we have μn ∈]0, n − 1[ and  dσy + |x − y|μn |Sn (x − y)| sup |φ| |vΩ [φ](x)| ≤ |x − y|μn ∂Ω ∂Ω  dσy μn (|Sn (a − b)| |a − b| ) sup sup |φ| . ≤ sup n 2 |x − y|μn ∂Ω x∈R (a,b)∈(∂Ω) , a=b ∂Ω

By the definition of Sn in (3.2) and by Lemma 4.21, we can verify that the first and + [·] is linear the second supremum in the right-hand side are finite. Accordingly, vΩ 0 0 and continuous from C (∂Ω) to C (Ω). n To complete the proof of (i), we observe that the kernels kj ≡ ∂S ∂xj are positively ∞ homogeneous of degree −(n − 1), odd, and of class C . Then Theorem 4.17 (i) of

4.4 The Single Layer Potential

Miranda implies that

159

+ ∂ ∂xj vΩ [φ] belongs 0,α

to C 0,α (Ω) for all j ∈ {1, . . . , n} and that

∂ the map from C 0,α (∂Ω) to C (Ω) that takes φ to ∂x v + [φ] ∈ C 0,α (Ω) is linear j Ω and continuous for all j ∈ {1, . . . , n}. Hence, statement (i) holds true. Then statement (ii) can be deduced applying statement (i) to the open bounded set Bn (0, r)\Ω of class C 1,α . Indeed the map from C 0,α (∂Ω) to C 0,α (∂(Bn (0, r)\Ω)) that takes φ to ! φ(x) if x ∈ ∂Ω , ˜ φ(x) ≡ 0 if x ∈ ∂Bn (0, r)

is linear and continuous and we have − vΩ [φ]|Bn (0,r)\Ω = vB+

n (0,r)\Ω

˜ [φ]

for all φ ∈ C 0,α (∂Ω).

 

When n ≥ 3, we have the following consequence of the previous statement. Corollary 4.26. Let n ∈ N, n ≥ 3, α ∈]0, 1[. Let Ω be a bounded open subset of − Rn of class C 1,α . Then the map that takes φ to vΩ [φ] is linear and continuous from 1,α 0,α − C (∂Ω) to Cb (Ω ). Proof. Let r ∈]0, +∞[ be such that Ω ⊆ Bn (0, r − 1). By Theorem 4.25 (ii), the − [·]|Bn (0,r)\Ω from C 0,α (∂Ω) to C 1,α (Bn (0, r) \Ω) is linear and continuous. map vΩ − We now show that the map vΩ [·]|Rn \Bn (0,r) is linear and continuous from 1,α 0,α n C (∂Ω) to Cb (R \ Bn (0, r)). By Lemma 2.48, Rn \ Bn (0, r) is regular in the sense of Whitney and thus Proposition 2.54 (i), (ii) implies that Cb2 (Rn \ Bn (0, r)) is continuously imbedded into Cb1,α (Rn \ Bn (0, r)). Thus it suffices to show that − [·]|Rn \Bn (0,r) is linear and continuous from C 0,α (∂Ω) to Cb2 (Rn \ Bn (0, r)). vΩ By the classical differentiability theorem for integrals depending on a parameter, by the equalities (3.2), (3.3) for Sn and ∇Sn , by inequality (3.4) for the partial derivatives of order two of Sn , and by the elementary inequality |x − y| ≥ |x| − (r − 1) ≥ r − (r − 1) = 1 we have − |Dη vΩ [φ](x)|

  = 

∀(x, y) ∈ (Rn \ Bn (0, r)) × (∂Ω) ,

  D Sn (x − y)φ(y) dσy  η

∂Ω

−1 −1 n ≤ max{(n − 2)−1 s−1 n , sn , ωn }mn−1 (∂Ω)φC 0,α (∂Ω) ∀x ∈ R \ Bn (0, r) − [·]|Rn \Bn (0,r) is for all β ∈ Nn such that |β| ≤ 2 and φ ∈ C 0,α (∂Ω). Hence, vΩ 0,α 2 n linear and continuous from C (∂Ω) to Cb (R \ Bn (0, r)). Next we note that if x ∈ Bn (0, r) \ Ω and y ∈ Rn \ Bn (0, r), then there exists ξ in the segment [x, y] such that ξ ∈ ∂Bn (0, r). Thus, if g belongs to Cb0 (Ω − ),

g|Bn (0,r)\Ω ∈ C 0,α (Bn (0, r) \ Ω) ,

and

g|Rn \Bn (0,r) ∈ Cb0,α (Rn \ Bn (0, r)) ,

160

4 Green Identities and Layer Potentials

then we have |g(x) − g(y)| ≤ |g(x) − g(ξ)| + |g(ξ) − g(y)| ≤ |g : Bn (0, r) \ Ω|α |x − ξ|α + |g : Rn \ Bn (0, r)|α |ξ − y|α ≤ max{|g : Bn (0, r) \ Ω|α , |g : Rn \ Bn (0, r)|α }(|x − ξ|α + |ξ − y|α ) ≤ 2 max{|g : Bn (0, r) \ Ω|α , |g : Rn \ Bn (0, r)|α }|x − y|α . − By applying the last inequality to the first order partial derivatives of vΩ [φ], we obtain − vΩ [φ]C 1,α (Ω − ) b

− − ≤ 2 max{vΩ [φ]|Bn (0,r)\Ω C 1,α (Bn (0,r)\Ω) , vΩ [φ]|Rn \Bn (0,r) C 1,α (Rn \Bn (0,r)) } b

− vΩ [·]|Rn \Bn (0,r)

for all φ ∈ C 0,α (∂Ω). Hence, the continuity of from C 0,α (∂Ω) 1,α − n [·] is to Cb (R \ Bn (0, r)) and Theorem 4.25 (ii) imply that the linear map vΩ 1,α 0,α   continuous from C (∂Ω) to Cb (Ω − ). Thus the proof is complete.

4.5 The Double Layer Potential Definition 4.27. Let α ∈]0, 1]. Let Ω be a bounded open subset of Rn of class C 1,α . If ψ ∈ C 0 (∂Ω), then we denote by wΩ [ψ] the double layer potential with moment (or density) ψ, i.e., the function from Rn to R defined by  ∂ wΩ [ψ](x) ≡ ψ(y) (Sn (x − y)) dσy ∀x ∈ Rn , ∂νΩy ∂Ω where ∂ (Sn (x − y)) ≡ −νΩ (y) · ∇Sn (x − y) ∂νΩy

∀(x, y) ∈ Rn × ∂Ω , x = y .

We note that if x ∈ Rn \ ∂Ω, then νΩ (y) · ∇Sn (x − y) is continuous in y ∈ ∂Ω and that accordingly ψ(y)νΩ (y) · ∇Sn (x − y) is integrable in y ∈ ∂Ω. If instead x ∈ ∂Ω, then Lemma 2.76 (iii) implies the existence of a constant cΩ,α > 0 such that |νΩ (y) · ∇Sn (x − y)| ≤

1 cΩ,α sn |x − y|(n−1)−α

∀y ∈ ∂Ω \ {x} .

(4.70)

Since ψ is measurable and essentially bounded, inequality (n − 1) − α < n − 1 ensures the integrability of the function ψ(y)νΩ (y) · ∇Sn (x − y) in y ∈ ∂Ω. We now introduce the following preliminary result.

4.5 The Double Layer Potential

161

Proposition 4.28. Let α ∈]0, 1]. Let Ω be a bounded open subset of Rn of class C 1,α . If ψ ∈ C 0 (∂Ω), then the following statements hold. (i) wΩ [ψ] is harmonic in Rn \ ∂Ω. (ii) limx→∞ wΩ [ψ](x) = 0 and accordingly wΩ [ψ] is harmonic at infinity. Proof. (i) Since Dxη (νΩ (y) · ∇Sn (x − y)ψ(y)) is continuous in (Rn \ ∂Ω) × ∂Ω for all η ∈ Nn , then the classical differentiation theorem for integrals depending on a parameter implies that wΩ [ψ] ∈ C ∞ (Rn \ ∂Ω) and that  η D wΩ [ψ](x) = − Dxη (νΩ (y) · ∇Sn (x − y)ψ(y)) dσy ∀x ∈ Rn \ ∂Ω ∂Ω

for all η ∈ Nn . Since Δx (νΩ (y) · ∇Sn (x − y)ψ(y)) = 0 for all x ∈ Rn \ ∂Ω and y ∈ ∂Ω, we conclude that wΩ [ψ] is harmonic in Rn \ ∂Ω. Now let r > 0 be such that Ω ⊆ Bn (0, r). Then we have     ∂  −  |ψ(y)|  (Sn (x − y)) dσy |wΩ [ψ](x)| ≤ ∂ν Ωy ∂Ω   |ψ(y)| 1 1 1 ≤ dσy ≤ |ψ(y)| dσy sn ∂Ω |x − y|n−1 sn ||x| − r|n−1 ∂Ω − for all x ∈ Rn \ Bn (0, r), and thus limx→∞ wΩ [ψ](x) = 0 and we conclude that −   wΩ [ψ] is harmonic at infinity.

We also note that the following known technical lemma holds (see Cialdea [51, p. 187], where (ii) is proved and interpreted in the theory of conjugate differential forms). Lemma 4.29. Let α ∈]0, 1]. Let Ω be a bounded open subset of Rn of class C 1,α . Then the following statements hold. (i) If ψ ∈ C 0,α (∂Ω), then we have wΩ [ψ](x) = −

  n  ∂ ψ(y)(νΩ (y))j Sn (x − y) dσy ∂xj ∂Ω j=1

for all x ∈ Rn \ ∂Ω. (ii) If ψ ∈ C 1,α (∂Ω), then we have  ∂ ∂ wΩ [ψ](x) = ∂xl ∂xj j=1 n



 Mlj [ψ](y)Sn (x − y) dσy ∂Ω

for all x ∈ Rn \ ∂Ω and l ∈ {1, . . . , n}. Proof. (i) Since Dxη (νΩ (y) · ∇Sn (x − y)ψ(y)) is continuous in (Rn \ ∂Ω) × ∂Ω for all η ∈ Nn , then the classical differentiation theorem for integrals depending on a parameter and the obvious equality

162

4 Green Identities and Layer Potentials

∂ ∂ (Sn (x − y)) = − (Sn (x − y)) ∂yj ∂xj

∀(x, y) ∈ Rn × Rn , x = y ,

imply the validity of statement (i). (ii) Since Sn (x − ·) is harmonic in Rn \ {x}, the above equality implies that ⎡ ⎤ n ∂ ∂ ⎣ (νΩ (y))j (Sn (x − y))⎦ ∂xl j=1 ∂yj =

n $ 

(νΩ (y))j

j=1

∂ ∂ − (νΩ (y))l ∂yl ∂yj

%

∂ (Sn (x − y)) ∂xj



for all x ∈ Rn \ ∂Ω and y ∈ ∂Ω. Now we fix x ∈ Rn \ ∂Ω. Since both ψ and ∂ 1 ∂xj Sn (x − ·) are of class C (∂Ω), the above equality, and Lemma 2.86, and the classical differentiation theorem for integrals depending on a parameter imply that ∂ wΩ [ψ](x) = ∂xl

 ψ(y) ∂Ω

=

=

j=1

$ Mjl,y

j=1

n   j=1 n 

n 

Mlj [ψ](y) ∂Ω

∂ ∂xj

% ∂ (Sn (x − y)) (y) dσy ∂xj

∂ Sn (x − y) dσy ∂xj

 Mlj [ψ](y)Sn (x − y) dσy ∂Ω

 

and thus the proof is complete.

Next we introduce the following known jump formula for the limiting value of the double layer potential on the boundary. Theorem 4.30. Let α ∈]0, 1]. Let Ω be a bounded open subset of Rn of class C 1,α . ˜ ∈ ∂Ω. Then we have the following jump formula for the Let ψ ∈ C 0,α (∂Ω). Let x double layer potential 1 x) + wΩ [ψ](˜ lim∓ wΩ [ψ](˜ x + tνΩ (˜ x)) = ± ψ(˜ x) 2 t→0 (cf. (2.1)). Proof. We only prove the formula for the limit in case t < 0, i.e., in the case where x ˜ + tνΩ (˜ x) ∈ Ω + = Ω. The proof for t > 0 follows same the lines and is accordingly omitted. By Lemma 4.29 (i), we have wΩ [ψ](x) = −

n  ∂ vΩ [ψ · (νΩ )j ](x) ∂x j j=1

∀x ∈ Ω .

4.5 The Double Layer Potential

163

Hence, the jump formula for the jth-partial derivative of a single layer potential of Theorem 4.24 (i) implies that x + tνΩ (˜ x)) = − lim wΩ [ψ](˜

t→0−

n  j=1

n $ 

lim

t→0−

∂ vΩ [ψ · (νΩ )j ](˜ x + tνΩ (˜ x)) ∂xj

%  ∂ 1 2 x)(νΩ (˜ x))j + Sn (˜ x − y)ψ(y)(νΩ (y))j dσy =− − ψ(˜ 2 ∂Ω ∂xj j=1   n 1 ∂ x) − = ψ(˜ Sn (˜ x − y)ψ(y)(νΩ (y))j dσy 2 ∂x j ∂Ω j=1 =

1 ψ(˜ x) + wΩ [ψ](˜ x) . 2  

Then we have the following classical Schauder regularity result for the restriction of the double layer potential to Ω + ≡ Ω and to the exterior Ω − of Ω. Theorem 4.31. Let α ∈]0, 1[. Let Ω be a bounded open subset of Rn of class C 1,α . + [ψ] of (i) If ψ ∈ C 1,α (∂Ω), then the restriction wΩ [ψ]|Ω extends to a function wΩ 1,α 1,α 1,α class C (Ω). Moreover, the map from C (∂Ω) to C (Ω) that takes ψ to + [ψ] is linear and continuous. wΩ − [ψ] (ii) If ψ ∈ C 1,α (∂Ω), then the restriction wΩ [ψ]|Ω − extends to a function wΩ 1,α − of Cloc (Ω ). Moreover, if r ∈]0, +∞[ and Ω ⊆ Bn (0, r), then the map from − [ψ]|Bn (0,r)\Ω is linear and C 1,α (∂Ω) to C 1,α (Bn (0, r) \ Ω) that takes ψ to wΩ continuous. ˜ ∈ ∂Ω, then (iii) If ψ ∈ C 1,α (∂Ω) and x + − νΩ (˜ x) · ∇wΩ [ψ](˜ x) = νΩ (˜ x) · ∇wΩ [ψ](˜ x)  n  ∂ p.v. (νΩ (˜ x))l Mlj [ψ](y) Sn (˜ x − y) dσy . = ∂xj ∂Ω l,j=1

Proof. (i) Since ψ ∈ C 1,α (∂Ω) ⊆ C 0,α (∂Ω) and (νΩ )j ∈ C 0,α (∂Ω), Theorem 4.25 (i) on the regularity of the single layer potential implies that the single layer + [ψ · (νΩ )j ] belongs to C 1,α (Ω). Then, by the formula of Lemma 4.29 potential vΩ (i), the function wΩ [ψ]|Ω equals the restriction to Ω of a function of class C 0,α (Ω). In particular, wΩ [ψ]|Ω admits a (unique) continuous extension to Ω and such an extension is of class C 0,α (Ω). Since the pointwise product by (νΩ )j is linear and continuous in C 0,α (∂Ω) and the Schauder regularity Theorem 4.25 (i) implies that + is linear and continuous from C 0,α (∂Ω) to C 1,α (Ω), the formula of Lemma 4.29 vΩ + [ψ] is linear (i) implies that the map from C 1,α (∂Ω) to C 0,α (Ω) that takes ψ to wΩ and continuous. If ψ ∈ C 1,α (∂Ω), then Mlj [ψ] ∈ C 0,α (∂Ω) and the Schauder regularity Theo+ [Mlj [ψ]] ∈ C 1,α (Ω). Then the formula of Lemma 4.29 rem 4.25 (i) implies that vΩ

164

4 Green Identities and Layer Potentials

∂ (ii) implies that ∂x wΩ [ψ]|Ω equals the restriction to Ω of a function of class l + 0,α [ψ] ∈ C 1,α (Ω). Since Mlj [·] is linear C (Ω) for all l ∈ {1, . . . , n}. Hence, wΩ + is linear and continuous from and continuous from C 1,α (∂Ω) to C 0,α (∂Ω) and vΩ + ∂ wΩ [·] is conC 0,α (∂Ω) to C 1,α (Ω), the formula of Lemma 4.29 (ii) implies that ∂x l 1,α 0,α tinuous from C (∂Ω) to C (Ω) for all l ∈ {1, . . . , n}. Then the above proved + + [·] implies that wΩ [·] is linear and continuous from C 1,α (∂Ω) to continuity of wΩ 1,α C (Ω). Then statement (ii) can be deduced applying statement (i) to the open bounded set Bn (0, r)\Ω of class C 1,α . Indeed the map from C 1,α (∂Ω) to C 1,α (∂(Bn (0, r)\Ω)) that takes ψ to ! ψ(x) if x ∈ ∂Ω , ˜ ψ(x) ≡ 0 if x ∈ ∂Bn (0, r)

is linear and continuous and we have − wΩ [ψ]|Bn (0,r)\Ω = −wB+

n (0,r)\Ω

˜ [ψ]

for all ψ ∈ C 1,α (∂Ω). (iii) By statement (i), by Lemma 4.29 (ii), by the jump formula for the derivatives of the single layer potential of Theorem 4.24, and by the skew-symmetry of the matrix (Mlj [ψ])l,j=1,...,n we have + x) · ∇wΩ [ψ](˜ x) = νΩ (˜

n 

(νΩ (˜ x))l lim−

l=1

=

n 

lim (νΩ (˜ x))l

t→0−

l,j=1 n 

t→0

∂ + w [ψ](˜ x + tνΩ (˜ x)) ∂xl Ω

∂ + v [Mlj [ψ]](˜ x + tνΩ (˜ x)) ∂xj Ω

1 (νΩ (˜ x))l (νΩ (˜ x))j Mlj [ψ](˜ x) 2 l,j=1  n  ∂ + (νΩ (˜ x))l p.v. Sn (˜ x − y)Mlj [ψ](y) dσy ∂x j ∂Ω l,j=1   n ∂ = p.v. (νΩ (˜ x))l Mlj [ψ](y) Sn (˜ x − y) dσy . ∂x j ∂Ω =−

lj=1

− The proof of the corresponding equality for wΩ [ψ] is similar and is accordingly omitted.  

By arguing exactly as in the proof of Corollary 4.26, we can verify the following consequence of the previous statement. Corollary 4.32. Let α ∈]0, 1[. Let Ω be a bounded open subset of Rn of class C 1,α . − [ψ] is linear and Then the map from C 1,α (∂Ω) to Cb1,α (Ω − ) that takes ψ to wΩ continuous.

4.6 A Regularizing Property of the Double Layer Potential on the Boundary

165

4.6 A Regularizing Property of the Double Layer Potential on the Boundary In this section, we present some results of Schauder [256, 257] on the restriction of the double layer potential on the boundary. For extensions to potentials corresponding to arbitrary second order elliptic operators with constant coefficients and for an account of other contributions on the topic, we refer to the paper [89] with Dondi. In particular, we wish to mention the contribution of von Wahl [272] in the frame of Sobolev spaces and the corresponding developments of Heinemann [120] for Schauder spaces. We begin with the following result of Schauder [256, Hilfsatz VII, p. 112]. For the sake of completeness, we include a proof. Theorem 4.33. Let α ∈]0, 1[. Let Ω be a bounded open subset of Rn of class C 1,α . Then the following statements hold. (i) Let ψ ∈ L∞ (∂Ω). Then the function WΩ [ψ] from ∂Ω to R defined by  WΩ [ψ](x) ≡ − ψ(y) νΩ (y) · ∇Sn (x − y) dσy ∀x ∈ ∂Ω (4.71) ∂Ω 0,α

belongs to C (∂Ω). (ii) The operator WΩ from L∞ (∂Ω) to C 0,α (∂Ω) that takes ψ to WΩ [ψ] is linear and continuous. Proof. We first prove statement (i). Since ψ ∈ L∞ (∂Ω), inequality (4.70) implies that the function ∂ν∂Ω (Sn (x − y))ψ(y) is Lebesgue integrable in y ∈ ∂Ω for all y x ∈ ∂Ω and the Vitali Convergence Theorem A.11 and Lemma 4.21 imply that WΩ [ψ] is continuous on ∂Ω. By inequality (4.70) and by Lemma 2.53 (i), we have sup |WΩ [ψ]| ≤ ∂Ω

cΩ,α cΩ,(n−1)−α sn

ψL∞ (∂Ω) .

(4.72)

We now turn to estimate the H¨older quotient of WΩ [ψ]. Let x , x ∈ ∂Ω. By Remark 2.20, there is no loss of generality in assuming that 0 < |x − x | ≤ 1. Then we have |WΩ [ψ](x ) − WΩ [ψ](x )| (4.73)     ∂ ≤  (Sn (x − y))ψ(y) dσy  ∂ν    Ωy B (x ,2|x −x |)∩∂Ω  n    ∂   + (Sn (x − y))ψ(y) dσy  B (x ,2|x −x |)∩∂Ω ∂νΩy  n  % $   ∂ ∂ + (Sn (x − y)) − (Sn (x − y)) ψ(y) dσy  . ∂νΩy ∂Ω\Bn (x ,2|x −x |) ∂νΩy By inequality (4.70) and Lemma 2.53 (ii) with λ ≡ (n − 1) − α, we have

166

4 Green Identities and Layer Potentials

   

Bn (x ,2|x −x |)∩∂Ω

   

≤ ψL∞ (∂Ω)

  ∂  (Sn (x − y))ψ(y) dσy  ∂νΩy

cΩ,α cΩ,(n−1)−α

Bn (x ,2|x −x |)∩∂Ω



≤ ≤

2α |x − x |α and sn   ∂ (Sn (x − y))ψ(y) dσy  ∂νΩy    ∂      ∂νΩ (Sn (x − y))|ψ(y)| dσy

Bn (x ,3|x −x |)∩∂Ω y cΩ,α cΩ,(n−1)−α α  ψL∞ (∂Ω) 3 |x sn

− x |α .

(4.74)

Now we consider the third term in the right-hand side of inequality (4.73), and we note that   % $   ∂ ∂    (Sn (x − y)) − (Sn (x − y)) ψ(y) dσy   ∂νΩy ∂Ω\Bn (x ,2|x −x |) ∂νΩy      1  νΩ (y) · (x − x ) (4.75) ≤ ψ(y) dσy    n sn ∂Ω\Bn (x ,2|x −x |) |x − y|   $ %  1 1 1   +  − (y) · (x − y)ψ(y) dσ ν Ω y .  n  n sn ∂Ω\Bn (x ,2|x −x |) |x − y| |x − y| By Lemmas 2.76 (iii), 2.77, and 2.53 (iii), we have    1  νΩ (y) · (x − x )  ψ(y) dσ y   n sn ∂Ω\Bn (x ,2|x −x |) |x − y|   1 |νΩ (y) − νΩ (x )||x − x | ≤ sn ∂Ω\Bn (x ,2|x −x |) |x − y|n  |νΩ (x ) · (x − x )| + |ψ(y)| dσy |x − y|n  ψL∞ (∂Ω) |νΩ |α |x − y|α ≤ sn |x − y|n ∂Ω\Bn (x ,2|x −x |)

(4.76)

cΩ,α |x − x |α dσy |x − x | |x − y|n  ψL∞ (∂Ω)  dσy |x − x |(|νΩ |α + cΩ,α ) ≤  n−α sn ∂Ω\Bn (x ,2|x −x |) |x − y| +

≤

ψL∞ (∂Ω) α−1  (|νΩ |α + cΩ,α )c |x − x |α . Ω,n−α 2 sn

By Lemma 2.52 (i), we have |x − y| ≤ 2|x − y|

4.6 A Regularizing Property of the Double Layer Potential on the Boundary

167

for all y ∈ ∂Ω \ Bn (x , 2|x − x |) and thus Lemma 2.52 (iii) with λ = −n, Lemma 2.53 (iii) with λ = n − α, and Lemma 2.76 imply that there exists q−n ∈ ]0, +∞[ such that   $ %  1 1  1  −  νΩ (y) · (x − y)ψ(y) dσy    n n sn ∂Ω\Bn (x ,2|x −x |) |x − y| |x − y|  ψL∞ (∂Ω) 1+α 1 ≤ 2 q−n |x − x |cΩ,α dσy  − y|n−α sn |x    ∂Ω\Bn (x ,2|x −x |) ≤

cΩ,α ψL∞ (∂Ω) 1+α α−1  2 q−n c |x − x |α . Ω,n−α 2 sn

(4.77)

Then, by combining inequalities (4.73)–(4.77), we deduce that WΩ [ψ] is α-H¨older continuous and that statement (i) holds. Statement (ii) follows by the same inequalities and by inequality (4.72) (see also Remark 2.20).   In particular, Theorem 4.33 implies that WΩ maps continuous functions to H¨older continuous functions. We note that if ψ ∈ C 0 (∂Ω), then WΩ [ψ](x) = wΩ [ψ](x)

∀x ∈ ∂Ω .

We now investigate what happens when we apply WΩ to continuously differentiable functions and to functions of class C 1,β , with β ∈]0, 1]. To do so, we first introduce an auxiliary operator in the following Lemma 4.34. In what follows, we denote by D∂Ω the subset of ∂Ω × ∂Ω consisting of the pairs (x, y) with x = y. Namely, D∂Ω ≡ {(x, x) ∈ ∂Ω × ∂Ω} .

Lemma 4.34. Let r ∈ {1, . . . , n}. Then the following statements hold. (i) Let θ ∈]0, 1]. Let Ω be a bounded open Lipschitz subset of Rn . Let (f, μ) belong to C 0,θ (Ω) × C 0 (∂Ω). Let Hr [f ] be the function from (Ω × ∂Ω) \ D∂Ω to R defined by Hr [f ](x, y) ≡ (f (x) − f (y))

∂ (Sn (x − y)) ∀(x, y) ∈ (Ω × ∂Ω) \ D∂Ω . ∂xr

If x ∈ Ω, then the function Hr [f ](x, ·) is Lebesgue integrable in ∂Ω and the function Qr [f, μ] from Ω to R defined by  Qr [f, μ](x) ≡ Hr [f ](x, y)μ(y) dσy ∀x ∈ Ω ∂Ω

is continuous. (ii) Let α ∈]0, 1[, and β, θ ∈]0, 1]. Let Ω be of class C 1,α . If (f, μ) belongs to C 0,θ (Ω) × C 0,β (∂Ω), then Qr [f, μ] ∈ C 0,min{α,β,θ} (Ω).

168

4 Green Identities and Layer Potentials

Proof. Since we have

∂Sn ∂xr (x

1 xr −yr sn |x−y|n

− y) =

|Hr [f ](x, y)| ≤

and f is H¨older continuous with exponent θ,

1 |f |θ sn |x − y|(n−1)−θ

∀(x, y) ∈ (Ω × ∂Ω) \ D∂Ω .

Then Hr [f ](x, ·) is Lebesgue integrable in ∂Ω for all x ∈ Ω and the Vitali Convergence Theorem A.11 and Lemma 4.21 imply that Qr [f, μ] is continuous in Ω. We now consider statement (ii). By treating separately the case where x ∈ ∂Ω and the case where x ∈ Ω and by exploiting the jump formulas for the partial derivatives of the single layer potential of Theorem 4.24 and the Schauder regularity Theorem 4.25 for the single layer potential, we have Qr [f, μ](x) = f (x)

∂ + ∂ + v [μ](x) − v [f μ](x) ∂xr Ω ∂xr Ω

∀x ∈ Ω .

Then, by the continuity of the pointwise product in H¨older spaces of Proposition 2.24 and by Theorem 4.25 (i), we have Qr [f, μ] ∈ C 0,min{α,β,θ} (Ω).   Then we can prove the following theorem on the action of Qr on the boundary. Theorem 4.35. Let r ∈ {1, . . . , n}. Let α ∈]0, 1[ and β ∈]0, 1]. Let Ω be a bounded open subset of Rn of class C 1,α . Then the following statements hold. (i) Let (g, μ) be a pair of functions in C 0,α (∂Ω) × C 0,β (∂Ω). Let Hr [g] be the function from (∂Ω × ∂Ω) \ D∂Ω to R defined by Hr [g](x, y) ≡ (g(x) − g(y))

∂ (Sn (x − y)) ∀(x, y) ∈ (∂Ω × ∂Ω) \ D∂Ω . ∂xr

If x ∈ ∂Ω, then the function Hr [g](x, ·) is Lebesgue integrable in ∂Ω, and the function Qr [g, μ] from ∂Ω to R defined by  Qr [g, μ](x) ≡ Hr [g](x, y)μ(y) dσy ∀x ∈ ∂Ω ∂Ω

belongs to C 0,α (∂Ω). (ii) The bilinear map Qr from C 0,α (∂Ω) × C 0,β (∂Ω) to C 0,α (∂Ω) that takes a pair (g, μ) to Qr [g, μ] is continuous. Proof. Let ρ ∈]0, +∞[ be such that Ω ⊆ Bn (0, ρ). Let ‘˜’ be the extension operator defined on C 1,α (∂Ω) as in Theorem 2.85 (ii) with Ω  = Bn (0, ρ), h = 0. Since g , μ] on ∂Ω, the integrability of Hr [g](x, ·) and the continuity of Qr [g, μ] = Qr [˜ Qr [g, μ] follow by Lemma 4.34 (i). Since ∂Sn 1 x r − yr (x − y) = ∂xr sn |x − y|n and g is H¨older continuous with exponent α, Lemma 2.53 (i) implies that

4.6 A Regularizing Property of the Double Layer Potential on the Boundary

sup |Qr [g, μ]| ≤

cΩ,(n−1)−α sn

∂Ω

|g|α μC 0 (∂Ω) .

169

(4.78)

We now turn to estimate the H¨older quotient of Qr [g, μ]. By the H¨older continuity of g, we have |(g(x ) − g(y))(xr − yr ) − (g(x ) − g(y))(xr − yr )| ≤ |g|α {|x − y|α |x − y| + |x − y|α |x − y|}

(4.79)

for all x , x , y ∈ ∂Ω. Now let x , x ∈ ∂Ω. Then we have |Qr [g, μ](x ) − Qr [g, μ](x )| (4.80)     Hr [g](x , y)μ(y) dσy  ≤  Bn (x ,2|x −x |)∩∂Ω     + Hr [g](x , y)μ(y) dσy  B (x ,2|x −x |)∩∂Ω  n       + [Hr [g](x , y) − Hr [g](x , y)]μ(y) dσy  . ∂Ω\Bn (x ,2|x −x |)

By Lemma 2.53 (ii) with λ ≡ (n − 1) − α, we have       Hr [g](x , y)μ(y) dσy   Bn (x ,2|x −x |)∩∂Ω

≤ μL∞ (∂Ω) and

   

|g|α  c 2α |x − x |α sn Ω,(n−1)−α

(4.81)

  Hr [g](x , y)μ(y) dσy  Bn (x ,2|x −x |)∩∂Ω      Hr [g](x , y)|μ(y)| dσy ≤   

Bn (x ,3|x −x |)∩∂Ω

≤ μL∞ (∂Ω)

|g|α  c 3α |x − x |α . sn Ω,(n−1)−α

(4.82)

Next we turn to estimate the third integral in the right-hand side of inequality (4.80). Since C 0,β1 (∂Ω) is continuously embedded into C 0,β2 (∂Ω) whenever 0 < β2 ≤ β1 , then we can assume that α + β < 1. We observe that the third integral in the right-hand side of inequality (4.80) equals J1 (x , x ) + μ(x )J2 (x , x ) where J1 (x , x )

(4.83)

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4 Green Identities and Layer Potentials

 ≡

∂Ω\Bn (x ,2|x −x |)

J2 (x , x ) ≡

[Hr [g](x , y) − Hr [g](x , y)](μ(y) − μ(x )) dσy ,



∂Ω\Bn (x ,2|x −x |)

[Hr [g](x , y) − Hr [g](x , y)] dσy .

In order to estimate J1 , we observe that     Hr [g](x , y) − Hr [g](x , y)   ≤

(4.84)

(4.85)

1 1 |(g(x ) − g(y))(xr − yr ) − (g(x ) − g(y))(xr − yr )|  sn |x − y|n     1 1 1 . + |(g(x ) − g(y))(xr − yr )|  − n  n sn |x − y| |x − y| 

Now, we have |x − x | ≤ |x − y| and Lemma 2.52 (i) implies that |x − y| ≤ 2|x − y|. Then, combining Lemma 2.52 (iii) with λ = −n and inequality (4.79), we conclude that there exists q−n ∈]0, +∞[ such that the right-hand side of (4.85) is less or equal to |g|α |x − y|α |x − y| + |x − y|α |x − y| (4.86) sn |x − y|n |g|α  + |x − y|α+1 q−n |x − x ||x − y|−n−1 sn 21+α |g|α 1 ≤ (2 + q−n )  . sn |x − y|(n−1)−α Then Lemma 2.53 (iii) with λ = n − (α + β) and inequalities (4.85), (4.86) imply that     J1 (x , x ) (4.87)    21+α |g|α 1 ≤ |μ|β (2 + q−n )  dσy (n−1)−α−β s |x − y|    n ∂Ω\Bn (x ,2|x −x |) ≤

21+α (2 + q−n ) α+β  |g|α |μ|β c |x − x |α+β . Ω,n−(α+β) 2 sn

We now consider J2 (x , x ) and we note that  ∂ (g(x) − g(y)) (Sn (x − y)) dσy (4.88) Qr [g, 1](x) = ∂xr ∂Ω  ∂ ∂ = g(x)p.v. (Sn (x − y)) dσy − p.v. (Sn (x − y))g(y) dσy ∂Ω ∂xr ∂Ω ∂xr for all x ∈ ∂Ω, where the last two integrals exist in the sense of the principal value. Since Ω is of class C 1,α and g ∈ C 0,α (∂Ω), the following equality holds

4.6 A Regularizing Property of the Double Layer Potential on the Boundary

∂ + 1 vΩ [g](x) = − (νΩ )r (x)g(x) + p.v. ∂xr 2

 ∂Ω

171

∂ (Sn (x − y))g(y) dσy (4.89) ∂xr

for all x ∈ ∂Ω (see Theorems 4.24, 4.25). By the Schauder regularity Theorem 4.25 + (i) for the single layer potential, the function ∂x∂ r vΩ [g](x) of the variable x ∈ ∂Ω 0,α 0,α is of class C . Since (νΩ )r g ∈ C (∂Ω), then equality (4.89) implies that the principal value  ∂ p.v. (Sn (x − y))g(y) dσy ∂Ω ∂xr defines a function of class C 0,α in the variable x ∈ ∂Ω. The same clearly holds if we replace g with the constant 1. Hence, equalities (4.88) and (4.89) imply that Qr [g, 1] ∈ C 0,α (∂Ω). Next we note that |J2 (x , x )| ≤ |Qr [g, 1](x ) − Qr [g, 1](x )|       + Hr [g](x , y) dσy  B (x ,2|x −x |)∩∂Ω  n    + Hr [g](x , y) dσy  .   

(4.90)

Bn (x ,2|x −x |)∩∂Ω

Since Bn (x , 2|x − x |) ⊆ Bn (x , 3|x − x |), inequalities (4.81) and (4.82) with μ = 1 imply that |J2 (x , x )| ≤ |Qr [g, 1] : ∂Ω|α |x −x |α +2

|g|α  c 3α |x −x |α . (4.91) sn Ω,(n−1)−α

Combining inequalities (4.87) and (4.91), we conclude that        [Hr [g](x , y) − Hr [g](x , y)]μ(y) dσy  

(4.92)

∂Ω\Bn (x ,2|x −x |)  1+α

2

(2 + q−n ) α+β |g|α |μ|β c (diam Ω)β Ω,n−(α+β) 2 sn $ % |g|α  +μL∞ (∂Ω) |Qr [g, 1] : ∂Ω|α + 31+α cΩ,(n−1)−α |x − x |α . sn ≤

Combining inequalities (4.78), (4.80), (4.81), (4.82), and (4.92), we conclude that Qr [g, μ] is α-H¨older continuous and that statement (ii) holds true.   We now exploit Theorem 4.33 to prove a classical formula for the tangential derivatives of WΩ . Proposition 4.36. Let α ∈]0, 1[, β ∈]0, α]. Let Ω be an open bounded subset of Rn of class C 1,α . If ψ ∈ C 1,β (∂Ω), then WΩ [ψ] ∈ C 1,β (∂Ω) and  n   Mij [WΩ [ψ]] = Qr [(νΩ )i , Mjr [ψ]] − Qr [(νΩ )j , Mir [ψ]] r=1

+WΩ [Mij [ψ]]

on ∂Ω ,

(4.93)

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4 Green Identities and Layer Potentials

for all i, j ∈ {1, . . . , n}. Proof. In this proof we find convenient to simplify our notation and set νi ≡ (νΩ )i

∀i ∈ {1, . . . , n} .

By the jump formula of Theorem 4.30, by the Schauder regularity Theorem 4.31 for the double layer potential, and by the formula of Lemma 4.29 (ii) for the partial + derivatives of the double layer potential in Ω, we have wΩ [ψ] ∈ C 1,β (Ω) and 1 + WΩ [ψ] = − ψ + wΩ [ψ]|∂Ω 2 n  ∂ + ∂ + wΩ [ψ] = v [Mij [ψ]] ∂xi ∂xj Ω j=1

on ∂Ω , in Ω

for all i ∈ {1, . . . , n}. In particular, WΩ [ψ] ∈ C 1,β (∂Ω) (cf. Theorem 4.31 (i)). Hence, Mij [WΩ [ψ]]

(4.94) 1 + [ψ]|∂Ω ] = − Mij [ψ] + Mij [wΩ 2 ∂ + ∂ + 1 = − Mij [ψ] + νi wΩ [ψ] − νj w [ψ] 2 ∂xj ∂xi Ω  n   ∂ + ∂ + 1 = − Mij [ψ] + v [Mjr [ψ]] − νj v [Mir [ψ]] , νi 2 ∂xr Ω ∂xr Ω r=1

on ∂Ω. By the jump formulas for the partial derivatives of the single layer potential of Theorem 4.24, we have  ∂ + ∂ 1 vΩ [φ](x) = − νr (x)φ(x)+p.v. (Sn (x−y))φ(y) dσy ∀x ∈ ∂Ω , ∂xr 2 ∂Ω ∂xr for all φ ∈ C 0,β (∂Ω). Hence, Mij [WΩ [ψ]](x) 1 1 = − Mij [ψ](x) − 2 2   n  + νi (x)p.v. r=1



− νj (x)p.v. ∂Ω

 n r=1

Mjr [ψ](x)νr (x)νi (x) −

n 

(4.95)  Mir [ψ](x)νr (x)νj (x)

r=1

∂ (Sn (x − y))Mjr [ψ](y) dσy ∂x r ∂Ω  ∂ (Sn (x − y))Mir [ψ](y) dσy ∀x ∈ ∂Ω . ∂xr

We now consider the first term in braces in the right-hand side of (4.95) and we note that

4.6 A Regularizing Property of the Double Layer Potential on the Boundary

 n

Mjr [ψ]νr νi −

r=1

n 

173

 Mir [ψ]νr νj

(4.96)

r=1

 % ∂ψ ∂ψ 2 ∂ψ 2 ∂ψ νr νi − νr νi − νi νr νj − νr νj = νj ∂xr ∂xj ∂xr ∂xi r=1 % n $  ∂ψ ∂ψ νi + νr2 νj = −νr2 ∂xj ∂xi r=1 n $ 

= −Mij [ψ]

on ∂Ω .

Next we consider the second term in braces in the right-hand side of (4.95) and we note that νi (x)Mjr [ψ](y) − νj (x)Mir [ψ](y)

(4.97)

= [νi (x)Mjr [ψ](y) − νi (y)Mjr [ψ](y)] + [νi (y)Mjr [ψ](y) − νj (y)Mir [ψ](y)] +[νj (y)Mir [ψ](y) − νj (x)Mir [ψ](y)] ∀x, y ∈ ∂Ω and that [νi Mjr [ψ] − νj Mir [ψ]] ∂ψ ∂ψ ∂ψ ∂ψ − νi νr − νj νi + νj νr = −νr Mij [ψ] = νi νj ∂xr ∂xj ∂xr ∂xi

(4.98) on ∂Ω .

Then, combining (4.95)–(4.98), we deduce that Mij [WΩ [ψ]](x) n   ∂ = (Sn (x − y))Mjr [ψ](y)(νi (x) − νi (y)) dσy ∂x r r=1 ∂Ω n   ∂ − (Sn (x − y))Mij [ψ](y)νr (y) dσy ∂xr r=1 ∂Ω n   ∂ + (Sn (x − y))Mir [ψ](y)(νj (y) − νj (x)) dσy ∂x r r=1 ∂Ω =

n 

Qr [νi , Mjr [ψ]](x) + WΩ [Mij [ψ]](x)

r=1 n 

−

Qr [νj , Mir [ψ]](x)

∀x ∈ ∂Ω

r=1

and thus formula (4.93) holds true.

 

Formulas as (4.93) are known and could be deduced by others in the literature. In particular, we mention Cialdea [51, Proof of Theorem I, p. 186], Cialdea et al. [54,

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4 Green Identities and Layer Potentials

(11) p. 70]. See also Cialdea et al. [55, (66), p. 7] for the double layer hydrodynamic potential. For a generalization of formula (4.93) for the tangential derivatives of the double layer potential corresponding to second order elliptic homogeneous operators, we refer to Hofmann et al. [125, (6.2.6)]. For inhomogeneous second order elliptic operators, we refer to the paper [89, (9.2)] with Dondi. For the double layer potential corresponding to the fundamental solution of the heat equation, we refer to the papers [172, 173] with Luzzini. We now exploit formula (4.93) for the tangential derivatives of WΩ to prove that the operator WΩ has a regularizing property on functions of class C 1,β (∂Ω). We do so by means of the following known result (see also Giraud [108, 109], Miranda [213, Theorem. 15.VI p. 42], and the paper [89, Theorem. 9.2] with Dondi). Theorem 4.37. Let α ∈]0, 1[, β ∈]0, α]. Let Ω be an open bounded subset of Rn of class C 1,α . If ψ ∈ C 1,β (∂Ω), then WΩ [ψ] ∈ C 1,α (∂Ω). Moreover, the operator WΩ from C 1,β (∂Ω) to C 1,α (∂Ω) that takes ψ to WΩ [ψ] is linear and continuous. Proof. We denote by Tij [ψ] the right-hand side of formula (4.93) for the tangential derivatives of WΩ [ψ]. By Lemma 2.87 (iii), it suffices to prove that the following two statements hold. (j) WΩ is continuous from C 1,β (∂Ω) to C 0 (∂Ω). (jj) Tij [·] is continuous from C 1,β (∂Ω) to C 0,α (∂Ω) for all i, j ∈ {1, . . . , n}. Theorem 4.33 (ii) implies the validity of statement (j). Since Mij is linear and continuous from C 1,β (∂Ω) to C 0,β (∂Ω) and C 0,β (∂Ω) is continuously embedded into L∞ (∂Ω), Theorem 4.33 (ii) implies that WΩ [Mij [·]] is linear and continuous from C 1,β (∂Ω) to C 0,α (∂Ω). By the membership of (νΩ )i , (νΩ )j in C 0,α (∂Ω) and by Theorem 4.35 (ii), the operator Qr [(νΩ )i , ·] − Qr [(νΩ )j , ·] is linear and continuous from C 0,β (∂Ω) to C 0,α (∂Ω). Since Mij is linear and continuous from C 1,β (∂Ω) to C 0,β (∂Ω), we conclude that the operator Qr [(νΩ )i , Mjr [·]] − Qr [(νΩ )j , Mir [·]] is linear and continuous from C 1,β (∂Ω) to C 0,α (∂Ω), and accordingly statement (jj) holds true.  

Chapter 5

Preliminaries on the Fredholm Alternative Principle

Abstract In the chapters that follow this one, we will exploit certain results of Wendland (Math. Z., 101:61–64, 1967; Methoden Verfahren Math. Phys. 3:141– 176,1970), on Fredholm operators in dual systems. The present chapter’s purpose is to introduce these results. We also recall some classical theorems of Functional Analysis in Banach spaces. We do not include proofs, however, as these can be found in monographs such as those of Kress (Linear integral equations, volume 82 of Applied Mathematical Sciences. Springer, New York, third edition, 2014.), Wloka et al. (Boundary value problems for elliptic systems. Cambridge University Press, Cambridge, 1995), as well as in the above-mentioned papers of Wendland.

5.1 Fredholm Alternative Let X be a normed space on R. We denote by X  the topological dual space L(X, R) of X, so that X  consists of the linear continuos operators from X to R. As usual, the elements of X  are also called functionals. Definition 5.1. Let X and Y be normed spaces. If L is a linear map from X to Y , then the transpose Lt of L is the linear map from Y  to X  that takes a functional ψ ∈ Y  to the functional Lt [ψ] ∈ X  defined by  t  L [ψ] [φ] = ψ [L[φ]] ∀φ ∈ X. As is well known, for every bounded operator L there exists a unique transpose operator Lt and it is bounded. In addition, if L is compact (i.e. if it maps bounded subsets of X to subsets of Y with compact closure), then also Lt is compact. We now introduce the definition of Fredholm operator. Definition 5.2 (Fredholm Operator). We say that a bounded linear operator L from a Banach space X to a Banach space Y is Fredholm if and only if both the following conditions hold. © Springer Nature Switzerland AG 2021 M. Dalla Riva et al., Singularly Perturbed Boundary Value Problems, https://doi.org/10.1007/978-3-030-76259-9 5

175

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5 Preliminaries on the Fredholm Alternative Principle

(i) The null space Ker L has finite dimension. (ii) The co-kernel Y /Im L has finite dimension. If L is a Fredholm operator, then the index of L is the integer defined by index L ≡ dim Ker L − dim(Y /Im L) . So for example, if a bounded operator L is an isomorphism from X onto Y , then L is a Fredholm operator and has index equal to zero. We observe that the condition that Y /Im L has finite dimension implies that the image Im L of L is closed in Y (see, e.g., Wloka et al. [276, Prop. 9.2]). We shall exploit the following theorems (for a proof we refer, e.g., to Wloka et al. [276, Chap. 9]). Theorem 5.3 (Compact Perturbation of a Fredholm Operator). Let X and Y be Banach spaces. If L is a Fredholm operator from X to Y and K is a compact operator from X to Y , then L + K is a Fredholm operator from X to Y and index (L + K) = index L. So for example, if a bounded operator L is an isomorphism from X onto Y and K is a compact operator from X to Y , then L + K is a Fredholm operator of index zero. Theorem 5.4 (Fredholm Alternative). Let X and Y be Banach spaces. Let L be a Fredholm operator of index 0 from X to Y . Then exactly one of the following statements hold true. (i) The operator L is an isomorphism (i.e., a linear homeomorphism) from X onto Y and the transpose operator Lt is an isomorphism from Y  onto X  . (ii) The null spaces Ker L and Ker Lt have the same nonzero finite dimension and Im L = {φ ∈ Y : ψ[φ] = 0 for all ψ ∈ Ker Lt } , Im Lt = {ψ ∈ X  : ψ[φ] = 0 for all φ ∈ Ker L} .

5.2 Fredholm Alternative in a Dual System In the following Definitions 5.5, 5.6, and 5.7 we introduce the notion of bilinear form, dual system, and transpose with respect to a dual system, respectively. Definition 5.5. Let X1 and X2 be real Banach spaces. A map ·, · from X1 × X2 to R is a bilinear form if c1 ψ1 + c2 ψ2 , φ1  = c1 ψ1 , φ1  + c2 ψ2 , φ1  and ψ1 , c1 φ1 + c2 φ2  = c1 ψ1 , φ1  + c2 ψ1 , φ2 

5.2 Fredholm Alternative in a Dual System

177

for all c1 , c2 ∈ R, ψ1 , ψ2 ∈ X1 , and φ1 , φ2 ∈ X2 . A bilinear form is non-degenerate if for all ψ ∈ X1 \ {0} there exists φ ∈ X2 such that ψ, φ = 0 and for all φ ∈ X2 \ {0} there exists ψ ∈ X1 such that ψ, φ = 0. Definition 5.6. A dual system consists of a pair of Banach spaces X1 and X2 and of a non-degenerate bilinear form ·, · from X1 × X2 to R and it is denoted by X1 , X2 . So for example, X2 , X2  together with the evaluation map that takes a pair (ψ, φ) of X2 × X2 to ψ[φ] is a dual system, the so-called canonical dual system between X2 and its dual X2 . Definition 5.7. Let X1 , X2  be a dual system. Let A be a linear operator from X1 to itself. Let B be a linear operator from X2 to itself. Then the operator A is said to be transpose to the operator B with respect to the dual system X1 , X2  if A[ψ], φ = ψ, B[φ] for all ψ ∈ X1 and all φ ∈ X2 . So for example, if X1 = X2 , then B t is transpose to B with respect to the canonical dual system X2 , X2 . Then, in a dual system we have the following Fredholm Alternative Theorem. For a proof, we refer the reader to Wendland [273], [274] and to Kress [149, Chapt. 4]. Theorem 5.8 (Fredholm Alternative in a Dual System). Let X1 , X2  be a dual system. Let K1 be a linear compact operator from X1 to itself. Let K2 be a linear compact operator from X2 to itself. Assume that K1 is transpose to K2 with respect to the dual system X1 , X2 . Then exactly one of the following statements holds true. (i) The operator IX1 + K1 is an isomorphism from X1 to itself and the operator IX2 + K2 is an isomorphism from X2 to itself. (ii) The null spaces Ker(IX1 + K1 ) and Ker(IX2 + K2 ) have the same nonzero finite dimension and Im(IX1 + K1 ) = {ψ ∈ X1 : ψ, φ = 0 for all φ ∈ Ker(IX2 + K2 )} , Im(IX2 + K2 ) = {φ ∈ X2 : ψ, φ = 0 for all ψ ∈ Ker(IX1 + K1 )} .

Chapter 6

Boundary Value Problems and Boundary Integral Operators

Abstract In this chapter we show the existence of solutions to the basic boundary value problems for the Laplace equation using a classical approach based on Potential Theory. Specifically, we consider the Dirichlet problem, the Neumann problem, the Robin problem, the transmission problem, and a mixed problem. To do so, we carry out an analysis of the boundary integral operators associated with the single and double layer potentials in a Schauder space setting. Our presentation of the topic stems from that of Folland (Introduction to partial differential equations. Princeton University Press, Princeton, NJ, second edition, 1995, Chap. 3).

6.1 The Geometric Setting We introduce here some notation that will be used all throughout the chapter. We fix a natural number n ∈ N, n ≥ 2 , a real number α ∈]0, 1[ , and an open bounded subset Ω of Rn of class C 1,α (cf. Section 2.13). As usual, we denote by Ω − ≡ Rn \ Ω the exterior of Ω. Moreover, we assume that Ω has κ + connected components and that Ω − has κ − + 1 connected components. Then, the (bounded) connected components of Ω are denoted by Ω1 , . . . , Ωκ + , the unbounded connected component of Ω − is denoted by (Ω − )0 , and the bounded connected components of Ω − are denoted by (Ω − )1 , . . . , (Ω − )κ − (see Lemma 2.38). We also remark that, when no misunderstanding is possible, we will use the symbol I instead of IC 0,α (∂Ω) or IC 1,α (∂Ω) for the identity operator from C 0,α (∂Ω) to itself and from C 1,α (∂Ω) to itself, respectively. © Springer Nature Switzerland AG 2021 M. Dalla Riva et al., Singularly Perturbed Boundary Value Problems, https://doi.org/10.1007/978-3-030-76259-9 6

179

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6 Boundary Value Problems and Boundary Integral Operators

6.2 The Dirichlet and Neumann Boundary Value Problems We plan to study the following boundary value problems for the Laplace operator in the bounded open set Ω and on its exterior Ω − . The interior Dirichlet boundary value problem. Given g ∈ C 1,α (∂Ω), find u in C 1,α (Ω) ∩ C 2 (Ω) such that ! Δu = 0 in Ω , u=g on ∂Ω . The exterior Dirichlet boundary value problem. Given g ∈ C 1,α (∂Ω), find u in 1,α (Ω − ) ∩ C 2 (Ω − ) such that Cloc ! Δu = 0 in Ω − , u=g on ∂Ω − = ∂Ω . The interior Neumann boundary value problem. Given g ∈ C 0,α (∂Ω), find u in C 1,α (Ω) ∩ C 2 (Ω) such that ! Δu = 0 in Ω , ∂u on ∂Ω . ∂νΩ = g The exterior Neumann boundary value problem. Given g ∈ C 0,α (∂Ω), find u in 1,α (Ω − ) ∩ C 2 (Ω − ) such that Cloc ! Δu = 0 in Ω − , ∂u on ∂Ω − = ∂Ω . ∂νΩ = g Later, we will consider other boundary value problems. We observe that the requirement that the solutions of the problems above are of class C 2 in the interior of their domain could be omitted, provided that we understand the equation Δu = 0 in the sense of distributions. Indeed, by Weyl’s Lemma 3.14, continuous functions that satisfy the (weak) Laplace equation Δu = 0 are harmonic, and therefore real analytic, in the interior of their domain (see also Theorem 3.13). We will henceforth omit to write explicitly the membership in C 2 1,α of the interior and we will just write that the solutions are of class C 1,α (or Cloc ) on the closure of their domain. In particular, we look for solutions that satisfy the boundary conditions in the classical sense. In the next section we investigate the uniqueness of the solutions of the above boundary value problems.

6.3 Uniqueness for the Interior and Exterior Dirichlet and Neumann Boundary. . .

181

6.3 Uniqueness for the Interior and Exterior Dirichlet and Neumann Boundary Value Problems We first make some remarks on the uniqueness for the Dirichlet problem. As we have already seen, we have a uniqueness theorem based on the Maximum Principle for the interior Dirichlet problem (cf. Theorem 3.6). We have also seen that for the exterior Dirichlet problem, we cannot expect uniqueness (cf. Example 3.7). We can prove, however, a uniqueness result for the exterior Dirichlet problem provided that we restrict the class of solutions by imposing a condition at infinity as in the following statement. Proposition 6.1. Let u1 , u2 ∈ C 0 (Ω − ). If u1 , u2 are harmonic in Ω − and harmonic at infinity and if u1 = u2 on ∂Ω − , then u1 = u2 in Ω − . Proof. Possibly scaling and translating, we can assume that Bn (0, 1) ⊆ Ω. Then we apply an inversion with respect to the sphere ∂Bn (0, 1) to the set Ω − and obtain a set (Ω − )∗ such that (Ω − )∗ ⊆ Bn (0, 1) (cf. Definition 3.19 with r = 1). Moreover, U ≡ (Ω − )∗ ∪ {0} is an open subset of Rn containing 0, and ∂U = (∂Ω)∗ . Since u1 , u2 are harmonic at infinity, we know that the Kelvin transforms   x u ˜j (x) ≡ |x|2−n uj ∀j = 1, 2 , ∀x ∈ U \ {0} , |x|2 have a removable singularity at zero, and we denote the corresponding harmonic extensions to U by the same symbols u ˜1 , u ˜2 (cf. Definition 3.21). Since u1 = u2 on ∂Ω − , we have ˜2 on ∂U , u ˜1 = u and thus u ˜1 = u ˜2 in U by the uniqueness of the interior Dirichlet problem of Theo  rem 3.6. Accordingly, u1 = u2 in Ω − . Incidentally, we observe that in the proof of Proposition 6.1 we did not use the C 1,α regularity of Ω and the statement could be proved without any regularity assumption on Ω. We now consider the Neumann problem. Since by adding a constant to a solution of a Neumann problem we obtain another solution of the same problem, we cannot expect uniqueness, but we can prove the following. Theorem 6.2. Let g ∈ C 0,α (∂Ω). Then the following statements hold. (i) If u1 , u2 ∈ C 1,α (Ω) solve the interior Neumann problem, then u1 − u2 is constant in each connected component of Ω. In particular, all solutions of the interior Neumann problem in C 1,α (Ω) can be obtained by adding to u1 an arbitrary function which is constant on the closure of each connected component of Ω.

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6 Boundary Value Problems and Boundary Integral Operators

1,α (ii) If u1 , u2 ∈ Cloc (Ω − ) solve the exterior Neumann problem, and if u1 , u2 are harmonic at infinity, then u1 − u2 is constant in each connected component of Ω − . Moreover, if n ≥ 3, then u1 − u2 = 0 in the unbounded connected component (Ω − )0 . In particular, all the solutions of the exterior Neumann problem 1,α (Ω − ) which are harmonic at infinity can be obtained by adding to u1 in Cloc an arbitrary function which is constant on the closure of each connected component of Ω − and equals zero in (Ω − )0 when n ≥ 3.

Proof. We present two different proofs of statement (i), the first one based on the Hopf-Ole˘ınik Lemma 3.15 and the second one based on an energy estimate. So, let u ≡ u1 − u2 and let Ω1 be a connected component of Ω. We claim that u is constant on Ω1 . Indeed, if u is not constant then, by Corollary 3.5, it attains its maximum on a point x0 of the boundary ∂Ω and, by the strong Maximum Principle 3.4, we have u(x) < u(x0 ) for all x ∈ Ω1 . Accordingly, the Hopf-Ole˘ınik Lemma 3.15 implies that ∂u (x0 ) < 0 . ∂νΩ This inequality yields a contradiction, because ∂u ∂u1 ∂u2 (x0 ) = (x0 ) − (x0 ) = 0 . ∂νΩ ∂νΩ ∂νΩ We now prove statement (i) with an energy estimate. By applying the first Green Identity of Theorem 4.2 to the function u ≡ u1 − u2 , we obtain   ∂u 0= u dσ = |∇u|2 dx . ∂Ω ∂νΩ Ω Hence, |∇u| = 0 in Ω and thus u is constant in each connected component of Ω. Finally, we show statement (ii) with an energy argument. As for (i), we set u ≡ u1 − u2 . Since u is harmonic at infinity, the first Green Identity for exterior domains of Corollary 4.7 implies that   ∂u |∇u|2 dx = − u dσ = 0 . Ω− ∂Ω ∂νΩ Since |∇u| is continuous in Ω − , we have ∇u = 0 in Ω − and thus u is constant in each connected component of Ω − . Moreover, if n ≥ 3 the assumption that u is   harmonic at infinity implies that u must vanish in (Ω − )0 (cf. Theorem 3.22). On account of the previous Proposition 6.1 and of statement (ii) of Theorem 6.2, we will consider the exterior Dirichlet and Neumann problems with the condition of harmonicity at infinity on the solutions. For a mixed problem with a Dirichlet condition on a non-empty portion of the boundary and a Neumann condition on the rest of it, one can prove uniqueness results that resemble those for the Dirichlet problem. We need, however, to introduce some assumptions on the connectedness of Ω and Ω − .

6.4 The Boundary Integral Operators Associated to the Single and Double Layer Potentials 183

Proposition 6.3. Let Σ ⊆ ∂Ω, Σ = 0. Let f ∈ C 1,α (∂Ω) and g ∈ C 0,α (∂Ω). Then the following statements hold. (i) Assume that u1 , u2 ∈ C 1,α (Ω) solve the mixed boundary value problem ⎧ ⎪ ⎨Δu = 0 in Ω , u=f on Σ , ⎪ ⎩ ∂u = g on ∂Ω \ Σ . ∂νΩ If Ω is connected, then u1 = u2 . 1,α (Ω − ) solve the (ii) Assume that u1 , u2 ∈ Cloc problem ⎧ Δu = 0 ⎪ ⎪ ⎪ ⎨u = f ∂u ⎪ =g ⎪ ⎪ ⎩ ∂νΩ u is harmonic at infinity .

exterior mixed boundary value in Ω − , on Σ , on ∂Ω \ Σ ,

If Ω − is connected, then u1 = u2 . Proof. Let u1 , u2 be as in (i). Since (u1 − u2 ) ∂ν∂Ω (u1 − u2 ) vanishes on the whole of ∂Ω, the energy argument used in the proof of Theorem 6.2 implies that the difference u1 − u2 is locally constant on Ω. Then the condition that Ω is connected implies that u1 −u2 is constant on Ω. Since there exists at least a point x ∈ Σ ⊆ ∂Ω where u1 (x) = u2 (x) = f (x), it follows that u1 − u2 = 0 in Ω and statement (i) holds true. Statement (ii) can be verified by a similar argument.  

6.4 The Boundary Integral Operators Associated to the Single and Double Layer Potentials In this section we introduce the boundary integral operators that we use to study the boundary value problems of Section 6.2. We begin with VΩ [φ] ≡ vΩ [φ]|∂Ω

∀φ ∈ C 0 (∂Ω) .

Then Theorem 4.25 implies the validity of the following. Theorem 6.4. The linear operator VΩ from C 0,α (∂Ω) to C 1,α (∂Ω) that takes φ to VΩ [φ] is bounded. Then we set WΩ [ψ] ≡ wΩ [ψ]|∂Ω

∀ψ ∈ C 1,α (∂Ω) ,

i.e.,  WΩ [ψ](x) ≡ −

ψ(y) νΩ (y) · ∇Sn (x − y) dσy ∂Ω

∀x ∈ ∂Ω

184

6 Boundary Value Problems and Boundary Integral Operators

for all ψ ∈ C 1,α (∂Ω) (see also (4.71)). By Theorems 4.30 and 4.31, the operator WΩ maps C 1,α (∂Ω) to itself and is bounded. We observe that in Definition 4.27 the operator WΩ was defined in a larger domain. In this chapter and in the rest of the book, however, we will always understand that, if not otherwise specified, the domain of WΩ is C 1,α (∂Ω). In the following theorem, we introduce a transpose of WΩ . Theorem 6.5. The transpose operator of WΩ with respect to the dual system C 0,α (∂Ω), C 1,α (∂Ω) induced by the non-degenerate bilinear form that takes a pair (φ, ψ) ∈ C 0,α (∂Ω)× C 1,α (∂Ω) to  φ, ψ =

φψ dσ

(6.1)

∂Ω

is the operator WΩt from C 0,α (∂Ω) to itself defined by the formula  WΩt [φ](x) ≡ φ(y) νΩ (x) · ∇Sn (x − y) dσy ∀x ∈ ∂Ω

(6.2)

∂Ω

for all φ ∈ C 0,α (∂Ω). Proof. By Theorem 4.24, the integral which defines WΩt [φ](x) is convergent for all x ∈ ∂Ω. By the jump formula for the normal derivative of a single layer potential of Theorem 4.24, we have WΩt [φ] =

∂ + 1 vΩ [φ] + φ ∂νΩ 2

∀φ ∈ C 0,α (∂Ω) .

(6.3)

+ By Theorem 4.25 (i), the operator vΩ [·] is bounded from C 0,α (∂Ω) to C 1,α (Ω) and accordingly the right-hand side of the above equation (6.3) defines a bounded operator from C 0,α (∂Ω) to C 0,α (∂Ω). By Lemma 2.76 (iii), there exists cΩ,α > 0 such that

|φ(x)ψ(y)νΩ (x) · ∇Sn (x − y)| cΩ,α sup∂Ω |φ| sup∂Ω |ψ| ≤ |x − y|(n−1)−α

∀(x, y) ∈ (∂Ω)2 , x = y .

Then the inequality (n − 1) − α < (n − 1) and the Fubini-Tonelli Theorem imply that   φ, WΩ [ψ] = − φ(x) νΩ (y) · ∇Sn (x − y)ψ(y) dσy dσx ∂Ω ∂Ω   = ψ(y) νΩ (y) · ∇Sn (y − x)φ(x) dσx dσy = WΩt [φ], ψ ∂Ω

∂Ω

6.4 The Boundary Integral Operators Associated to the Single and Double Layer Potentials 185

for all (φ, ψ) ∈ C 0,α (∂Ω) × C 1,α (∂Ω). Hence, WΩt is transpose to WΩ as in the statement.   Then we can rewrite the jump formulas of Theorems 4.24 (ii), 4.30, and 4.31 (iii) in terms of the boundary operators WΩ and WΩt . Theorem 6.6. Let φ ∈ C 0,α (∂Ω) and ψ ∈ C 1,α (∂Ω), then the following equalities hold on ∂Ω. 1 + [φ]|∂Ω = − φ + WΩt [φ] , νΩ · ∇vΩ 2 1 − νΩ · ∇vΩ [φ]|∂Ω = φ + WΩt [φ] , 2 1 + wΩ [ψ]|∂Ω = ψ + WΩ [ψ] , 2 1 − wΩ [ψ]|∂Ω = − ψ + WΩ [ψ] , 2 + − [ψ]|∂Ω = νΩ · ∇wΩ [ψ]|∂Ω . νΩ · ∇wΩ Theorem 6.6 relates the operators ± 21 I + WΩ and their transpose operators ± 12 I + WΩt to the boundary conditions of the interior and exterior Dirichlet and Neumann problems of Section 6.2. Our plan now is to use these operators and the Fredholm Theory of Chapter 5 to deduce existence results for the boundary value problems. As a first step, we show that WΩ and WΩt are compact in C 1,α (∂Ω) and C 0,α (∂Ω), respectively, and that accordingly we can apply the Fredholm Alternative of Theorem 5.8 in the dual systems C 0,α (∂Ω) , C 1,α (∂Ω) with K1 = 2WΩ and K2 = 2WΩt . To prove that WΩ and WΩt are compact we exploit the compact embedding of C 0,α (∂Ω) in C 0,β (∂Ω) and of C 1,α (∂Ω) in C 1,β (∂Ω) which holds for all β ∈]0, α[ and the regularizing properties of WΩ and WΩt . Indeed, we have already proved that the operator WΩ has a regularizing effect on the functions of L∞ (∂Ω) and C 1,β (∂Ω) (cf. Theorems 4.33 and 4.37). We now show that also the transpose operator WΩt has a regularizing effect on C 0,β (∂Ω). Namely, we prove the following. Theorem 6.7. Let β ∈]0, α]. Then the operator WΩt is linear and continuous from C 0,β (∂Ω) to C 0,α (∂Ω). Proof. If μ ∈ C 0,β (∂Ω), then we have  t WΩ [μ](x) = νΩ (x) · ∇Sn (x − y)μ(y) dσy ∂Ω

=

n   l=1

+

((νΩ )l (x) − (νΩ )l (y)) ∂Ω

n   l=1

=

n  l=1

(νΩ )l (y) ∂Ω

∂Sn (x − y)μ(y) dσy ∂xl

∂Sn (x − y)μ(y) dσy ∂xl

Ql [(νΩ )l , μ](x) − WΩ [μ](x)

∀x ∈ ∂Ω

186

6 Boundary Value Problems and Boundary Integral Operators

(for the definition of Ql , see Theorem 4.35). Since the components of νΩ are of class C 0,α , the validity of the statement follows by Theorem 4.35, by the continuity   of the embedding of C 0,β (∂Ω) into L∞ (∂Ω), and by Theorem 4.33. For an extension of the previous statement to potentials corresponding to the fundamental solution of an arbitrary elliptic operator with constant coefficients, we refer to the paper [89, Theorem 10.1] with Dondi. By Theorems 4.37 and 6.7 on the regularizing properties of WΩ and WΩt and by the compactness of the embeddings of C 1,α (∂Ω) into C 1,β (∂Ω) and of C 0,α (∂Ω) into C 0,β (∂Ω) for all β ∈]0, α[ (cf. Proposition 2.25 and Theorem 2.83), we can deduce the validity of the following classical result of Schauder [256, 257]. Theorem 6.8. The following statements hold. (i) The operator WΩ is compact from C 1,α (∂Ω) to itself. (ii) The operator WΩt is compact from C 0,α (∂Ω) to itself. We conclude this section with the following Theorems 6.9 and 6.10. The statements are an immediate consequence of the third Green Identities in Ω and Ω − of Corollary 4.6 and Theorem 4.10, respectively, and of the jump formulas of Theorem 6.6. Theorem 6.9. If u ∈ C 1,α (Ω) is harmonic in Ω then ⎧ ⎪ ⎨u(x) wΩ [u|∂Ω ](x) − vΩ [νΩ · ∇u|∂Ω ](x) = 12 u(x) ⎪ ⎩ 0

if x ∈ Ω , if x ∈ ∂Ω , if x ∈ Ω − .

1,α Theorem 6.10. Let u ∈ Cloc (Ω − ) be harmonic in Ω − and be harmonic at infinity. Let u∞ ≡ limx→∞ u(x). (Note that u∞ = 0 if n ≥ 3.) Then ⎧ ⎪ if x ∈ Ω − , ⎨u(x) 1 −wΩ [u|∂Ω ](x) + vΩ [νΩ · ∇u|∂Ω ](x) + u∞ = 2 u(x) if x ∈ ∂Ω , ⎪ ⎩ 0 if x ∈ Ω .

6.5 The Null Spaces of 12 I + WΩ and 12 I + WΩt As mentioned in the previous section, we plan to use the operators ± 12 I + WΩ and ± 12 I + WΩt and the Fredholm Alternative of Theorem 5.8 to obtain existence results for the interior and exterior Dirichlet and Neumann boundary value problems and accordingly we want to better understand the properties of these operators. If, for example, we can prove that 12 I +WΩ is an isomorphism (that is, a linear homeomorphim) from C 1,α (∂Ω) to itself, then the jump formula for the double layer potential

6.5 The Null Spaces of

1 I 2

+ WΩ and

1 I 2

t + WΩ

187

of Theorem 6.6 guarantees that a solution of the interior Dirichlet problem can be found in the form + [ψ] u = wΩ for some ψ ∈ C 1,α (∂Ω). We shall, however, see that 12 I + WΩ is an isomorphism only when Ω − is connected and, in general, the operators ± 12 I + WΩ and ± 12 I + WΩt have nontrivial null spaces. Therefore, we will exploit statement (ii) of the Fredholm Alternative Theorem 5.8 –instead of statement (i)– and, to handle the corresponding orthogonality conditions, we will have to characterize the null spaces of these operators. In this section we consider the operators with the plus sign, that is + 12 I + WΩ and + 12 I + WΩt , which are transpose to one another. Then, in the next Section 6.6, we turn to − 12 I + WΩ and − 12 I + WΩt , which are also transpose to one another. Our plan is the following. First we show that VΩ maps Ker( 12 I + WΩt ) into Ker( 12 I + WΩ ), then we prove that VΩ is actually an isomorphism (i.e., a linear homeomorphism) from Ker( 12 I + WΩt ) to Ker( 12 I + WΩ ), and finally we will be ready to introduce an explicit characterization of Ker( 12 I + WΩ ). We begin with the following technical lemma. Lemma 6.11. Let φ ∈ C 0,α (∂Ω). Then     1 I + WΩt [φ] dσ = φ dσ . 2 ∂Ω ∂Ω Proof. We have        1 1 I + WΩt [φ] dσ = φ dσ + − I + WΩt [φ] dσ 2 2 ∂Ω ∂Ω ∂Ω and



 ∂Ω

  1 + t νΩ · ∇vΩ [φ] dσ = 0 − I + WΩ [φ] dσ = 2 ∂Ω  

by the first Green Identity 4.2 (see also Theorem 6.6). We now show that VΩ maps

Ker( 12 I

+

WΩt )

into

Ker( 12 I

+ WΩ ).

Proposition 6.12. If μ ∈ Ker( 12 I + WΩt ) then VΩ [μ] ∈ Ker( 12 I + WΩ ).  Proof. If μ ∈ Ker( 12 I + WΩt ) then Lemma 6.11 implies that ∂Ω μ dσ = 0 and − accordingly vΩ [μ] is harmonic at infinity and has limit equal to 0 at infinity (cf. Theorem 4.23). Then the third Green Identity of Theorem 6.10 implies that − −wΩ [VΩ [μ]](x) + vΩ [νΩ · ∇vΩ [μ]|∂Ω ](x) = 0

for all x ∈ Ω. Then, by Theorem 4.31 on the continuous extendibility of the double layer to Ω, we have + + − −wΩ [VΩ [μ]](x) + vΩ [νΩ · ∇vΩ [μ]|∂Ω ](x) = 0

188

6 Boundary Value Problems and Boundary Integral Operators

for all x ∈ Ω and, by the membership of μ in Ker( 12 I +WΩt ) and the jump formulas in Theorem 6.6, it follows that   1 I + WΩ [VΩ [μ]] 2 + + − + 1 [VΩ [μ]]|∂Ω = vΩ [νΩ · ∇vΩ [μ]|∂Ω ]|∂Ω = vΩ [ μ + WΩt [μ]] = 0 . = wΩ 2   Then we prove that VΩ is an isomorphism from Ker( 12 I+WΩt ) to Ker( 12 I+WΩ ). Proposition 6.13. The map from Ker( 12 I + WΩt ) to Ker( 12 I + WΩ ) that takes μ to VΩ [μ] is an isomorphism. Proof. By Proposition 6.12, VΩ maps Ker( 12 I +WΩt ) into Ker( 12 I +WΩ ). By Theorem 6.8, WΩ and WΩt are compact, and thus the Fredholm Alternative Theorem 5.8 implies that the null spaces Ker( 12 I + WΩt ) and Ker( 12 I + WΩ ) have the same finite dimension. As a consequence, it suffices to show that the map that takes μ to VΩ [μ] is injective to prove that it is an isomorphism. If μ ∈ Ker( 12 I + WΩt ) and + [μ] = 0 on the whole VΩ [μ] = 0, then by the uniqueness Theorem 3.6 we have vΩ + of Ω and accordingly νΩ · ∇vΩ [μ] = 0 on ∂Ω. Moreover, by the jump formula for the normal derivative of a single layer of Theorem 6.6 and by the membership of μ − [μ] = 0. Then, again by the jump formula for in Ker( 12 I + WΩt ), we have νΩ · ∇vΩ the normal derivatives of a single layer of Theorem 6.6, we deduce that − + μ = νΩ · ∇vΩ [μ] − νΩ · ∇vΩ [μ] = 0 .

  We are now ready to describe the null space of 12 I + WΩ . Theorem 6.14. If κ − > 0, then the null space Ker( 12 I + WΩ ) consists of the functions from ∂Ω to R that are constant on ∂(Ω − )j for all j ∈ {1, . . . , κ − } and that are identically equal to 0 on ∂(Ω − )0 . If κ − = 0, then Ker( 12 I + WΩ ) = {0}. Proof. If ψ ∈ Ker( 12 I + WΩ ), then Proposition 6.13 implies that there exists μ ∈ Ker( 12 I + WΩt ) such that ψ = VΩ [μ]. Then the jump formula for the normal − [μ] = 0 derivative of a single layer potential of Theorem νΩ · ∇vΩ  that  6.6 implies 1 t on ∂Ω. Moreover, Lemma 6.11 implies that ∂Ω μ dσ = ∂Ω ( 2 I + WΩ )μ dσ = 0. − Then, by Theorem 4.23, vΩ [μ] vanishes at infinity. As a consequence, the unique− [μ] is locally ness Theorem 6.2 (ii) for the exterior Neumann problem implies that vΩ − − − constant on Ω and equal to 0 on (Ω )0 . Accordingly, when κ = 0 we have that ψ = VΩ [μ] is 0 and we conclude that Ker( 12 I + WΩ ) is trivial. If instead κ − > 0, then ψ = VΩ [μ] is constant on ∂(Ω − )j for all j ∈ {1, . . . , κ − } and equal to 0 on ∂(Ω − )0 . It remains to show that, for κ − > 0, all functions that are constant on ∂(Ω − )j for all j ∈ {1, . . . , κ − } and equal to 0 on ∂(Ω − )0 belong to Ker( 12 I + WΩ ). So,

6.5 The Null Spaces of

1 I 2

+ WΩ and

1 I 2

t + WΩ

189

let ψ be one such function. Then there exists a function u constant on (Ω − )j for all j ∈ {1, . . . , κ − } and equal to 0 on (Ω − )0 such that ψ = u|∂Ω . Hence ∇u = 0 and, by the Green Identity in Ω − of Theorem 6.10, we have −wΩ [ψ](x) = −wΩ [u|∂Ω ](x) + vΩ [νΩ · ∇u|∂Ω ](x) = 0

∀x ∈ Ω .  

+ [ψ]|∂Ω = 0 by the jump formulas of Theorem 6.6. Then ( 12 I + WΩ )[ψ] = wΩ

We observe that Proposition 6.13 and Theorem 6.14 imply that Ker( 12 I + WΩ ) and Ker( 12 I + WΩt ) have dimension κ − . Since WΩ and WΩt are compact (cf. Theorem 6.8), the Fredholm Alternative of Theorem 5.8 implies the validity of the following. Corollary 6.15. If κ − = 0, then 12 I + WΩ is an isomorphism from C 1,α (∂Ω) onto itself and 12 I + WΩt is an isomorphism from C 0,α (∂Ω) onto itself. We conclude this section with the following result, which shows that there are no nonzero elements of Ker( 12 I + WΩt ) that are orthogonal to Ker( 12 I + WΩ ).  Theorem 6.16. Assume that κ − > 0. If φ ∈ Ker( 12 I + WΩt ) and ∂(Ω − )j φ dσ = 0 for all j in {1, . . . , κ − } (i.e., φ is orthogonal to Ker( 12 I + WΩ )), then φ = 0.  Proof. Let φ ∈ Ker( 12 I + WΩt ) and ∂(Ω − )j φ dσ = 0 for all j ∈ {1, . . . , κ − }. By the membership of φ in Ker( 12 I + WΩt ) and by the jump properties of the single layer potential of Theorem 6.6, we have − νΩ · ∇vΩ [φ] = 0 and

+ νΩ · ∇vΩ [φ] = −φ .

(6.4)

Moreover, by the characterization of Ker( 12 I + WΩ ) of Theorem 6.14 we have  φψ dσ = 0 for all ψ ∈ Ker( 12 I + WΩ ). Accordingly, the Fredholm Alternative ∂Ω Theorem 5.8 implies that there exists μ ∈ C 0,α (∂Ω) such that   1 t I + WΩ [μ] . φ= 2 By the jump properties of the single layer potential of Theorem 6.6 it follows that − νΩ · ∇vΩ [μ] = φ .

(6.5)

In addition, Lemma 6.11 together with the membership of φ in Ker( 12 I + WΩt ) implies that ∂Ω φ dσ = 0 and that      1 t I + WΩ [μ] dσ = μ dσ = φ dσ = 0 . 2 ∂Ω ∂Ω ∂Ω − − [φ] and vΩ [μ] are harmonic at infinity (cf. TheoHence, both the single layers vΩ rem 4.23), and thus the second Green Identity for exterior domains of Corollary 4.8 implies that

190

6 Boundary Value Problems and Boundary Integral Operators

 ∂Ω

− − − − (νΩ · ∇vΩ [φ]) vΩ [μ] − vΩ [φ] (νΩ · ∇vΩ [μ]) dσ = 0 .

(6.6)

− + Now, by (6.4)–(6.6), by equality vΩ [φ]|∂Ω = vΩ [φ]|∂Ω , and by the first Green Identity in Ω, we deduce that  − − − − 0= (νΩ · ∇vΩ [φ]) vΩ [μ] − vΩ [φ] (νΩ · ∇vΩ [μ]) dσ ∂Ω   − − − vΩ [φ] (νΩ · ∇vΩ [μ]) dσ = − vΩ [φ]φ dσ =− ∂Ω ∂Ω    + 2 + + ∇v [φ] dx . vΩ [φ] (νΩ · ∇vΩ [φ]) dσ = = Ω ∂Ω

Ω

+ + Hence, vΩ [φ] is locally constant on Ω. Thus νΩ · ∇vΩ [φ] = 0 and by the second equality of (6.4) we deduce that φ = 0.  

As a consequence of Theorem 6.16, we have the following Corollary 6.17 (for a similar statement in L2 (∂Ω) we refer to Folland [102, Corollary 3.39]). Corollary 6.17. We have  C

0,α

(∂Ω) = Im 

and C 1,α (∂Ω) = Im

1 I + WΩt 2

1 I + WΩ 2



 ⊕ Ker



 ⊕ Ker

1 I + WΩt 2

1 I + WΩ 2

 (6.7)

 ,

(6.8)

where the sums are direct but not necessarily orthogonal. Proof. By Theorems 5.3 and 6.8 (ii) we can see that 12 I + WΩt is a Fredholm operator of index zero and, by Theorem 6.14, the null space of 12 I + WΩt is a closed subspace C 0,α (∂Ω) of dimension κ − . We deduce that the image of 12 I + WΩt has 0,α (see Section 5.1). Then, to prove (6.7), co-dimension κ − and is closed    1 in C t (∂Ω) it suffices to show that Im 2 I + WΩ and Ker 12 I + WΩt have a trivial inter- section, a fact that follows from Theorem 6.16 and by noting that Im 12 I + WΩt 0,α coincides  1 witht the subspace of C (∂Ω) of the functions that are orthogonal to Ker 2 I + WΩ (see also the Fredholm Alternative Theorem 5.8). 1 To prove (6.8) we observe that, by Theorems 5.3 and 6.8  (i), 2 I + WΩ is a 1 Fredholm operator of index zero. Moreover, Ker 2 I + WΩ is closed and has dimension κ − by Theorem 6.14. Then the image of 12 I + WΩ is a closed subspace of C 1,α κ − (see Section 5.1). So, it remains to show that 1  1(∂Ω) andhas co-dimension Im 2 I + WΩ and Ker 2 I + WΩ have a trivial intersection. Let     1 1 φ ∈ Im I + WΩ ∩ Ker I + WΩ . 2 2

t 6.6 The Null Spaces of − 21 I + WΩ and − 21 I + WΩ

191

0,α Since φ ∈ C 1,α (∂Ω) implies  that there exist (and are  1 ⊆ C t (∂Ω), equality (6.7)  unique) φ1 ∈ Im 2 I + WΩ and φ2 ∈ Ker 12 I + WΩt such that φ = φ1 + φ2. But then φ1 , φ = ∂Ω φ1 φdσ = 0 by the membership of φ in Ker 12I + WΩ  and φ2 , φ = ∂Ω φ2 φdσ = 0 by the membership of φ in Im 12 I + WΩ (see also the Fredholm Alternative of Theorem 5.8). Hence, φ, φ = 0 and φ = 0.  

6.6 The Null Spaces of − 12 I + WΩ and − 12 I + WΩt We turn to the operator − 12 I + WΩ and its transpose − 12 I + WΩt . Our plan to describe their null spaces is similar to the one used in Section 6.5 and accordingly we begin by showing that the image of Ker(− 12 I + WΩt ) under VΩ is contained in Ker(− 12 I + WΩ ). In this case, however, we can prove that VΩ restricts to an isomorphism between Ker(− 12 I + WΩt ) and Ker(− 12 I + WΩ ) only when the dimension n is ≥ 3. For n = 2 we can just show that VΩ is injective on the subspace of Ker(− 12 I + WΩt ) of the functions with zero integral mean. Then, to obtain an explicit description of Ker(− 12 I +WΩ ) we will pass through an intermediate proposition where we show that Ker(− 12 I +WΩ ) is the direct sum of the space of constant functions on ∂Ω and of the image under VΩ of the subspace of Ker(− 12 I + WΩt ) consisting of functions with zero integral mean. The reason why this extra difficulty appears in dimension n = 2 is that for a function μ ∈ Ker(− 12 I + WΩt ) we cannot  − [μ] is harmonic at infinity unless ∂Ω μdσ = 0 (cf. Theorem 4.23). claim that vΩ − Instead, if μ ∈ Ker( 12 I + WΩt ), then Lemma 6.11 ensures that vΩ [μ] is harmonic at infinity, a fact that we could exploit in Section 6.5 but not in this section. Then we begin with the following. Proposition 6.18. If μ ∈ Ker(− 12 I + WΩt ) then VΩ [μ] ∈ Ker(− 12 I + WΩ ). + Proof. By applying the third Green Identity of Theorem 6.9 to the function vΩ [μ] in Ω, we have + [μ]|∂Ω ](x) = 0 wΩ [VΩ [μ]](x) − vΩ [νΩ · ∇vΩ

for all x ∈ Ω − . Then, by Theorem 4.31 we have − − + wΩ [VΩ [μ]](x) − vΩ [νΩ · ∇vΩ [μ]|∂Ω ](x) = 0

for all x ∈ Ω − and, by the membership of μ in Ker(− 12 I + WΩt ) and the jump formulas in Theorem 6.6, it follows that   1 − [VΩ [μ]]|∂Ω − I + WΩ [VΩ [μ]] = wΩ 2 $  % 1 + = VΩ [νΩ · ∇vΩ [μ]|∂Ω ] = VΩ − I + WΩt [μ] = 0 . 2  

192

6 Boundary Value Problems and Boundary Integral Operators

In addition, if n ≥ 3, then VΩ is an isomorphism between Ker(− 12 I + WΩt ) and Ker(− 12 I + WΩ ). Proposition 6.19. If n ≥ 3, then the map from Ker(− 12 I +WΩt ) to Ker(− 12 I +WΩ ) that takes μ to VΩ [μ] is an isomorphism. Proof. By Proposition 6.18, VΩ maps Ker(− 12 I + WΩt ) into Ker(− 12 I + WΩ ). By Theorem 6.8, WΩ and WΩt are compact, and thus the Fredholm Alternative of Theorem 5.8 implies that the null spaces Ker(− 12 I + WΩt ) and Ker(− 12 I + WΩ ) have the same finite dimension. As a consequence, it suffices to show that the map that takes μ to VΩ [μ] is injective to show that it is an isomorphism. If μ ∈ − [μ] is harmonic Ker(− 12 I +WΩt ) and VΩ [μ] = 0, then Theorem 4.23 implies that vΩ at infinity and the uniqueness of the solution of the exterior Dirichlet problem of − [μ] = 0 on the whole of Ω − . Accordingly, we have Proposition 6.1 implies that vΩ − νΩ · ∇vΩ [μ] = 0. Then the jump formulas of Theorem 6.6 and the membership of μ in Ker(− 12 I + WΩt ) imply that − + μ = νΩ · ∇vΩ [μ] − νΩ · ∇vΩ [μ] = 0 .

  1

Let us recall the definition of formula (2.2): if X is a subspace of L (∂Ω), then X0 is the subspace of X consisting of the functions which have 0 integral mean. In general, for n ≥ 2, we have to restrict VΩ to Ker(− 12 I + WΩt )0 to obtain an injective map. Proposition 6.20. The map from Ker(− 12 I + WΩt )0 to Ker(− 12 I + WΩ ) that takes μ to VΩ [μ] is injective.  Proof. By the membership of μ in Ker(− 12 I + WΩt )0 we have ∂Ω μ dσ = 0, and − thus limx→∞ vΩ [μ](x) = 0 by Theorem 4.23. Then the proof follows by the same argument of the proof of Proposition 6.19.   In Proposition 6.23 we will describe Ker(− 12 I + WΩ ) as the direct sum of the space of constant functions on ∂Ω and of VΩ [Ker(− 12 I + WΩt )0 ]. Meanwhile, we introduce two preliminary lemmas. In the first one we show that locally constant functions on Ω have a restriction to ∂Ω that belongs to Ker(− 12 I + WΩ ). Lemma 6.21. The null space Ker(− 12 I + WΩ ) contains the functions from ∂Ω to R which are constant on ∂Ωj for all j ∈ {1, . . . , κ + }. Proof. If ψ is constant on ∂Ωj for all j ∈ {1, . . . , κ + }, then there exists a function u that is constant on Ωj for all j ∈ {1, . . . , κ + } and such that ψ = u|∂Ω . Hence, by the third Green Identity of Theorem 6.9 we have wΩ [ψ](x) = wΩ [ψ](x) − vΩ [νΩ · ∇u|∂Ω ](x) = 0 − [ψ](x) = 0 for all x ∈ Ω − (cf. Theorem 4.31). for all x ∈ Ω − and accordingly wΩ − 1 Then we have (− 2 I + WΩ )ψ = wΩ [ψ]|∂Ω = 0 by the jump formulas in Theorem 6.6.  

t 6.6 The Null Spaces of − 21 I + WΩ and − 21 I + WΩ

193

Moreover, we have the following. Lemma 6.22. If φ ∈ Ker(− 12 I + WΩt )0 and VΩ [φ] is constant on ∂Ω, then φ = 0. Proof. By the jump formula for the normal derivative of a single layer potential of Theorem 6.6, we have   1 − t I + WΩ [φ] = φ . νΩ · ∇v [φ]|∂Ω = (6.9) 2  − [φ] is harmonic at infinity, and Since ∂Ω φ dσ = 0, Theorem 4.23 implies that vΩ thus the first Green Identity for exterior domains of Corollary 4.7 implies that    − − 2 |∇vΩ [φ]| dx = − VΩ [φ] νΩ · ∇vΩ [φ] dσ = −VΩ [φ] φ dσ = 0 . Ω−

∂Ω

∂Ω

It follows that v − [φ] is locally constant on Ω − . Then equality (6.9) implies that φ = 0.   Then we are ready to prove the following. Proposition 6.23. The null space Ker(− 12 I + WΩ ) is the topological direct sum of $   % 1 t VΩ Ker − I + WΩ 2 0 and of the space of constant functions from ∂Ω to R. 1 t Proof. The closed subspace Ker(− 12 I + WΩt )0 of Ker(−  2 I + WΩ ) is defined by the vanishing of the linear functional that takes φ to ∂Ω φ dσ, therefore it has co-dimension smaller than or equal to one. By Theorem 6.8, WΩ and WΩt are compact, and thus the Fredholm Alternative of Theorem 5.8 implies that the null spaces Ker(− 12 I + WΩt ) and Ker(− 12 I + WΩ ) have the same finite dimension. Then by Proposition 6.20, the subspace Ker(− 12 I + WΩt )0 of Ker(− 12 I + WΩt ) and the subspace VΩ [Ker(− 12 I + WΩt )0 ] of Ker(− 12 I + WΩ ) have the same (finite) dimension and co-dimension. In particular, VΩ [Ker(− 12 I + WΩt )0 ] has codimension less than or equal to one in Ker(− 12 I + WΩ ). By Lemma 6.21, the constant functions on ∂Ω are contained in Ker(− 12 I + WΩ ). Instead, Lemma 6.22 implies that VΩ [Ker(− 12 I + WΩt )0 ] does not contain non-trivial constant functions. Since the space of constant functions on ∂Ω has dimension one, we conclude that VΩ [Ker(− 12 I + WΩt )0 ] has co-dimension exactly equal to one and that Ker(− 12 I + WΩ ) is the algebraic direct sum of VΩ [Ker(− 12 I + WΩt )0 ] and of the space of constant functions. Finally, since all the spaces involved have finite dimension, they are closed and the algebraic direct sum is topological by Theorem 2.6 (see also Theorem 2.3).  

In the following Theorem 6.24 we obtain an explicit description of the null space of − 12 I + WΩ .

194

6 Boundary Value Problems and Boundary Integral Operators

Theorem 6.24. The null space Ker(− 12 I + WΩ ) consists of the functions from ∂Ω to R which are constant on ∂Ωj for all j ∈ {1, . . . , κ + }. Proof. If ψ ∈ Ker(− 12 I + WΩ ), then Proposition 6.23 implies that there exist μ ∈ Ker(− 12 I + WΩt )0 and a constant function ρ on ∂Ω such that ψ = VΩ [μ] + ρ. Then, by the jump properties of the single layer potential (cf. Theorem 6.6) we have + [μ] = 0 on ∂Ω and, by the first Green Identity of Theorem 4.2, it follows νΩ · ∇vΩ + that vΩ [μ] is locally constant on Ω. Accordingly, ψ = VΩ [μ] + ρ is constant on ∂Ωj for all j ∈ {1, . . . , κ + }. Conversely, Lemma 6.21 implies that every function on ∂Ω which is constant on the ∂Ωj ’s belongs to Ker(− 12 I + WΩ ). Thus, the theorem is proved.   We observe that Theorem 6.24 implies that the null spaces Ker(− 12 I + WΩ ) and Ker(− 12 I + WΩt ) have dimension κ + (see also Theorems 5.8 and 6.8). We conclude this subsection with the following theorem, which shows that there are no nonzero elements of Ker(− 12 I + WΩt ) orthogonal to Ker(− 12 I + WΩ ).  Theorem 6.25. If φ ∈ Ker(− 12 I +WΩt ) and ∂Ωj φ dσ = 0 for all j ∈ {1, . . . , κ + } (i.e., φ is orthogonal to Ker(− 12 I + WΩ )), then φ = 0.  Proof. Let φ ∈ Ker(− 12 I + WΩt ) and ∂Ωj φ dσ = 0 for all j ∈ {1, . . . , κ + }. By the membership of φ ∈ Ker(− 12 I + WΩt ) and by the jump properties of the single layer potential of Theorem 6.6, we have + νΩ · ∇vΩ [φ] = 0 and

− νΩ · ∇vΩ [φ] = φ .

(6.10)

Moreover, by the characterization of Ker(− 12 I + WΩ ) of Theorem 6.24 we have  φψ dσ = 0 for all ψ ∈ Ker(− 12 I + WΩ ). Accordingly, by the Fredholm Alter∂Ω native of Theorem 5.8 there exists μ ∈ C 0,α (∂Ω) such that   1 φ = − I + WΩt [μ] . 2 By the jump properties of the single layer potential of Theorem 6.6 it follows that + νΩ · ∇vΩ [μ] = φ .

Then, by the second Green Identity of Theorem 4.5, we have  + + + + (νΩ · ∇vΩ [φ]) vΩ [μ] − vΩ [φ] (νΩ · ∇vΩ [μ]) dσ ∂Ω  + + + + = (ΔvΩ [φ])vΩ [μ] − vΩ [φ]ΔvΩ [μ] dx = 0 . Ω

Moreover, we have



κ   +

φ dσ = ∂Ω

j=1

φ dσ = 0 , ∂Ωj

(6.11)

(6.12)

6.7 The Dirichlet Problem in Ω

195

− and by Theorem 4.23 we deduce that vΩ [φ] is harmonic at infinity. Hence, by (6.10), (6.11), and (6.12) and by the first Green Identity for exterior domains of Corollary 4.7 it follows that  + + + + (νΩ · ∇vΩ [φ]) vΩ [μ] − vΩ [φ] (νΩ · ∇vΩ [μ]) dσ 0= ∂Ω     − 2 + − − ∇v [φ] dx . =− vΩ [φ] φ dσ = − vΩ [φ] (νΩ · ∇vΩ [φ]) dσ = Ω ∂Ω

Ω−

∂Ω

− − As a consequence, vΩ [φ] is locally constant on Ω − . Thus, νΩ · ∇vΩ [φ] = 0 and the second equality of (6.10) implies that φ = 0.  

By Theorem 6.25 we can deduce the following Corollary 6.26. The proof can be obtained by a straightforward modification of that of Corollary 6.17 and using Theorems 6.24 and 6.25 instead of Theorems 6.14 and 6.16 (for a similar statement in L2 (∂Ω) we refer to Folland [102, Corollary 3.39]). Corollary 6.26. We have 

C

0,α

1 (∂Ω) = Im − I + WΩt 2 

and C

1,α

1 (∂Ω) = Im − I + WΩ 2







1 ⊕ Ker − I + WΩt 2



  1 ⊕ Ker − I + WΩ , 2

(6.13)

where the sums are direct but not necessarily orthogonal.

6.7 The Dirichlet Problem in Ω We are now ready to prove the existence of a solution for the Dirichlet problem in the bounded domain Ω. As observed in the introduction of Section 6.5, when the exterior of Ω is connected and κ − = 0, we can obtain the solution in the form of a double layer potential. Indeed, if that is the case, then the operator 12 I + WΩ is an isomorphism from C 1,α (∂Ω) to itself by Corollary 6.15. Instead, if Ω − has more than one connected component (that is, if κ − > 0), then the null space of 12 I+WΩ is nontrivial (cf. Theorem 6.14). Then, the Fredholm Alternative of Theorem 5.8 tells us that also the null space of the transpose operator 12 I + WΩt is nontrivial and, as a consequence, we cannot expect every function of C 1,α (∂Ω) to be in the image of 1 2 I +WΩ . In such a case, the double layer potential alone will not suffice to write the solution of the Dirichlet problem for any possible boundary datum g ∈ C 1,α (∂Ω). To overcome this difficulty we will exploit the direct sum in (6.8). Accordingly, we will split a function g ∈ C 1,α (∂Ω) into the sum of a function g1 in the image of 1 1 2 I + WΩ and a function g2 in the null of space of 2 I + WΩ . Then the Dirichlet problem with boundary datum g1 has a solution that can be written as a double layer

196

6 Boundary Value Problems and Boundary Integral Operators

potential, and, by Proposition 6.13, the Dirichlet problem with boundary datum g2 has a solution that can be written as a single layer potential with a density that belongs to the null space of 12 I + WΩt . By linearity, the sum of these two solutions is the solution of the problem with boundary datum g. We use this argument in the proof the following Theorem 6.27, where we show an existence result for the interior Dirichlet problem. Theorem 6.27. If g ∈ C 1,α (∂Ω), then the boundary value problem ! Δu = 0 in Ω , u=g on ∂Ω

(6.14)

has one and only one solution u ∈ C 1,α (Ω). Proof. By the direct sum in (6.8) there exist a function g1 ∈ Im( 12 I + WΩ ) and a function g2 ∈ Ker( 12 I + WΩ ) such that g = g1 + g2 (here g2 = 0 when κ − = 0). Then there exists ψ ∈ C 1,α (∂Ω) such that 1 ψ + WΩ [ψ] = g1 2 and by the jump formula for the double layer potential of Theorem 6.6 it follows that + [ψ]|∂Ω = g1 . (6.15) wΩ Moreover, by the membership of g2 in Ker( 12 I + WΩ ) and by Proposition 6.13, there exists a function μ ∈ Ker( 12 I + WΩt ) ⊆ C 0,α (∂Ω) such that VΩ [μ] = vΩ [μ]|∂Ω = g2 .

(6.16)

Then, by equalities (6.15) and (6.16), by Theorem 4.22 (i) and Proposition 4.28 (i) on the harmonicity of the single and double layer potentials, and by the regularity Theorems 4.25 (i) and 4.31 (i) for the single and double layer potentials, we can verify that + + [ψ] + vΩ [μ] u ≡ wΩ is a solution of (6.14) and belongs to C 1,α (Ω). The uniqueness of u is a consequence of Theorem 3.6.   To work out some of the perturbation problems that are the objective of this book, we need to show in a constructive way how to obtain the solution of a Dirichlet problem solving a specific set of boundary integral equations. For this purpose, we can exploit the argument in the proof of Theorem 6.27 and write the solution as the sum of a single and a double layer potential. We still have, however, to identify

6.7 The Dirichlet Problem in Ω

197

the corresponding densities as the solution of a specific system of integral equations, and to do so we have to write a suitable expression for the functions g1 and g2 . Then, with this goal in mind, we introduce some convenient bases for Ker( 12 I + WΩ ) and Ker( 12 I + WΩt ). We begin with some notation. If Σ is a subset of ∂Ω, then we denote by ζΣ the map from ∂Ω to R defined by ! 1 if x ∈ Σ , (6.17) ζΣ (x) ≡ 0 if x ∈ ∂Ω \ Σ . So that ζ∂(Ω − )l (x) = δl,j

∀x ∈ ∂(Ω − )j , l, j ∈ {1, . . . , κ − },

where δl,j denotes the Kronecker delta function defined by δl,j = 1 if l = j and δl,j = 0 if l = j, for all l, j ∈ N. Then Theorem 6.14 implies that for κ − > 0 the set {ζ∂(Ω − )1 , . . . , ζ∂(Ω − )κ− } is a basis for Ker( 12 I + WΩ ). Moreover, by 1 − t Theorem 6.16, the linear  map fromthe κ -dimensional space Ker( 2 I + WΩ ) to − is an isomorphism. Then we deduce Rκ that takes τ to ∂(Ω − )j τ dσ j=1,...,κ −

the validity of the following. Lemma 6.28. If κ − > 0, then for each l ∈ {1, . . . , κ − } there exists a unique function τl ∈ C 0,α (∂Ω) such that  1 ( I + WΩt )[τl ] = 0 and τl dσ = δl,j ∀j ∈ {1, . . . , κ − } 2 − ∂(Ω )j and the set {τ1 , . . . , τκ − } is a basis for Ker( 12 I + WΩt ). Moreover, by Proposition 6.13, we have the following. Lemma 6.29. If κ − > 0, then {VΩ [τ1 ], . . . , VΩ [τκ − ]} is a basis for Ker( 12 I +WΩ ). Now we have two bases for Ker( 12 I + WΩ ). Namely, {ζ∂(Ω − )1 , . . . , ζ∂(Ω − )κ− }

and

{VΩ [τ1 ], . . . , VΩ [τκ − ]} .

Thus, we can consider the corresponding transition matrix from one basis to the other. Since the basis {ζ∂(Ω − )1 , . . . , ζ∂(Ω − )κ− } is orthogonal in L2 (∂Ω), an elementary argument of linear algebra shows the validity of the following Lemma 6.30. − − − 2 Lemma 6.30. Let κ − > 0. Let ΛΩ − ≡ (λl,j Ω − )(l,j)∈{1,...,κ } be the real κ × κ matrix with entries λl,j Ω − defined by   1 − ) dσ = − ≡ V [τ ] ζ VΩ [τl ] dσ λl,j − Ω l ∂(Ω j Ω mn−1 (∂(Ω − )j ) ∂Ω ∂(Ω − )j

for all (l, j) ∈ {1, . . . , κ − }2 . Then ΛΩ − is invertible and we have

198

6 Boundary Value Problems and Boundary Integral Operators −

VΩ [τl ] =

κ 

− λl,j Ω − ζ∂(Ω )j

on ∂Ω

j=1

for all l ∈ {1, . . . , κ − }.

 (We recall that mn−1 (M ) = M dσ for all (n − 1)-dimensional manifolds M of class C 1 embedded in Rn .) We are now ready to deduce the validity of the following Theorem 6.31, where we write the solution of the interior Dirichlet problem as a linear combination of single and double layer potentials whose densities solve a certain system of boundary integral equations. Theorem 6.31. Let g ∈ C 1,α (∂Ω). Then the following statements hold. (i) If κ − = 0, then there exists one and only one function ψg ∈ C 1,α (∂Ω) such that   1 I + WΩ [ψg ] = g . 2 If κ − > 0, then there exists one and only one function ψg ∈ C 1,α (∂Ω) such that  !  κ −  ( 12 I + WΩ )[ψg ] = g − l=1 ∂Ω gτl dσ ζ∂(Ω − )l ,  ψ dσ = 0 ∀j ∈ {1, . . . , κ − }. ∂(Ω − )j g (6.18) (ii) If u ∈ C 1,α (Ω) is the unique solution of problem (6.14), then we have κ   −

u=

+ [ψg ] wΩ

+

l,j=1

∂Ω

 + gτl dσ (Λ−1 Ω − )l,j vΩ [τj ] ,

(6.19)

with the τl ’s as in Lemma 6.28 and ΛΩ − as in Lemma 6.30 and where we understand that the sum is zero for κ − = 0. Proof. (i) The first part of the statement on the case where κ − = 0 is a consequence of Corollary 6.15. To prove the second part for κ − > 0 we observe that, by Lemma 6.28, the function κ  

 gτl dσ ζ∂(Ω − )l

−

g1 ≡ g −

l=1

∂Ω

in the right-hand side of the first equation of (6.18) satisfies the equalities  τj g1 dσ = 0 ∀j ∈ {1, . . . , κ − } ∂Ω

and it is therefore orthogonal to Ker( 12 I + WΩt ) with respect to the dual system C 0,α (∂Ω), C 1,α (∂Ω) induced by the non-degenerate bilinear form (6.1). Accord-

6.7 The Dirichlet Problem in Ω

199

ingly, the Fredholm Alternative of Theorem 5.8 implies that g1 is in the image of 1 2 I + WΩ . Since {ζ∂(Ω − )1 , . . . , ζ∂(Ω − )κ− } is a basis of Ker( 12 I + WΩ ) and is orthogonal in L2 (∂Ω) (cf. Theorem 6.14), we can also verify that a function ψg that satisfies both the equations of (6.18) exists and is unique. (We observe that here we have introduced an expression for the functions g1 and g2 = g − g1 of the proof of Theorem 6.27.) (ii) Let u ˜ be the function in the right-hand side of (6.19). By the regularity Theorems 4.25 and 4.31 for the single and the double layer potentials, u ˜ is of class C 1,α (Ω). By the definition of ψg in statement (i) and by the jump formula for the double layer potential of Theorem 6.6 we have κ  

 gτl dσ ζ∂(Ω − )l ,

−

+ wΩ [ψg ]|∂Ω = g −

l=1

∂Ω

where we omit the sum for κ − = 0. If κ − > 0, then Lemma 6.30 implies that κ  



−

l,j=1

gτl dσ ∂Ω

κ  

 gτl dσ ζ∂(Ω − )l .

−

+ (Λ−1 Ω − )l,j vΩ [τj ]|∂Ω

=

l=1

∂Ω

Then u ˜|∂Ω = g. We conclude that u ˜ is a solution of the Dirichlet problem (6.14) and thus u = u ˜ by the uniqueness Theorem 3.6.   We shall see in what follows that there are many ways to represent the solution of a boundary value problem for the Laplace equation Δu = 0 by means of layer potentials. Depending on the problem under consideration and the specific properties that one wishes to emphasize, one might choose one representation or another. For example, the expression (6.19) obtained in Theorem 6.31 will be very useful to study a Dirichlet problem in a planar domain with one or more small holes which shrink to points as we do in Sections 8.4 and 10.4. As a consequence of Theorem 6.31, when κ − = 0 the image of the map + C 1,α (∂Ω)  ψ → wΩ [ψ] ∈ C 1,α (Ω)

(6.20)

coincides with the set of the harmonic functions of C 1,α (Ω). By Theorem 6.31 we can also prove the following result, which characterizes the image of the map in (6.20) for κ − > 0. + Theorem 6.32. Let κ − > 0. Let u ∈ C 1,α (Ω) be harmonic in Ω. Then u = wΩ [ψ] 1,α for some ψ ∈ C (∂Ω) if and only if  ν(Ω − )j · ∇u dσ = 0 ∀j ∈ {1, . . . , κ − } . (6.21) ∂(Ω − )j

200

6 Boundary Value Problems and Boundary Integral Operators

+ Proof. We first prove that (6.21) holds for all u = wΩ [ψ] with ψ ∈ C 1,α (∂Ω). To do so, we verify that  + ν(Ω − )j · ∇wΩ [ψ] dσ = 0 ∀j ∈ {1, . . . , κ − } , ψ ∈ C 1,α (∂Ω) . ∂(Ω − )j

(6.22) Indeed, if j ∈ {1, . . . , κ − }, then by the jump formulas of Theorem 6.6 we have + − ν(Ω − )j · ∇wΩ [ψ] = ν(Ω − )j · ∇wΩ [ψ]

on ∂(Ω − )j .

(6.23)

− [ψ] is harmonic in Ω − and, by Theorem 4.31 By Proposition 4.28 (i) the function wΩ 1,α − − (ii), it belongs to Cloc (Ω ). Then the restriction of wΩ [ψ] to (Ω − )j is harmonic in − 1,α (Ω )j and belongs to C ((Ω − )j ) for all j ∈ {1, . . . , κ − }. Here we note that we can omit the subscript ‘loc’ of C 1,α because (Ω − )j is bounded for j = 0. Then the first Green Identity of Theorem 4.2 implies that  − ν(Ω − )j · ∇wΩ [ψ] dσ = 0 (6.24) ∂(Ω − )j

for all j ∈ {1, . . . , κ − }. The validity of (6.22) follows by (6.23) and (6.24). We now prove the other implication: we assume that (6.21) holds true and we + [ψ] for some ψ ∈ C 1,α (∂Ω). We shall exploit formula (6.19) show that u = wΩ for the solution of the Dirichlet problem in Ω in the case where g = u|∂Ω . If i ∈ {1, . . . , κ − }, then we verify that   + − 0= ν(Ω )i · ∇u dσ = ν(Ω − )i · ∇wΩ [ψg ] dσ ∂(Ω − )i

∂(Ω − )i

  gτl dσ (Λ−1 ) Ω − l,j

κ   −

+

l,j=1

By (6.22) we have

∂Ω

(6.25) ∂(Ω − )i

+ ν(Ω − )i · ∇vΩ [τj ] dσ .

 ∂(Ω − )

+ ν(Ω − )i · ∇wΩ [ψg ] dσ = 0 .

(6.26)

i

Moreover, by the jump relations of Theorem 6.6 and by the definition of τj in Lemma 6.28 we have   + + ν(Ω − )i · ∇vΩ [τj ] dσ = − νΩ · ∇vΩ [τj ] dσ ∂(Ω − )i ∂(Ω − )i   − =− νΩ · ∇vΩ [τj ] dσ + τj dσ (6.27) ∂(Ω − )i ∂(Ω − )i  − = ν(Ω − )i · ∇vΩ [τj ] dσ + δij . ∂(Ω − )i

6.8 The Dirichlet Problem in Ω −

201

1,α − Since the function vΩ [τj ] is harmonic in Ω − and belongs to Cloc (Ω − ) (cf. The− orems 4.22 (i) and 4.25 (ii)), the restriction of vΩ [τj ] to the bounded set (Ω − )i is harmonic in (Ω − )i and belongs to C 1,α ((Ω − )i ). Then the first Green Identity of Theorem 4.2 implies that  − ν(Ω − )i · ∇vΩ [τj ] dσ = 0 (6.28) ∂(Ω − )i

for all i ∈ {1, . . . , κ−}. Then, by (6.27) and (6.28), we can write  + ν(Ω − )i · ∇vΩ [τj ] dσ = δji

(6.29)

∂(Ω − )i

and, by (6.25), (6.26), and (6.29), we conclude that κ   −

l=1

Then

∂Ω

 gτl dσ (Λ−1 Ω − )l,j = 0

κ   −

l,j=1

∂Ω

∀j ∈ {1, . . . , κ − } .

 + gτl dσ (Λ−1 Ω − )l,j vΩ [τj ] = 0

+ and formula (6.19) of Theorem 6.31 (ii) implies that u = wΩ [ψg ] with ψg ∈ 1,α   C (∂Ω).

A variant of Theorem 6.32 can be obtained from Theorem 8.20 of McLean [206], where the author characterizes the null space of the (bounded self-adjoint) boundary operator corresponding to the normal derivative of the double layer potential (see, e.g., [60] with Costabel and Dauge).

6.8 The Dirichlet Problem in Ω − We now turn to the Dirichlet problem in the exterior domain Ω − . We have the same kind of difficulty we met with the interior problem: If the boundary datum g is in the image of − 12 I + WΩ , then we can obtain the solution as a double layer potential. The image of − 12 I + WΩ , however, never fills the whole of C 1,α (∂Ω) and so the double layer potential is not in general sufficient (this is different for the interior problem, where 12 I + WΩ is surjective for κ − = 0). Indeed, any non-empty domain Ω has at least one connected component, thus we always have κ + > 0 and, by Theorem 6.24 and by the Fredholm Alternative of Theorem 5.8, we can see that the null spaces of − 12 I +WΩ and − 12 I +WΩt are non-trivial and that Im(− 12 I +WΩ ) is properly contained in C 1,α (∂Ω). Then we resort to the same strategy employed in the interior case and we exploit the direct sum of (6.13) to split g into a sum g1 + g2

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6 Boundary Value Problems and Boundary Integral Operators

with g1 in the image of − 12 I + WΩ and g2 in the null space of the same operator. The solution of the Dirichlet problem with boundary datum g1 can be obtained as a double layer potential and the solution of the problem with datum g2 as a single layer if n ≥ 3, and as the sum of a single layer and a constant function if n = 2 (cf. Propositions 6.19 and 6.23). Then the solution with boundary datum g is given by the sum of the solutions for g1 and g2 . With this argument we now prove the following existence theorem. Theorem 6.33. Let g ∈ C 1,α (∂Ω). The boundary value problem ⎧ ⎪ in Ω − , ⎨Δu = 0 u=g on ∂Ω , ⎪ ⎩ u is harmonic at infinity

(6.30)

1,α (Ω − ). has one and only one solution u ∈ Cloc

Proof. By the direct sum in (6.13) there exist a function g1 ∈ Im(− 12 I + WΩ ) and a function g2 ∈ Ker(− 12 I + WΩ ) such that g = g1 + g2 . We first consider the Dirichlet problem with boundary datum g1 . Since g1 belongs to Im(− 12 I + WΩ ), there exists ψ ∈ C 1,α (∂Ω) such that 1 − ψ + WΩ [ψ] = g1 2 and the jump formula for the double layer potential of Theorem 6.6 implies that − wΩ [ψ]|∂Ω = g1 . − [ψ] is harmonic and harmonic at infinity by ProposiMoreover, the function wΩ 1,α (Ω − ) by the regularity Theorem 4.31 (ii). tion 4.28 and belongs to Cloc We now turn to the problem with datum g2 and we consider separately the cases of dimension n ≥ 3 and n = 2. For n ≥ 3, the membership of g2 in the null space of − 12 I + WΩ and Proposition 6.19 imply that there exists a function μ in Ker(− 12 I + WΩt ) ⊆ C 0,α (∂Ω) such that

VΩ [μ] = vΩ [μ]|∂Ω = g2 . − [μ] is harmonic, harmonic Then, by Theorems 4.22 and 4.23 (i) the function vΩ at infinity, and solves the Dirichlet problem with boundary datum g2 . Moreover, 1,α − [μ] ∈ Cloc (Ω − ) by the regularity Theorems 4.25 (ii). Hence, vΩ − − u ≡ wΩ [ψ] + vΩ [μ] 1,α is a solution of (6.30) and belongs to Cloc (Ω − ).

6.8 The Dirichlet Problem in Ω −

203

Finally, we consider the case when n = 2. By Proposition 6.23, there exist a function μ ∈ Ker(− 12 I + WΩt )0 ⊆ C 0,α (∂Ω)0 and a constant function ρ on ∂Ω such that VΩ [μ] + ρ = vΩ [μ]|∂Ω + ρ = g2 . − − [μ] By Theorems 4.22 and 4.23 (ii), the function vΩ  is harmonic in Ω , harmonic − [μ] at infinity, and equals g2 − ρ on ∂Ω (we note that ∂Ω μ dσ = 0). Moreover, vΩ 1,α − belongs to Cloc (Ω ) by the regularity Theorem 4.25 (ii). Then, if we still denote by ρ the constant extension of ρ to Ω − , we have that − − u ≡ wΩ [ψ] + vΩ [μ] + ρ 1,α (Ω − ). is a solution of (6.30) and belongs to Cloc Both for n ≥ 3 and for n = 2, the uniqueness of the solution u is a consequence of Theorem 6.1  

We can exploit the argument in the proof of Theorem 6.33 to write the solution of an exterior Dirichlet problem as the sum of a single and a double layer potential and, for n = 2, a constant function. But, to identify the corresponding densities as the solutions of a specific set of boundary integral equations, we need to write a suitable expression for the functions g1 and g2 . To do so, we now introduce some convenient bases for the null spaces of the operators − 21 I + WΩ and − 12 I + WΩt . With the notation introduced in (6.17) we have ∀x ∈ ∂Ωj , l, j ∈ {1, . . . , κ + } ζ∂Ωl (x) = δl,j and Theorem 6.24 implies that {ζ∂Ω1 , . . . , ζ∂Ωκ+ } is a basis for the null space of + − 12 I + WΩ . Moreover, by Theorem 6.25 the linear  map from  the κ -dimensional + is an isomorspace Ker(− 12 I + WΩt ) to Rκ that takes τ to ∂Ωj τ dσ j=1,...,κ +

phism and thus we deduce the validity of the following. Lemma 6.34. For each l ∈ {1, . . . , κ + } there exists a unique function ςl in C 0,α (∂Ω) such that  1 t (− I + WΩ )[ςl ] = 0 and ςl dσ = δl,j ∀j ∈ {1, . . . , κ + } 2 ∂Ωj and the set {ς1 , . . . , ςκ + } is a basis for Ker(− 12 I + WΩt ). Then, for n ≥ 3 we have the following Lemma 6.35, which follows from Proposition 6.19 Lemma 6.35. If n ≥ 3, then for each l ∈ {1, . . . , κ + } there exists a unique function ψl ∈ Ker(− 12 I + WΩt ) such that VΩ [ψl ] = ζ∂Ωl . In addition, by Proposition 6.23 we have the following.

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6 Boundary Value Problems and Boundary Integral Operators

Lemma 6.36. For each l ∈ {1, . . . , κ + } there exists a unique function φl in Ker(− 12 I + WΩt )0 and a unique constant ρl ∈ R such that VΩ [φl ] + ρl = ζ∂Ωl . We are now ready to deduce the validity of the following. Theorem 6.37. Let g ∈ C 1,α (∂Ω). Then the following statements hold. (i) There exists one and only one function ψg ∈ C 1,α (∂Ω) such that  !  κ +  (− 12 I + WΩ )[ψg ] = g − l=1 ∂Ω gςl dσ ζ∂Ωl ,  ψ dσ = 0 ∀j ∈ {1, . . . , κ + }. ∂Ωj g (6.31) 1,α (Ω − ) is the unique solution of (6.30), then we have (ii) If n ≥ 3 and u ∈ Cloc u=

− [ψg ] wΩ

+

κ +   ∂Ω

l=1

 − gςl dσ vΩ [ψl ] ,

(6.32)

where the ψl are as in Lemma 6.35. 1,α (Ω − ) is the unique solution of (6.30), then we have (iii) If n = 2 and u ∈ Cloc κ   +

u=

− [ψg ] wΩ

+

l=1

∂Ω

 − gςl dσ (vΩ [φl ] + ρl ) ,

(6.33)

where φl and ρl are as in Lemma 6.36. Proof. (i) By Lemma 6.34 we can verify that the function κ  

 gςl dσ ζ∂Ωl

+

g1 ≡ g −

l=1

∂Ω

in the right-hand side of the first equation in (6.31) is orthogonal to the null space Ker(− 12 I + WΩt ) with respect to the dual system C 0,α (∂Ω), C 1,α (∂Ω) induced by the non-degenerate bilinear form (6.1). Then by Theorem 6.5 and by the Fredholm Alternative of Theorem 5.8 we deduce the existence of ψg . The uniqueness of ψg is a consequence of Theorem 6.24. (We observe that g1 and g2 ≡ g − g1 are the functions that appear in the proof of Theorem 6.33.) (ii) Let u ˜ be the function in the right hand side of (6.32). By the regularity Theo1,α (Ω − ). By Theorems 4.22 (i), 4.23 (i), rems 4.25 (ii) and 4.31 (ii), u ˜ is of class Cloc − and Proposition 4.28, u ˜ is harmonic in Ω and harmonic at infinity. By statement (i) and by the jump formula for the double layer potential of Theorem 6.6 we have κ  

 gςl dσ ζ∂Ωl .

+

− wΩ [ψg ]|∂Ω

=g−

l=1

∂Ω

By Theorem 4.22 (ii) and by Lemma 6.35, we have

6.8 The Dirichlet Problem in Ω − κ   +

l=1

∂Ω

205

κ +    − gςl dσ vΩ [ψl ]|∂Ω = l=1

 gςl dσ ζ∂Ωl . ∂Ω

Hence, u ˜|∂Ω = g. Then u ˜ is a solution of (6.30) and by the uniqueness Proposition 6.1 it follows that u ˜ coincide with the solution u of Theorem 6.33. (iii) The proof of statement (iii) is very similar to that of statement (ii), we just need to use Lemma 6.36 instead of Lemma 6.35 and Theorem 4.23 (ii) instead of Theorem 4.23 (i).   We observe that, to prove Theorem 6.37, we did not follow the same path as for the analogous Theorem 6.31 for the interior problem. In particular, we did not introduce a transition matrix similar to ΛΩ − . Theorem 6.37 is, however, presented only for the sake of completeness, and so we opted for the fastest proof. On the contrary, Theorem 6.31 is largely used in the book and we exploit the specific form of formula (6.19) to analyze certain perturbation problems (cf. Chapters 8 and 10). If Ω is connected (i.e., κ + = 1), then we have the following variant of Theorem 6.37 (i). Lemma 6.38. Let Ω be connected. Let g ∈ C 1,α (∂Ω). Then there exists one and only pair (ψg , cg ) ∈ C 1,α (∂Ω)0 × R such that 1 (− I + WΩ )[ψg ] + cg = g . 2

(6.34)

Proof. By Theorem 6.24 the null space of − 12 I + WΩ consists of the constant functions on ∂Ω. Then, by the direct sum in (6.13), there  exist a pair (φg , cg ) ∈ C 1,α (∂Ω) × R that satisfies (6.34). Taking ψg ≡ φg − −∂Ω φg dσ, we obtain a pair (ψg , cg ) as in the statement. The uniqueness  of (ψg , cg ) follows by the direct sum   (6.13), by Theorem 6.24, and by condition ∂Ω ψg dσ = 0. Then, when n = 2 and Ω is connected we do not need the single layer potential to write the solution of the exterior Dirichlet problem. Indeed, we have the following. Proposition 6.39. Assume that n = 2 and Ω is connected. Let g ∈ C 1,α (∂Ω) and 1,α (Ω − ) be the unique solution of the boundary value problem (6.30). let u ∈ Cloc Then we have − − [ψg ] + cg = wΩ [ψg ] + lim u(y) , u = wΩ y→∞

where (ψg , cg ) ∈ C 1,α (∂Ω)0 × R is the unique solution of (6.34). Proof. The first equality follows from Lemma 6.38, from the jump formula for − wΩ [ψg ] in Theorem 6.6, and from the uniqueness of the solution of the Dirichlet − [ψg ](y) problem in Ω − of Proposition 6.1. For the second equality we note that wΩ vanishes as y → ∞ by Proposition 4.28 (ii).  

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6 Boundary Value Problems and Boundary Integral Operators

6.9 The Neumann Problem in Ω and Ω − For the interior and exterior Neumann problems to have solutions, the boundary datum g has to satisfy certain compatibility conditions, as we show in the following Lemmas 6.40 and 6.41. Lemma 6.40. Let g ∈ C 0,α (∂Ω). If the interior Neumann problem ! Δu = 0 in Ω , νΩ · ∇u = g on ∂Ω , has a solution u ∈ C 1,α (Ω), then  g dσ = 0

∀j ∈ {1, . . . , κ+ } .

(6.35)

(6.36)

∂Ωj

Proof. If u ∈ C 1,α (Ω) is a solution of (6.35) and j ∈ {1, . . . , κ+ }, then by the first Green Identity of Theorem 4.2 we have     g dσ = 1 νΩ · ∇u dσ = ∇1 · ∇u dx + Δu dx = 0 ∂Ωj

∂Ωj

Ωj

Ωj

 

and the lemma is proved. Lemma 6.41. Let g ∈ C 0,α (∂Ω). If the exterior Neumann problem ⎧ ⎪ in Ω − , ⎨Δu = 0 on ∂Ω , νΩ · ∇u = g ⎪ ⎩ u is harmonic at infinity, 1,α (Ω − ), then has a solution u ∈ Cloc  g dσ = 0 ∂(Ω − )j

∀j ∈ {1, . . . , κ− } .

(6.37)

(6.38)

In addition, if n = 2, then we also have  g dσ = 0 .

(6.39)

∂(Ω − )0

Proof. The proof of (6.38) is very similar to that of (6.36). To show the validity of (6.39) we observe that, when n = 2, the constant function identically equal to 1 on (Ω − )0 is harmonic and harmonic at infinity (cf. Theorem 3.22). Then, by the first Green Identity in exterior domains of Corollary 4.7, we deduce that

6.9 The Neumann Problem in Ω and Ω −

207







g dσ = ∂(Ω − )0

∂(Ω − )0

1 νΩ ·∇u dσ = − lim

R→+∞

(Ω − )0 ∩B2 (0,R)

∇1·∇u dx = 0 .  

We now prove the existence of solutions to the interior and exterior Neumann problems. We begin with Theorem 6.42, where we consider the interior case. Theorem 6.42. If g ∈ C 0,α (∂Ω) satisfies the compatibility condition (6.36), then the integral equation   1 (6.40) − I + WΩt [φ] = g 2 has a solution φ ∈ C 0,α (∂Ω) and the single layer + vΩ [φ] ∈ C 1,α (Ω)

is a solution of the interior Neumann boundary value problem (6.35). The set of all solutions of (6.35) consists of the functions u ∈ C 1,α (Ω) such that the difference + [φ] is constant on each connected component Ω1 , . . . , Ωκ+ of Ω. u − vΩ Proof. If g ∈ C 0,α (∂Ω) satisfies the compatibility condition (6.36), then, by Theorem 6.24, it is orthogonal to the null space of the integral operator − 12 I + WΩ . Accordingly, the Fredholm Alternative of Theorem 5.8 implies that equation (6.40) + [φ] is harmonic has a solution φ ∈ C 0,α (∂Ω). Then, Theorem 4.22 implies that vΩ 1,α in Ω and Theorem 4.25 implies that it is a function of C (Ω). By the jump formula for the normal derivative of single layer potential of Theorem 6.6, it follows + [φ] is a solution of (6.35). The last statement of the theorem is a consequence that vΩ of the uniqueness result of Theorem 6.2 (i).   For the exterior Neumann problem we find convenient to describe separately the case of dimension n ≥ 3 and the case of dimension n = 2. Theorem 6.43. If n ≥ 3 and g ∈ C 0,α (∂Ω) satisfies the compatibility condition (6.38), then the integral equation   1 t I + WΩ [φ] = g 2 has a solution φ ∈ C 0,α (∂Ω) and the single layer 1,α − vΩ [φ] ∈ Cloc (Ω − )

is a solution of the exterior Neumann boundary value problem (6.37). The set of all 1,α solutions of (6.37) consists of the functions u ∈ Cloc (Ω − ) such that the difference − u − vΩ [φ] is constant on each bounded connected component (Ω − )1 , . . . , (Ω − )κ− of Ω − and is identically equal to 0 on the unbounded connected component (Ω − )0 . Proof. The proof is very similar to that of Theorem 6.42. We just have to use The− orem 6.14 instead of Theorem 6.24 and notice that vΩ [φ] is harmonic at infinity by

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6 Boundary Value Problems and Boundary Integral Operators

Theorem 4.23 (i). The last statement of the theorem follows by the uniqueness result of Theorem 6.2 (ii).   Finally, for the two-dimensional case we have the following. Theorem 6.44. If n = 2 and g ∈ C 0,α (∂Ω) satisfies the compatibility conditions (6.38) and (6.39), then the integral equation   1 I + WΩt [φ] = g (6.41) 2 has a solution φ ∈ C 0,α (∂Ω)0 and the single layer 1,α − vΩ [φ] ∈ Cloc (Ω)

is a solution of the exterior Neumann boundary value problem (6.37). Moreover, 1,α the set of all solutions of (6.37) consists of the functions u ∈ Cloc (Ω − ) such that − the difference u − vΩ [φ] is constant on each (bounded and unbounded) connected component (Ω − )0 , . . . , (Ω − )κ− of Ω − . Proof. The proof is very similar to that of Theorems 6.42 and 6.43. We only observe that, if g satisfies (6.38) and (6.39) and φ is a solution of (6.41), then Lemma 6.11 implies that      1 t 0= I + WΩ [φ] dσ = g dσ = φ dσ . 2 ∂Ω ∂Ω ∂Ω − [φ] is harmonic at infinity by Theorem 4.23 (ii). Accordingly, vΩ

 

6.10 Further Mapping Properties of VΩ We have already seen in Theorem 6.4 that VΩ is bounded from C 0,α (∂Ω) to C 1,α (∂Ω). In this section we show that if n ≥ 3, then VΩ is actually an isomorphism and that, both for n = 2 and n ≥ 3, the map that takes a function φ ∈ C 0,α (∂Ω) with zero integral mean and a real number ρ to the sum VΩ [φ] + ρ is an isomorphism between C 0,α (∂Ω)0 × R and C 1,α (∂Ω). The results of this section will be useful to analyze the Dirichlet problem in a domain with a small hole, especially in the case where the dimension n is three or bigger, problems with mixed boundary conditions, transmission problems, and problems with Robin boundary conditions. We begin with the following lemma, where we show that any function of C 1,α (∂Ω) can be written as the sum of a single layer potential and a real number, which can be 0 for n ≥ 3. Lemma 6.45. If f ∈ C 1,α (∂Ω), then there exist φ ∈ C 0,α (∂Ω) and ρ ∈ R such that VΩ [φ] + ρ = f . If n = 2, then we can choose φ and ρ so that ∂Ω φ dσ = 0. If n ≥ 3, then we can take φ and ρ so that ρ = 0.

6.10 Further Mapping Properties of VΩ

209

Proof. By the solvability Theorem 6.27 for the interior Dirichlet problem, there exists a function u+ ∈ C 1,α (Ω) that is harmonic in Ω and such that u+ = f on ∂Ω. Hence, the third Green Identity in the form of Theorem 6.9 implies that 1 f = WΩ [f ] − VΩ [νΩ · ∇u+ |∂Ω ] . 2

(6.42)

By the solvability Theorem 6.33 for the exterior Dirichlet problem, there exists a 1,α function u− ∈ Cloc (Ω − ) that is harmonic in Ω − , harmonic at infinity, and such − that u = f on ∂Ω. Then the third Green Identity for exterior domains in the form of Theorem 6.10 implies that 1 f = −WΩ [f ] + VΩ [νΩ · ∇u− |∂Ω ] + ρ 2

(6.43)

with ρ = limx→∞ u− (x). Now, taking the sum of (6.42) and (6.43) we verify that + . In addition, ρ = 0 for n ≥ 3. VΩ [φ] + ρ = f with φ ≡ νΩ · ∇u− |∂Ω − νΩ · ∇u|∂Ω  Thus, to complete the proof we have to show that ∂Ω φ dσ = 0 when n = 2. By  the first Green Identity of Theorem 4.2 we have ∂Ω νΩ · ∇u+ |∂Ω dσ = 0. Then we observe that for n = 2 the function identically equal to 1 is harmonic in Ω − and harmonic at infinity. Thus ∂Ω νΩ · ∇u− |∂Ω dσ = 0 by the Green Identity for exterior domains of Corollary 4.7. Hence,    − φ dσ = νΩ · ∇u|∂Ω dσ − νΩ · ∇u+ |∂Ω dσ = 0 . ∂Ω

∂Ω

∂Ω

  For n ≥ 3 we can exploit Lemma 6.45 to show that VΩ is an isomorphism. Theorem 6.46. If n ≥ 3, then VΩ is an isomorphism from C 0,α (∂Ω) to C 1,α (∂Ω). Proof. By the Open Mapping Theorem 2.2 and by the boundedness of VΩ (cf. Theorem 4.25), it suffices to show that VΩ is a bijection. By Lemma 6.45 we already know that VΩ is surjective. Thus it suffices to prove that VΩ is injective. If + [φ] = 0 on Ω by the uniqueness of the solution of the Dirichlet VΩ [φ] = 0, then vΩ + [φ] problem of Theorem 3.6. By the jump formula for the normal derivative of vΩ + 1 t of Theorem 6.6, it follows that (− 2 I + WΩ )[φ] = νΩ · ∇vΩ [φ]|∂Ω = 0. Then Proposition 6.19 implies that φ = 0.   In general, for all dimensions n ≥ 2 we have the following. Theorem 6.47. The map from C 0,α (∂Ω)0 × R to C 1,α (∂Ω) that takes (φ, ρ) to VΩ [φ] + ρ is an isomorphism. Proof. Since VΩ is bounded from C 0,α (∂Ω) to C 1,α (∂Ω) (cf. Theorem 4.25), the map in the statement is bounded from C 0,α (∂Ω)0 × R to C 1,α (∂Ω). Then by the Open Mapping Theorem 2.2 it suffices to show that it is a bijection to conclude that it

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6 Boundary Value Problems and Boundary Integral Operators

is an isomorphism. We first prove that it is injective. We observe that if VΩ [φ] + ρ = + [φ] = −ρ on Ω by the uniqueness of the solution of the Dirichlet problem 0, then vΩ of Theorem 3.6. By the jump formula of Theorem 6.6 for the normal derivative of + + [φ], it follows that (− 12 I + WΩt )[φ] = νΩ · ∇vΩ [φ]|∂Ω = 0. Then the single layer vΩ + 1 t φ ∈ Ker(− 2 I + WΩ )0 and we have φ = 0 by Lemma 6.22. Hence, 0 = vΩ [φ] = −ρ and (φ, ρ) = (0, 0). To show that the map in the statement is surjective, we consider separately the case when n = 2 and when n ≥ 3. Indeed, for n = 2 the surjectivity follows directly from Lemma 6.45. Instead, for n ≥ 3 we argue as follows. If f ∈ C 1,α (∂Ω), then Lemma 6.45 implies that there exists φ ∈ C 0,α (∂Ω) such that VΩ [φ] = f . By Theorem 6.24 (or Lemma 6.21) the constant function that is identically equal to 1 belongs to Ker(− 12 I + WΩ ). By Proposition 6.19 it follows ˜ is equal to the constant 1 that there exists φ˜ ∈ Ker(− 12 I + WΩt ) such that VΩ [φ]  ˜ on ∂Ω. Then Lemma 6.22 implies that ∂Ω φ dσ = 0, otherwise it would be φ˜ = 0, ˜ = 1. Then we can take contrary to VΩ [φ]  φ dσ ρ ≡ ∂Ω φ˜ dσ ∂Ω

and we have ˜ + VΩ [ρφ] ˜ = VΩ [φ − ρφ] ˜ +ρ=f VΩ [φ − ρφ]

 ˜ dσ = 0. (φ − ρφ)

and ∂Ω

  We conclude this section with the following Theorem 6.48, where we show that harmonic functions in a bounded open set and in the exterior of a bounded open set can be written as a sum of a single layer and a constant function. The proof can be deduced by Theorems 6.46 and 6.47 and by the uniqueness of the solution of the interior and exterior Dirichlet problems of Theorems 3.6 and 6.1, respectively. Theorem 6.48. The following statements hold. + (i) The map that takes a pair (μ, c) to vΩ [μ] + c is a linear bijection from 0,α C (∂Ω)0 × R to the space of the functions of C 1,α (Ω) that are harmonic in Ω. − [μ] is a linear bijection from C 0,α (∂Ω) (ii) Let n ≥ 3. The map that takes μ to vΩ 1,α to the space of those functions of Cloc (Ω − ) that are harmonic in Ω − and harmonic at infinity. − [μ] + c is a linear bijection (iii) Let n = 2. The map that takes a pair (μ, c) to vΩ 1,α 0,α from C (∂Ω)0 × R to the space of those functions of Cloc (Ω − ) that are − harmonic in Ω and harmonic at infinity.

6.11 A Mixed Boundary Value Problem

211

6.11 A Mixed Boundary Value Problem We see here an application of the results obtained in Section 6.10. We consider two (nonempty) open subsets Ω i and Ω o of Rn , both of class C 1,α and with connected exteriors Ω i− and Ω o− . We also assume that Ω i ⊆ Ω o . So that the superscript ‘i’ stands for ‘inner’ and ‘o’ stands for ‘outer.’ Then we take a function g ∈ C 0,α (∂Ω i ) and a function f ∈ C 1,α (∂Ω o ) and we consider the following mixed boundary value problem for a function u ∈ C 1,α (Ω o \ Ω i ), ⎧ ⎪ in Ω o \ Ω i , ⎨Δu = 0 (6.44) νΩ i · ∇u = g on ∂Ω i , ⎪ ⎩ o u=f on ∂Ω . By Proposition 6.3 we know that if a solution u of (6.44) exists, then it is unique. In the following Proposition 6.49 we show that such a solution actually exists. We observe that the specific integral representation of u that we use in the proof of Proposition 6.49 (cf. formula (6.45)) will present some advantage when we will study a perturbed version of problem (6.44) in Section 9.2. Proposition 6.49. There exists a solution u ∈ C 1,α (Ω o \ Ω i ) of problem (6.44). Proof. We look for a solution u in the form − + i o u(x) = vΩ i [φ ](x) + vΩ o [φ ](x) + ρ

∀x ∈ Ω o \ Ω i

(6.45)

with φi ∈ C 0,α (∂Ω i ), φo ∈ C 0,α (∂Ω o )0 , and ρ ∈ R. Then, by the jump formulas for the single layer potential of Theorem 6.6, u is a solution of (6.44) if and only if both the following equations hold:   1 t I + WΩ i [φi ](x) (6.46) 2  + νΩ i (x) · φo (y) ∇Sn (x − y) dσy = g(x) ∀x ∈ ∂Ω i , o ∂Ω  o VΩ o [φ ](x) + ρ + Sn (x − y) φi (y) dσy = f (x) ∀x ∈ ∂Ω o . (6.47) ∂Ω i

We denote by L1 the linear operator from C 0,α (∂Ω i ) × C 0,α (∂Ω o )0 × R to C 0,α (∂Ω i ) that takes (φi , φo , ρ) to the left-hand side of (6.46). We denote by L2 the linear operator from C 0,α (∂Ω i ) × C 0,α (∂Ω o )0 × R to C 1,α (∂Ω o ) that takes (φi , φo , ρ) to the left-hand side of (6.47). Then we let L be the linear operator from C 0,α (∂Ω i ) × C 0,α (∂Ω o )0 × R to C 0,α (∂Ω i ) × C 1,α (∂Ω o ) defined by L ≡ (L1 , L2 ) . We observe that 12 I + WΩt i is an isomorphism from C 0,α (∂Ω i ) to itself (cf. Corollary 6.15) and (φo , ρ) → VΩ o [φo ] + ρ is an isomorphism from C 0,α (∂Ω o )0 × R to

212

6 Boundary Value Problems and Boundary Integral Operators

C 1,α (∂Ω o ) (cf. Theorem 6.47). Hence, the map from C 0,α (∂Ω i )×C 0,α (∂Ω o )0 ×R to C 0,α (∂Ω i ) × C 1,α (∂Ω o ) that takes (φi , φo , ρ) to 12 φi + WΩt i [φi ], VΩ o [φo ] + ρ is an isomorphism. Moreover, by the compact embedding of C 0,α (∂Ω o ) into L1 (∂Ω o ) (cf. Proposition 2.25) and by Theorem A.28 (ii) on the properties of integral operators with real analytic kernel, the map from C 0,α (∂Ω o )0 to C 1,α (∂Ω i ) that takes φi to the function  νΩ i (x) · φo (y) ∇Sn (x − y) dσy ∀x ∈ ∂Ω i ∂Ω o

is compact. Since C 1,α (∂Ω i ) is continuously imbedded into C 0,α (∂Ω i ), the same map is compact from C 0,α (∂Ω o )0 to C 0,α (∂Ω i ). Similarly, we can show that the map from C 0,α (∂Ω i ) to C 1,α (∂Ω o ) that takes φi to  Sn (x − y) φi (y) dσy ∀x ∈ ∂Ω o ∂Ω i

is compact. Accordingly, L is a compact perturbation of an isomorphism and therefore a Fredholm operator of index zero (cf. Theorem 5.3). Then we can show that it is a linear homeomorphism by proving that it is injective (cf. Theorem 5.4). So we assume that L[φi , φo , ρ] = (0, 0). Then, following backward the argument above, we can see that the corresponding function u, defined by (6.45), is the unique solution of the homogeneous mixed problem ⎧ ⎪ in Ω o \ Ω i , ⎨Δu = 0 νΩ i · ∇u = 0 on ∂Ω i , ⎪ ⎩ u=0 on ∂Ω o . Proposition 6.3 implies that u = 0 and, as a consequence, we have that − + i o vΩ i [φ ](x) + vΩ o [φ ](x) + ρ = 0

∀x ∈ Ω o \ Ω i .

Since vΩ i [φi ] and vΩ o [φo ] are continuous in Rn and harmonic on Ω i , we deduce by the uniqueness of the solution of the interior Dirichlet problem of Theorem 6.1 that + + i o vΩ i [φ ](x) + vΩ o [φ ](x) + ρ = 0

∀x ∈ Ω i .

Then, by the jump formulas for the single layer potential of Theorem 6.6, we have − + i i o φi = νΩ · ∇(vΩ i [φ ](x) + vΩ o [φ ](x) + ρ) + + i i o − νΩ · ∇(vΩ i [φ ](x) + vΩ o [φ ](x) + ρ) = 0 . + o i o o It follows that vΩ o [φ ] + ρ = 0 on Ω \ Ω and thus that VΩ o [φ ] + ρ = 0, which o in turn implies that (φ , ρ) = (0, 0) by Theorem 6.47. We conclude that L is an  isomorphism and we deduce the existence of (φi , φo , ρ) as in (6.46) and (6.47). 

t 6.12 The Operators I + λWΩ and I + λWΩ

213

6.12 The Operators I + λWΩ and I + λWΩt In the following Lemma 6.50 we show that I + λWΩt is injective for all λ ∈] − 2, 2[. The result is due to Kellogg and the proof that we present is adapted from that in [137, Chap. XI, Sect. 11]. Lemma 6.50. Let λ ∈] − 2, 2[. If φ ∈ C 0,α (∂Ω) and (I + λWΩt )[φ] = 0, then φ = 0. Proof. Since (I + λWΩt )[φ] = 0, Lemma 6.11 implies that  0= (I + λWΩt )[φ] dσ ∂Ω     1 λ = 1− I + WΩt [φ] dσ φ dσ + λ 2 2 ∂Ω    ∂Ω   λ λ φ dσ + λ φ dσ = 1 + φ dσ , = 1− 2 2 ∂Ω ∂Ω ∂Ω  − [φ] is harmonic at infinity (cf. Theoand accordingly, ∂Ω φ dσ = 0. Then vΩ rem 4.23) and there exist λ+ and λ− in [0, +∞[ such that   + − λ+ = |∇vΩ [φ]|2 dx and λ− = |∇vΩ [φ]|2 dx . (6.48) Ω−

Ω

(We note that the first Green Identity for exterior domains of Corollary 4.7 implies that λ− is actually finite.) In addition, by the first Green Identity (cf. Theorem 4.2 and Corollary 4.7) and by the jump formula for the normal derivative of the single layer potential of Theorem 6.6 we have    1 + t λ = VΩ [φ] − I + WΩ [φ] dσ , 2 ∂Ω    1 − t VΩ [φ] − I − WΩ [φ] dσ . λ = 2 ∂Ω Then, a straightforward computation shows that  λ+ + λ− = − VΩ [φ] φ dσ ,  ∂Ω VΩ [φ] WΩt [φ] dσ λ+ − λ− = 2 ∂Ω

and thus equality (I + λWΩt )[φ] = 0 implies that 2(λ+ + λ− ) = λ(λ+ − λ− ) . Hence, (2 − λ)λ+ + (2 + λ)λ− = 0. Since λ belongs to ] − 2, 2[, both (2 − λ) and (2 + λ) are strictly positive and accordingly both λ+ and λ− must be equal to zero.

214

6 Boundary Value Problems and Boundary Integral Operators

+ − Then by (6.48) we deduce that vΩ [φ] and vΩ [φ] are locally constant in Ω and Ω − , respectively. Finally, the jump formula for the normal derivative of the single layer potential of Theorem 6.6 implies that φ = νΩ · ∇v − [φ]|∂Ω − νΩ · ∇v − [φ]|∂Ω = 0.  

Then, by the compactness of WΩ and WΩt of Theorem 6.8, by the Fredholm Alternative of Theorem 5.8, and by Lemma 6.50 that we have just proved, we have the following. Theorem 6.51. If λ ∈] − 2, 2[, then I + λWΩ is an isomorphism from C 1,α (∂Ω) onto itself and I + λWΩt is an isomorphism from C 0,α (∂Ω) onto itself. Theorem 6.51 could be reformulated by saying that every real number ρ with |ρ| > 1/2 is in the resolvent set of WΩ and WΩt . With an argument similar to the one in the proof of Lemma 6.50, Kellogg also proved that all complex numbers with nonzero imaginary part are in the resolvent set of WΩ and WΩt (cf. [137, Chap. XI, Sect. 11]). It follows that the spectrum of WΩ and WΩt is contained in the real interval [−1/2, 1/2] and, thanks to Theorems 6.14 and 6.24, we can also say that 1/2 is an eigenvalue whereas −1/2 is an eigenvalue only when Ω − has at least two connected components, that is, if κ − ≥ 1. (Then, for κ − = 0, the spectrum is contained in ] − 1/2, 1/2].) At this point, it would be natural to ask ourselves what are the other eigenvalues of WΩ and WΩt and how they are related to the shape of the domain Ω. This is a classical question and there is a large literature about it, also in view of the applications to conformal mappings and inverse scattering theory. The early works on the topic date back to the seminal paper of Poincar´e [244] and later contributions include those of Plemelj [243] and Carleman [39]. We also mention the works of Ahlfors [3] and the papers of Schiffer, as for example [258]. We will not, however, attempt a complete list. The interested reader can find a ‘modern’ treatment of the problem in the paper of Khavinson, Putinar, and Shapiro [138] and for applications to scattering theory we refer, for example, to Ammari and Kang [8].

6.13 A Linear Transmission Problem To illustrate some applications of the results in Sections 6.10 and 6.12, we study in this section a transmission problem in Ω. We fix a strictly positive real number a and we take two functions f ∈ C 1,α (∂Ω) and g ∈ C 0,α (∂Ω). Then we consider 1,α (Ω − ), the following transmission problem for a pair (u+ , u− ) ∈ C 1,α (Ω) × Cloc ⎧ + ⎪ in Ω , ⎪ ⎪Δu = 0 ⎪ − ⎪ ⎪ = 0 in Ω− , Δu ⎨ (6.49) on ∂Ω , a u+ − u− = f ⎪ ⎪ ⎪νΩ · ∇u+ − νΩ · ∇u− = g on ∂Ω , ⎪ ⎪ ⎪ ⎩u− is harmonic at infinity.

6.13 A Linear Transmission Problem

215

Problems like (6.49) play a role in the mathematical description of composite materials with a physical field governed by the Laplace equation. Typical examples are those of the electric field, the temperature field, and the longitudinal flow. The boundary conditions model the contact between different media. The constant a expresses the contrast parameter of the physical properties (e.g., the conductivity or the permeability) of the materials that occupy the domain Ω and the domain Ω − . We observe that, in dimension two, problem (6.49) can have a solution only if the function g satisfies a compatibility condition. Indeed we have the following. Proposition 6.52. If n = 2 and the transmission problem (6.49) has a solution, then  g dσ = 0 . ∂Ω

Proof. If problem (6.49) has a solution, then we have     νΩ · ∇u+ − νΩ · ∇u− dσ g dσ = ∂Ω

∂Ω



κ   −

νΩ · ∇u dσ − +

= ∂Ω

j=1

∂Ωj−

νΩ − · ∇u− dσ − j

 ∂Ω0−

νΩ − · ∇u− dσ . 0

 By Identity of Theorem 4.2 we have ∂Ω νΩ · ∇u+ dσ = 0 and  the first Green ν − · ∇u− dσ = 0 for all j ∈ {1, . . . , κ − }. Since u− and the function ∂Ωj− Ωj identically equal to 1 are harmonic at infinity (in dimension  two), the second Green Identity in exterior domains of Corollary 4.8 implies that ∂Ω − νΩ − · ∇u− dσ = 0. 0 0 Hence, the validity of the statement follows.   Remark 6.53. If n = 2, then the solution of the transmission problem (6.49), can1,α (Ω − ) is a solutions of the not be unique. Indeed, if (u+ , u− ) ∈ C 1,α (Ω) × Cloc + − transmission problem (6.49), then (u + c, u + ac) solves the same problem for all c ∈ R. Then we have the following uniqueness result. − + − Proposition 6.54. If (u+ 1 , u1 ) and (u2 , u2 ) are solutions of the transmission prob1,α + 1,α lem (6.49) in C (Ω) × Cloc (Ω − ), then there exists c ∈ R such that u+ 1 = u2 + c − − and u1 = u2 + ac. If n ≥ 3, then c = 0. + − − + − Proof. Let u+ ≡ u+ ≡ u− 1 − u2 and u 1 − u2 . Then (u , u ) is a solution of (6.49) with f = g = 0. Since u− is harmonic at infinity, the first Green Identities of Theorem 4.2 and Corollary 4.7 imply that   a |∇u+ |2 dx + |∇u− |2 dx Ω Ω−   +  a u νΩ · ∇u+ − u− νΩ · ∇u− dx = 0 . = ∂Ω

216

6 Boundary Value Problems and Boundary Integral Operators

  Since a is strictly positive, we deduce that both Ω |∇u+ |2 dx and Ω − |∇u− |2 dx are equal to zero. It follows that u+ and u− are locally constant on Ω and on Ω − , respectively. We now find convenient to denote by u the function from Rn to R that is equal to au+ on Ω and to u− on Ω − . We show that u is constant on Rn . Since Rn is connected it suffices to prove that u is locally constant. So let x ∈ Rn . We may either have x ∈ Rn \ ∂Ω or x ∈ ∂Ω. If x ∈ Rn \ ∂Ω then there exists i ∈ {1, . . . , κ + } or j ∈ {0, . . . , κ − } such that x ∈ Ωi or x ∈ (Ω − )j . Since u is constant on each Ωi and (Ω − )j , we deduce that there exists an open neighborhood of x where u is constant. If instead x ∈ ∂Ω, then there exist a coordinate cylinder C around x (see Lemma 2.38) and indexes i ∈ {1, . . . , κ + } and j ∈ {0, . . . , κ − } such that C ∩ Ω ⊆ Ωi and C ∩ Ω − ⊆ (Ω − )j . Since u is constant on Ωi it follows that u is identically equal to au+ (x) on C ∩ Ω = C ∩ Ωi . Since u = u− on (Ω − )j , u− is constant on (Ω − )j , and the transmission condition au+ − u− = 0 holds on ∂Ω, it follows that u is identically equal to au+ (x) on C ∩ Ω − = C ∩ (Ω − )j . Hence u is equal to the constant value au+ (x) on the whole of C. Doing so, we have shown the existence of an open neighborhood of x where u is constant and we can conclude that u is constant on Rn . Now, if we take c ∈ R such that u(x) = ac for all x ∈ Rn we deduce that + − − u1 = u+ 2 + c and u1 = u2 + ac. To complete the proof we observe that if the dimension n is bigger than or equal to three, then the condition that u− is harmonic at infinity implies that limx→∞ u− (x) = 0 (cf. Theorem 3.22). Hence   ac = limx→∞ u− (x) = 0 and, since a > 0, we conclude that c = 0. Finally, exploiting Theorem 6.47 on the representation of a function on the boundary as a sum of a single layer and a constant and the isomorphism Theorem 6.51 for I + λWΩt , we can prove the existence of solutions for the transmission problem (6.49) and that such solutions can be written in terms of single layer potentials and constant functions.  Theorem 6.55. If the dimension n is two, assume that ∂Ω g dσ = 0. Then there exists one and only one triple (φ, μ, ρ) in C 0,α (∂Ω) × C 0,α (∂Ω)0 × R such that the pair + − [φ + μ] + ρ, avΩ [φ]) (u+ , u− ) ≡ (vΩ is a solution of the transmission problem (6.49). Proof. By the continuity of the single layer potential (cf. Theorem 4.22) the pair + − (u+ , u− ) ≡ (vΩ [φ+μ]+ρ, avΩ [φ]) satisfies the boundary condition au+ −u− = f if and only if (6.50) aVΩ [μ] + aρ = f . Since a > 0, Theorem 6.47 implies that (6.50) has a unique solution (μ, ρ) ∈ C 0,α (∂Ω)0 × R. By the jump properties of Theorem 6.6, the pair of functions + − (vΩ [φ + μ] + ρ, avΩ [φ])

satisfies the boundary condition νΩ · ∇u+ − νΩ · ∇u− = g if and only if

6.14 A Robin Problem



   1 1 − (1 + a)I + (1 − a)WΩt [φ] = g − − I + WΩt [μ] . 2 2

217

(6.51)

Since 2(1−a)/(1+a) ∈]−2, 2[, Theorem 6.51 implies that − 12 (1+a)I +(1−a)WΩt is an isomorphism from C 0,α (∂Ω) to itself. As a consequence, we  deduce that there exists a unique φ ∈ C 0,α (∂Ω) that satisfies (6.51). Moreover, if ∂Ω g dσ = 0, then a computation based on Lemma 6.11 and equality (6.51) shows that ∂Ω φ dσ = 0. − Accordingly limx→∞ vΩ [φ] = 0 both in dimension n = 2 and in dimension n ≥ 3 − [φ] is harmonic at infinity and the theorem is (cf. Theorem 4.23). It follows that vΩ proved.    Incidentally, we observe that in Theorem 6.55 condition ∂Ω g dσ = 0 implies  that ∂Ω φ dσ = 0 both in dimension n = 2 and in dimension  n ≥ 3. In addition, in dimension n ≥ 3 the constant ρ can be omitted if we allow ∂Ω μ dσ to be different from zero (cf. Theorem 6.46). In such a case, however, we could no longer infer that  φ dσ is necessarily equal to 0. ∂Ω

6.14 A Robin Problem We fix two functions g, b ∈ C 0,α (∂Ω) and then we consider the following interior Robin problem  Δu(x) = 0 ∀x ∈ Ω , (6.52) ∂ u(x) + b(x)u(x) = g(x) ∀x ∈ ∂Ω ∂νΩ in the unknown u ∈ C 1,α (Ω) and the corresponding exterior Robin problem ⎧ ∀x ∈ Ω − , ⎨ Δu(x) = 0 ∂ u(x) + b(x)u(x) = g(x) ∀x ∈ ∂Ω , (6.53) ⎩ ∂νΩ u is harmonic at infinity, 1,α in the unknown u ∈ Cloc (Ω − ). As a first step, we prove the following theorem, which ensures the uniqueness of a solution of the Robin problems (6.52) and (6.53) under suitable assumptions.

Theorem 6.56. The following statements hold.  (i) If b ≥ 0 on ∂Ω, ∂Ωj b dσ = 0 for all j ∈ {1, . . . , κ + }, and u1 , u2 ∈ C 1,α (Ω) solve the interior Robin problem (6.52), then u1 = u2 .  (ii) If n ≥ 3, b ≤ 0 on ∂Ω, ∂(Ω − )j b dσ = 0 for all j ∈ {1, . . . , κ − }, and u1 , 1,α u2 ∈ Cloc (Ω − ) solve the exterior Robin problem (6.53), then u1 = u2 . (iii) If n = 2, b ≤ 0 on ∂Ω, ∂(Ω − )j b dσ = 0 for all j ∈ {0, . . . , κ − } (including 1,α for j = 0), and u1 , u2 ∈ Cloc (Ω − ) solve the exterior Robin problem (6.53), then u1 = u2 .

Proof. We first prove (i). We set u ≡ u1 − u2 , and we show that ∇u = 0 in Ω. By the first Green Identity of Theorem 4.2, we have

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6 Boundary Value Problems and Boundary Integral Operators





0≤



|∇u| dx =

u νΩ · ∇udσ = −

2

Ωj

∂Ωj

b u2 dσ ≤ 0

(6.54)

∂Ωj

for all j ∈ {1, . . . , κ + } (here we have used the assumption that b ≥ 0). Then we have ∇u = 0 in Ωj and there exists cj ∈ R such that u(x) = cj for all x ∈ Ωj and all j ∈ {1, . . . , κ + }. By (6.54) and, again, by inequality b ≥ 0, we have   0= b u2 dσ = c2j b dσ . ∂Ωj

∂Ωj

 Since we assume that ∂Ωj b dσ = 0, we deduce that cj = 0 for all j ∈ {1, . . . , κ + }. We now consider statements (ii) and (iii). As in (i), we set u ≡ u1 − u2 . Since u is harmonic at infinity, the first Green Identity for exterior domains of Corollary 4.7 in Ω − implies that   2 0≤ |∇u| dx = − u νΩ · ∇u dσ Ω−



∂Ω

κ   −

2

=

b u dσ = ∂Ω

j=0

∂(Ω − )j

b u2 dσ ≤ 0

(we note that b ≤ 0 by assumption). Accordingly, we have ∇u = 0 in Ω − and thus u is constant in all connected components of Ω − . Then, arguing as in the proof of statement (i), we conclude that u = 0 in all the bounded connected components of Ω − . If n ≥ 3 the condition that u is harmonic at infinity implies that u vanishes on   the unbounded connected component (Ω − )0 . Inspired by the representation Theorem 6.48 for harmonic functions, we look for a + solution of the interior Robin problem (6.52) in the form of a sum vΩ [μ] + c for + some μ ∈ C 0,α (∂Ω)0 and c ∈ R. We note that if vΩ [μ] + c solves problem (6.52), then the jump formula for the normal derivative of a single layer potential and the boundary condition of the interior Robin problem (6.52) imply that μ must solve the following integral equation   1 on ∂Ω . (6.55) − I + WΩt [μ] + b VΩ [μ] + b c = g 2 + [μ]+ Conversely, if (μ, c) ∈ C 0,α (∂Ω)0 ×R is a solution of equation (6.55), then vΩ c solves the Robin problem in (6.52). Then, we find convenient to introduce the operator LΩ,b,+ from C 0,α (∂Ω) × R to C 0,α (∂Ω), defined by   1 t LΩ,b,+ [μ, c] ≡ − I + WΩ [μ] + b VΩ [μ] + b c (6.56) 2

for all (μ, c) ∈ C 0,α (∂Ω) × R. Similarly, if n ≥ 3, we introduce the operator LΩ,b,− from C 0,α (∂Ω) to 0,α C (∂Ω) defined by

6.14 A Robin Problem

219

 LΩ,b,− [μ] ≡

 1 I + WΩt [μ] + b VΩ [μ] 2

(6.57)

for all μ ∈ C 0,α (∂Ω), and if n = 2, we introduce the operator LΩ,b,− from C 0,α (∂Ω) × R to C 0,α (∂Ω) defined by   1 I + WΩt [μ] + b VΩ [μ] + b c (6.58) LΩ,b,− [μ, c] ≡ 2 for all (μ, c) ∈ C 0,α (∂Ω) × R. Then, if n ≥ 3, a function μ ∈ C 0,α (∂Ω) satisfies equation (6.59) LΩ,b,− [μ] = g , − if and only if vΩ [μ] is a solution of the exterior Robin problem (6.53). If n = 2, a 0,α pair (μ, c) ∈ C (∂Ω)0 × R satisfies equation

LΩ,b,− [μ, c] = g

(6.60)

− if and only if vΩ [μ] + c is a solution of the exterior Robin problem (6.53). To analyze the integral equations (6.55), (6.59) and (6.60), we introduce the following statement.

Proposition 6.57. The following statements hold.  (i) If n ≥ 2, b ≥ 0 on ∂Ω, and ∂Ωj b dσ = 0 for all j ∈ {1, . . . , κ + }, then the operator from C 0,α (∂Ω) × R to itself that takes a pair (μ, c) to    μ dσ LΩ,b,+ [μ, c], ∂Ω

is an isomorphism. In particular, LΩ,b,+ is an isomorphism from C 0,α (∂Ω)0 × R onto C 0,α (∂Ω).  (ii) If n ≥ 3, b ≤ 0 on ∂Ω, and ∂(Ω − )j b dσ = 0 for all j ∈ {1, . . . , κ − }, then LΩ,b,− is an isomorphism from C0,α (∂Ω) to itself. (iii) If n = 2, b ≤ 0 on ∂Ω, and ∂(Ω − )j b dσ = 0 for all j ∈ {0, . . . , κ − } (including j = 0), then the operator from C 0,α (∂Ω) × R to itself that takes a pair (μ, c) to    LΩ,b,− [μ, c],

μ dσ ∂Ω

is an isomorphism. In particular, LΩ,b,− is an isomorphism from C 0,α (∂Ω)0 × R onto C 0,α (∂Ω). Proof. Let us denote by JΩ,b,+ the operator of statement (i). We can write JΩ,b,+ = JˆΩ,b,+ + J˜Ω,b,+ with

220

6 Boundary Value Problems and Boundary Integral Operators

 JˆΩ,b,+ [μ, c] ≡

1 − μ, c 2





and J˜Ω,b,+ [μ, c] ≡



 WΩt [μ]

+ bVΩ [μ] + bc , −c +

μ dσ ∂Ω

for all (μ, c) ∈ C 0,α (∂Ω) × R. Then we can see that JˆΩ,b,+ is an isomorphism from C 0,α (∂Ω) × R to itself. Moreover, J˜Ω,b,+ is compact from C 0,α (∂Ω) × R to itself. Indeed, by Theorem 6.8 (ii), WΩt is compact in C 0,α (∂Ω) and, by Theorem 4.25, VΩ is bounded from C 0,α (∂Ω) to C 1,α (∂Ω), which is compactly embedded into C 0,α (∂Ω) (cf. Proposition 2.25 and Theorem 2.83). Being the sum of an isomorphism and a compact operator, it follows that JΩ,b,+ is a Fredholm operator of index zero (cf. Theorem 5.3). Accordingly, to show that it is a isomorphism it suffices to prove that it is injective. So, let (μ, c) ∈ C 0,α (∂Ω) × R satisfy equations  LΩ,b,+ [μ, c] = 0 , μ dσ = 0 . ∂Ω

We claim that (μ, c) = (0, 0). Indeed, the jump formula for the normal derivative of + the single layer potential of Theorem 6.6 implies that the function v = vΩ [μ] + c satisfies the interior Robin problem (6.52)with g = 0. Then the uniqueness Theorem 6.56 (i) implies that v = 0. Since ∂Ω μ dσ = 0, the representation Theorem 6.48 (i) for harmonic functions implies that (μ, c) = (0, 0). It follows that JΩ,b,+ is an isomorphism and we can verify the validity of statement (i). The proof of statements (ii) and (iii) follows the lines of that of statement (i), we only need to use Theorem 6.56 (ii), (iii) instead of Theorem 6.56 (i) and Theorem 6.48 (ii), (iii) instead of Theorem 6.48 (i) (see also Theorem 4.23).   We are now ready to state the following. Theorem 6.58 (Existence and Uniqueness for the Robin Problem). The following statements hold.  (i) If n ≥ 2, b ≥ 0 on ∂Ω, and ∂Ωj b dσ = 0 for all j ∈ {1, . . . , κ + }, then the interior Robin problem (6.52) has a unique solution u ∈ C 1,α (Ω). Such solution is delivered by the formula + u = vΩ [μ] + c ,

(6.61)

where (μ, c) is the only solution in C 0,α (∂Ω)0 ×R of the equation LΩ,b,+ [μ, c] = g.  (ii) If n ≥ 3, b ≤ 0 on ∂Ω, and ∂(Ω − )j b dσ = 0 for all j ∈ {1, . . . , κ − }, then 1,α the exterior Robin problem (6.53) has a unique solution u ∈ Cloc (Ω − ). Such solution is delivered by the formula − u = vΩ [μ] ,

(6.62)

where μ is the only solution in C 0,α (∂Ω) of the equation LΩ,b,− [μ] = g.

6.14 A Robin Problem

221

 (iii) If n = 2, b ≤ 0 on ∂Ω, and ∂(Ω − )j b dσ = 0 for all j ∈ {0, . . . , κ − } (including j = 0), then the exterior Robin problem (6.53) has a unique solution 1,α (Ω − ). Such solution is delivered by the formula u ∈ Cloc − u = vΩ [μ] + c ,

(6.63)

where the pair (μ, c) is the only solution in C 0,α (∂Ω)0 × R of the equation LΩ,b,− [μ, c] = g. Proof. By Proposition 6.57, the integral equations (6.55), (6.59), and (6.60) have a unique solution. Then the jump formulas form the normal derivative of the single layer potential of Theorem 6.6 imply that the functions in (6.61)–(6.63) are solutions of the interior Robin problem (6.52) and of the exterior Robin problem (6.53) both for n ≥ 3 and n = 2, respectively. Finally, the uniqueness of such solutions follows by Theorem 6.56.   In conclusion of this chapter we observe that, when dealing with the twodimensional case, one could exploit the fact that harmonic functions are real (or imaginary) parts of holomorphic functions. Then, to obtain integral representations of the solutions of boundary value problems for the Laplace equation, one could resort to the Cauchy formula and other complex analysis techniques. This program has been carried out for a variety of boundary value problems. Classic references on the topic are the books of Gakhov [105] and Muskhelishvili [227]. More recent references can be found in the monograph by Mityushev and Rogosin [224]. We also note that in the present book we focus exclusively on boundary value problems in domains of class C 1,α , but the regularity of the boundary may be relaxed. Indeed, there exists a large literature on boundary value problems in piecewise smooth domains and in Lipschitz domains. The reader interested in boundary value problems in piecewise smooth domains is referred, for example, to Borsuk and Kondratiev [33], Dauge [84], Grisvard [112], Plamenevskij [242], and to references therein. For the development of potential theoretic methods in Lipschitz domains we refer, for example, to Gewinner and Stephan [116], Maz’ya and Soloviev [205], McLean [206], Medkov´a [208], Mitrea and Mitrea [215], and to references therein.

Chapter 7

Poisson Equation and Volume Potentials

Abstract This chapter analyzes volume potentials and the solvability of the Poisson equation by means of such volume potentials. To go into detail, we first consider the case of weakly singular kernels, and then the case in which the first order partial derivatives of the kernel are also weakly singular. That been done, we turn to singular kernels and, after that, to weakly singular kernels with singular first order partial derivatives. Finally, we prove the basic properties of the corresponding volume potentials in spaces of H¨older continuous functions, in Schauder spaces, and in the Roumieu classes. Although similar properties in Schauder spaces can be found in the classical monographs of G¨unter (Potential theory and its applications to basic problems of mathematical physics. Translated from the Russian by John R. Schulenberger. Frederick Ungar Publishing Co., New York, 1967) and of Kupradze, Gegelia, Basheleishvili, and Burchuladze (Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. North-Holland series in applied mathematics and mechanics, vol 25, Russian edition. North-Holland Publishing Co., Amsterdam, New York, 1979. Edited by V. D. Kupradze), here we prove the corresponding statements with optimal H¨older exponents. To do so, we follow a proof of Miranda (Atti Accad Naz Lincei Mem Cl Sci Fis Mat Natur Sez I (8), 7, 303–336, 1965) and we exploit a known lemma (cf. Majda and Bertozzi, Vorticity and incompressible flow. Cambridge texts in applied mathematics, vol 27, Prop. 8.12, pp 348–350. Cambridge University Press, Cambridge, 2002) for which we provide a proof of Mateu, Orobitg, and Verdera (J Math Pures Appl (9), 91(4), 402–431, 2009). To conclude the chapter, we use the properties proven for the volume potentials to study the solvability of the basic boundary value problems for the Poisson equation. Unless otherwise specified, we understand that throughout this chapter n is a natural number bigger than or equal to two. Namely, n ∈ N, n ≥ 2. We also fix a real number α ∈]0, 1[ . © Springer Nature Switzerland AG 2021 M. Dalla Riva et al., Singularly Perturbed Boundary Value Problems, https://doi.org/10.1007/978-3-030-76259-9 7

223

224

7 Poisson Equation and Volume Potentials

7.1 Preliminary Remarks on the Poisson Equation We are interested in the existence of solutions of boundary value problems for the Poisson equation Δu = f (7.1) in an open subset Ω, which we assume to be at least of class C 1,α . The function f is given and the unknown of the problem is u. A good starting point to study the existence of solutions for the Poisson equation is the third Green Identity of Theorem 4.5, which shows that any u ∈ C 1,α (Ω) ∩ C 2 (Ω) such that Δu is integrable can be written as a sum of a harmonic function and of the Newtonian potential  Sn (x − y)Δu(y) dy ∀x ∈ Ω . Ω

Thus it is natural to think of looking for a solution of the Poisson equation (7.1) in the form of the Newtonian potential  Sn (x − y)f (y) dy ∀x ∈ Ω , (7.2) Pn [Ω, f ](x) ≡ Ω

which is a particular case of a volume potential (as in G¨unter [115]). By the classical Kellogg Theorem, if f ∈ C 0,α (Ω), then the corresponding Newtonian potential belongs to C 2,α (Ω) and solves the Poisson equation classically (see also Theorem 7.18). Then, if we have to show the existence of a solution for a boundary value problem for the Poisson equation (7.1), we can always introduce a new unknown v = u−Pn [Ω, f ] and obtain a new boundary value problem for the Laplace equation Δv = 0, as for example one of those considered in the previous chapter. Thus, we now turn to analyze volume potentials.

7.2 Volume Potentials As we shall see in Section 7.3, we can use the Newtonian potential (7.2) associated to the fundamental solution Sn of the Laplace operator and the single and double layer potentials of the previous Chapter 6 to represent the solutions of the basic boundary value problems for the Poisson equation in Ω. However, later in the book we need to consider volume potentials with integral kernels that are different from Sn . For this reason, we study in this section volume potentials associated to a general class of functions h, and not only to Sn . We begin with two subsections of preliminaries, where we follow the presentation of [69, §3]. In the first one we consider volume potentials for a class of weakly singular integral kernels and in the second one volume potentials for a class of kernels that are weakly singular together with their first order partial derivatives.

7.2 Volume Potentials

225

7.2.1 Volume Potentials with Weakly Singular Kernels We first introduce a class of continuous functions in a punctured ball that are singular at the center. Definition 7.1. Let λ ∈]0, +∞[. Let R ∈]0, +∞[. Then we denote by A0λ (R) the set of functions k ∈ C 0 ((Bn (0, R)) \ {0}) such that |k(x)| |x|λ < +∞ ,

sup x∈(Bn (0,R))\{0}

and we set kA0λ (R) ≡

sup

|k(x)| |x|λ

∀k ∈ A0λ (R) .

x∈(Bn (0,R))\{0}

We can readily verify that (A0λ (R),  · A0λ (R) ) is a Banach space. Then we say that the singularity of a function k defined in a punctured neighborhood of 0 is weak or that k is weakly singular (convolution) kernel at 0 provided that k is integrable in a punctured neighborhood of 0. If instead k is not integrable in any punctured neighborhood of zero, we say that k is a singular (convolution) kernel, or that k has a strong singularity at 0. If λ < n then the elements of A0λ (R) are integrable in a punctured neighborhood of 0 and accordingly have a weak singularity at 0. If instead λ ≥ n, then the elements of A0λ (R) may be singular. We now introduce the volume potential associated to a function of class A0λ (R) in the case where λ < n. We do so by means of the following. Proposition 7.2. Let λ ∈]0, n[. Let Ω be a bounded open subset of Rn . Then the following statements hold. (i) If (k, ϕ) ∈ A0λ (diam (Ω)) × L∞ (Ω) and if x ∈ Ω, then the function from Ω to R that takes y ∈ Ω to k(x − y)ϕ(y) is integrable. (ii) If (k, ϕ) ∈ A0λ (diam (Ω)) × L∞ (Ω), then the function  k(x − y)ϕ(y) dy ∀x ∈ Ω (7.3) P[k, ϕ](x) ≡ Ω

is continuous on Ω. (iii) P[k, ϕ] is bounded and sup |P[k, ϕ]| ≤ sn Ω

(diam (Ω))n−λ kA0λ (diam (Ω)) ϕL∞ (Ω) n−λ

for all (k, ϕ) ∈ A0λ (diam (Ω)) × L∞ (Ω). Proof. If (k, ϕ) ∈ A0λ (diam (Ω)) × L∞ (Ω), then we have |k(x − y)ϕ(y)| ≤ |k(x − y)|ϕL∞ (Ω)

for a.a. y ∈ Ω

(7.4)

226

7 Poisson Equation and Volume Potentials

for all x ∈ Ω. Then k(x − ·)ϕ(·) is integrable in Ω. Since Ω ⊆ Bn (x, diam (Ω)) for all x ∈ Ω, we have       k(x − y)ϕ(y) dy  ≤ |k(x − y)| dyϕL∞ (Ω)   Ω B (x,diam (Ω))  n dy ≤ kA0λ (diam (Ω)) ϕL∞ (Ω) |x − y|λ Bn (x,diam (Ω)) = kA0λ (diam (Ω)) sn

(diam (Ω))n−λ ϕL∞ (Ω) n−λ

∀x ∈ Ω .

Hence, inequality (7.4) follows. To show that P[k, ϕ] is continuous, we proceed by a standard argument. We take a function g ∈ C ∞ (R) such that ∀t ∈] − ∞, 1] ,

g(t) = 0

g(t) = 1

∀t ∈ [2, +∞[ .

(7.5)

Then we set gδ (t) ≡ g(t/δ)

∀t ∈ R ,

(7.6)



and uδ (x) ≡

gδ (|x − y|)k(x − y)ϕ(y) dy

∀x ∈ Ω ,

(7.7)

Ω

for all δ ∈]0, +∞[. We observe that the function that takes (x, y) ∈ Ω × Ω to gδ (|x − y|)k(x − y) is continuous. By the definition of gδ , we have   |gδ (|x − y|)k(x − y)ϕ(y)| ≤ gL∞ (R)

sup

|k| |ϕ(y)| , (7.8)

Bn (0,diam (Ω))\Bn (0,δ)

for all x ∈ Ω and for almost all y ∈ Ω. Since ϕ ∈ L1 (Ω), inequality (7.8) and the continuity theorem for integrals depending on a parameter imply that uδ is continuous in Ω. In order to prove that P[k, ϕ] is continuous, it suffices to show that lim uδ = P[k, ϕ]

δ→0

uniformly in Ω .

Since 1 − gδ (|x − y|) = 0 for |x − y| ≥ 2δ, we have |P[k, ϕ](x) − uδ (x)|       (1 − gδ (|x − y|))k(x − y)ϕ(y) dy  =   Bn (x,2δ)∩Ω  ≤ (1 + gL∞ (R) )ϕL∞ (Ω) |k(x − y)| dy B (x,2δ)∩Bn (x,diam (Ω))  n = (1 + gL∞ (R) )ϕL∞ (Ω) |k(y)| dy Bn (0,2δ)∩Bn (0,diam (Ω))

(7.9)

7.2 Volume Potentials

227

≤ (1 + gL∞ (R) )ϕL∞ (Ω)  ×

 |k(y)| |y|

sup

λ Bn (0,2δ)

y∈Bn (0,diam (Ω))\{0}

= (1 + gL∞ (R) )ϕL∞ (Ω) kA0λ (diam (Ω)) sn

|y|−λ dy

(2δ)n−λ , n−λ

for all x ∈ Ω and for all δ ∈]0, diam (Ω)/2]. Hence, (7.9) holds true and P[k, ϕ] is continuous.   The function P[k, ϕ] of (7.3) is said to be a volume potential of (convolution) kernel k and density (or moment) ϕ. Whenever necessary, we may write PΩ [k, ϕ] instead of P[k, ϕ] in order to emphasize that the domain of integration is Ω. As we can easily verify, the fundamental solution Sn of the Laplace operator belongs to A0max{n−2,1/2} (R) for all R ∈]0, +∞[ and accordingly Proposition 7.2 implies the validity of the following. Proposition 7.3. Let Ω be a bounded open subset of Rn . Then the operator from L∞ (Ω) to C 0 (Ω) that takes ϕ to P[Sn , ϕ] (= Pn [Ω, ϕ]) is linear and continuous (see (7.2)).

7.2.2 Volume Potentials with Kernels Which are Weakly Singular Together with Their First Order Partial Derivatives We now introduce a subclass A1λ (R) of A0λ (R) consisting of continuously differentiable functions with a controlled singular behavior at the origin (cf. Definition 7.1). Definition 7.4. Let λ ∈]0, +∞[. Let R ∈]0, +∞[. Then we denote by A1λ (R) the set of functions k ∈ C 1 ((Bn (0, R)) \ {0}) such that ∂k ∈ A0λ+1 (R) ∂xj

k ∈ A0λ (R) ,

∀j ∈ {1, . . . , n} ,

and we set kA1λ (R)

0 n 0  0 ∂k 0 0 0 ≡ kA0λ (R) + 0 ∂xj 0 j=1

∀k ∈ A1λ (R) . A0λ+1 (R)

 One can easily verify that A1λ (R),  · A1λ (R) is a Banach space. In the following 

proposition we consider the function P[k, ϕ] with (k, ϕ) in A1λ (diam (Ω))×L∞ (Ω) and λ smaller than (n − 1).

Proposition 7.5. Let λ ∈]0, n − 1[. Let Ω be a bounded open subset of Rn . Then the following statements hold.

228

7 Poisson Equation and Volume Potentials

(i) If (k, ϕ) ∈ A1λ (diam (Ω)) × L∞ (Ω) and if x ∈ Ω, then the functions from ∂k (x − y)ϕ(y), with Ω to R which take y ∈ Ω to k(x − y)ϕ(y) and to ∂x j j ∈ {1, . . . , n}, are integrable. (ii) If (k, ϕ) ∈ A1λ (diam (Ω)) × L∞ (Ω), then P[k, ϕ] ∈ C 1 (Ω) and ∂ ∂k P[k, ϕ] = P[ , ϕ] ∂xj ∂xj

in Ω .

(7.10)

Proof. Statement (i) is an immediate consequence of Proposition 7.2 applied to k ∂k and ∂x . j ∂k We now consider statement (ii). By Proposition 7.2 (ii), P[k, ϕ] and P[ ∂x , ϕ] j ∂ P[k, ϕ] are continuous in Ω for all j ∈ {1, . . . , n}. Thus it suffices to show that ∂x j exists in Ω and that (7.10) holds in Ω. We proceed by a standard argument, g ∈ C ∞ (R), gδ ∈ C ∞ (R), uδ be as in (7.5), (7.6), (7.7) for all δ ∈]0, +∞[. We already know that uδ ∈ C 0 (Ω). We also observe that the function that takes (x, y) ∈ Ω × Ω to gδ (|x − y|)k(x − y) is of class C 1 . We now show that uδ ∈ C 1 (Ω) by applying the classical theorem of differentiation for integrals depending on a parameter. We have     ∂   (7.11) {g (|x − y|)k(x − y)ϕ(y)}   ∂xj δ         x j − yj ∂k ≤ gδ (|x − y|) k(x − y)ϕ(y) + gδ (|x − y|) (x − y)ϕ(y) |x − y| ∂xj

for all x ∈ Ω and for almost all y ∈ Ω. The functions Bn (0, diam (Ω)) \ {0}. Hence, the functions

∂k ∂xj

∂k ∂xj

are continuous in the set

are bounded in the compact set

Bn (0, diam (Ω)) \ Bn (0, δ). Then the right-hand side of (7.11) is less than or equal to   1  g L∞ (R) |k| |ϕ(y)| (7.12) sup δ Bn (0,diam (Ω))\Bn (0,δ)     ∂k    +gL∞ (R) sup  ∂xj  |ϕ(y)| Bn (0,diam (Ω))\Bn (0,δ) for all x ∈ Ω and for almost all y ∈ Ω. Since ϕ ∈ L1 (Ω), inequalities (7.11), (7.12) and the differentiability theorem for integrals depending on a parameter imply that  ∂ ∂uδ (x) = [gδ (|x − y|)k(x − y)] ϕ(y) dy ∀x ∈ Ω ∂xj ∂x j Ω and that

∂uδ ∂xj

has a continuous extension to Ω. Hence, uδ ∈ C 1 (Ω). In order to prove

that P[k, ϕ] belongs to C 1 (Ω), it suffices to remember the validity of the limiting relation (7.9) and to show that

7.2 Volume Potentials

229

lim

δ→0

∂uδ ∂k = P[ , ϕ] ∂xj ∂xj

uniformly in Ω

(7.13)

for all j ∈ {1, . . . , n}. Since the support of gδ is contained in [δ, 2δ], the same argument we have exploited to prove (7.9) implies that    ∂k  ∂uδ P[   ∂xj , ϕ](x) − ∂xj (x)     ∂k  ≤  (1 − gδ (|x − y|)) (x − y)ϕ(y) dy  ∂xj Ω       |x − y| x j − yj 1   + g k(x − y)ϕ(y) dy  δ |x − y| Ω δ ≤ (1 + gL∞ (R) )ϕL∞ (Ω)      ∂k  λ+1 (2δ)n−λ−1   × sup sn  ∂xj (y) |y| n−λ−1 y∈Bn (0,diam (Ω))\{0}  1 |k(x − y)| dyϕL∞ (Ω) + g  L∞ (R) δ Bn (x,2δ)\Bn (x,δ) ≤ sn (1 + gL∞ (R) + g  L∞ (R) )ϕL∞ (Ω)      ∂k  λ+1 (2δ)n−λ−1   |y| × sup (y)   n−λ−1 x∈Bn (0,diam (Ω))\{0} ∂xj    n−λ 1 λ (2δ) |k(y)| |y| + sup ∀x ∈ Ω , δ y∈Bn (0,diam (Ω))\{0} n−λ for all δ ∈]0, diam (Ω)/2[. Since n − λ − 1 > 0, the limiting relation (7.13) follows.   As we can easily verify, the fundamental solution Sn of the Laplace operator belongs to A1max{n−2,1/2} (R) for all R ∈]0, +∞[, and accordingly Propositions 7.2 and 7.5 imply the validity of the following. Proposition 7.6. Let Ω be a bounded open subset of Rn . Then the operator from L∞ (Ω) to C 1 (Ω) that takes ϕ to P[Sn , ϕ] = Pn [Ω, ϕ] is linear and continuous (see (7.2)) and ∂ ∂Sn P[Sn , ϕ] = P[ , ϕ] in Ω , ∂xj ∂xj for all j ∈ {1, . . . , n}. Then we introduce the following classical formula for the first order partial derivatives of a volume potential with a differentiable density. Lemma 7.7. Let λ ∈]0, n − 1[. Let Ω be a bounded open Lipschitz subset of Rn . If (k, ϕ) ∈ A1λ (diam (Ω)) × C 1 (Ω) and j ∈ {1, . . . , n}, then

230

7 Poisson Equation and Volume Potentials

∂ ∂ϕ P[k, ϕ](x) = P[k, ](x) − ∂xj ∂xj

 k(x − y)ϕ(y)(νΩ )j (y) dσy

∀x ∈ Ω .

∂Ω

(7.14)

Proof. By Proposition 7.5 and by the Leibnitz rule, we have  ∂ ∂k P[k, ϕ](x) = (x − y)ϕ(y) dy ∂xj ∂x j Ω  ∂ =− (k(x − y))ϕ(y) dy ∂y j Ω   ∂ ∂ϕ =− (k(x − y)ϕ(y)) dy + k(x − y) (y) dy ∂y ∂y j j Ω Ω

(7.15)

∀x ∈ Ω .

Next we fix x ∈ Ω and we take x ∈]0, dist (x, ∂Ω)[. Then Bn (x, ) ⊆ Ω and the set Ω ≡ Ω \ Bn (x, ) is of Lipschitz class for all  ∈]0, x [. By the Divergence Theorem 4.1, we have  ∂ (k(x − y)ϕ(y)) dy (7.16) Ω ∂yj   ∂ ∂ (k(x − y)ϕ(y)) dy + (k(x − y)ϕ(y)) dy = Ω ∂yj Bn (x,) ∂yj   xj − yj = dσy k(x − y)ϕ(y)(νΩ )j (y) dσy + k(x − y)ϕ(y) |x − y| ∂Ω ∂Bn (x,)   ∂k ∂ϕ − (x − y)ϕ(y) dy + k(x − y) (y) dy ∂yj Bn (x,) ∂xj Bn (x,) for all  ∈]0, x [. We note that     xj − yj   dσy  k(x − y)ϕ(y)   ∂Bn (x,)  |x − y|  ≤

sup

|k(y)| |y|

(7.17)  λ



|y|−λ dσy

ϕL∞ (Ω)

y∈Bn (0,diam (Ω))\{0}

∂Bn (0,)

= kA0λ (diam (Ω)) ϕL∞ (Ω) sn n−1−λ for all  ∈]0, x [. By Proposition 7.5 (i), both the functions k(x −

∂ϕ (·) ·) ∂x j

∂k ∂xj (x

− ·)ϕ(·) and

are integrable in Ω and accordingly 

∂k (x − y)ϕ(y) dy = 0 , ∂x j Bn (x,)  ∂ϕ k(x − y) (y) dy = 0 . lim →0 B (x,) ∂y j n

lim

→0

(7.18)

7.2 Volume Potentials

231

By (7.17) and (7.18), we can take the limit as  tends to 0 in (7.16) and deduce that   ∂ (k(x − y)ϕ(y)) dy = k(x − y)ϕ(y)(νΩ )j (y) dσy . Ω ∂yj ∂Ω Then equality (7.15) implies the validity of formula (7.14).

 

7.2.3 Volume Potentials with Singular Kernels and with a Constant Density In Section 7.2.1 we have considered volume potentials of the form  k(x − y)ϕ(y) dy Ω

where ϕ is essentially bounded and k has a weak singularity at the origin, i.e., k is integrable in a neighborhood of the origin, a case in which k(x−y)ϕ(y) is integrable in the sense of Lebesgue in the variable y ∈ Ω for all x ∈ Rn . We now want to consider the case in which the singularity of k at the origin is strong, i.e., the case in which k is not integrable in a neighborhood of the origin. In such a case, the function k(x − y)ϕ(y) may not  have an integral in the sense of Lebesgue in the variable y ∈ Ω, but the integral Ω k(x − y)ϕ(y) dy may exist in the sense of the principal value. In the present subsection we mainly concentrate on the case where ϕ is constant. Moreover, we consider only the specific class of kernels which are positively homogeneous of degree −n, a class which includes the important case of the second order partial derivatives of the fundamental solution Sn of the Laplace operator, which is relevant for the present book. We start by introducing some preliminaries on the principal value. Definition 7.8. Let Ω be an open subset of Rn . Let xo ∈ Ω. Let f be a function from Ω \ {xo } to R such that the restriction f|Ω\Bn (xo ,) is integrable for all  ∈]0, +∞[. We say that f admits an integral in the sense of the principal value in Ω provided that the limit   f dx ≡ lim f dx p.v. Ω

→0

Ω\Bn (xo ,)

exists in R ∪ {−∞} ∪ {+∞}. We now look closely at the case of positively homogeneous functions of degree −n. Proposition 7.9. Let n ∈ N \ {0}. Let f ∈ C 0 (Rn \ {0}) be a positively homogeneous function of degree −n. Then the following statements hold. (i) If a, b ∈]0, +∞[, a < b, then

232

7 Poisson Equation and Volume Potentials



 f (x) dx =

f (u) dσu log(b/a) ,

Bn (0,b)\Bn (0,a)

(7.19)

∂Bn (0,1)

 where we understand that ∂Bn (0,1) f dσ ≡ f (1) + f (−1) in case n = 1. In particular, if  f dσ = 0 , (7.20) ∂Bn (0,1)

then the integral of f on the circular annulus Bn (0, b) \ Bn (0, a) equals 0. (ii) If condition (7.20) holds true, then   f (x) dx = lim f (x) dx = 0 p.v. →0

Bn (0,b)

Bn (0,b)\Bn (0,)



for all b ∈]0, +∞[. Moreover, p.v. Bn (0,b) f (x) dx can exist finite for some b in ]0, +∞[ only if condition (7.20) holds true. Proof. Let n ≥ 2. By integrating on the spheres, we have 



b



f (x) dx =

f (ξ) dσξ dρ

Bn (0,b)\Bn (0,a)



b

a



∂Bn (0,ρ)



b



f (η) dση ρ−n+n−1 dρ

f (ρη) dση ρn−1 dρ =

= 

a

∂Bn (0,1)

=

∂Bn (0,1)

a

f (η) dση log(b/a) . ∂Bn (0,1)

and thus statement (i) holds true. If instead n = 1, then Bn (0, b) \ Bn (0, a) = ] − b, −a] ∪ [a, b[ and we have 

 Bn (0,b)\Bn (0,a)



b

= 

−b



a

f (x) dx a

b

f (x) dx a

b

=

b

f (x) dx +

f (−x) dx + a



−a

f (x) dx =

f (−1)|x|−1 dx +



b

f (1)|x|−1 dx

a

= (f (−1) + f (1)) log(b/a) , and thus statement (i) holds true also for n = 1. Statement (ii) is an immediate consequence of statement (i).   By the previous statement, one can prove the following lemma for volume potentials with constant density. Lemma 7.10. Let Ω be an open subset of Rn of finite measure. Let the function k ∈ C 0 (Rn \ {0}) be positively homogeneous of degree −n. Let

7.2 Volume Potentials

233

 k dσ = 0 .

(7.21)

∂Bn (0,1)

Then the following statements hold. (i) (Cancellation property for singular integrals) If x ∈ Ω, then the function k(x − y) is Lebesgue integrable in the variable y ∈ Ω \ Bn (x, ) for all  in ]0, d(x)] and   k(x − y) dy = k(x − y) dy ∀ ∈]0, d(x)] , Ω\Bn (x,)

Ω\Bn (x,d(x))

where d(x) ≡ dist(x, ∂Ω) . If x ∈ Rn \Ω, then the function k(x − y) is Lebesgue integrable in the variable y ∈ Ω. (ii) If x ∈ Ω, then   k(x − y) dy = k(x − y) dy . p.v. Ω

If x ∈ Rn \ Ω, then p.v.

Ω\Bn (x,d(x))

 Ω

k(x − y) dy =

 Ω

k(x − y) dy.

Proof. (i) If x ∈ Ω, then Bn (x, d(x)) \ Bn (x, ) ⊆ Ω

∀ ∈]0, d(x)]

and  χΩ\B

n (x,)

 (y)k(x − y) ≤

sup |k| |x − y|−n ≤ −n ∂Bn (0,1)

sup |k| ∂Bn (0,1)

for all y ∈ Ω \ Bn (x, ) and  ∈]0, d(x)]. Since the measure of Ω is finite, the function k(x − y) is Lebesgue integrable in the variable y ∈ Ω \ Bn (x, ) for all  ∈]0, d(x)]. Then formula (7.19) and assumption (7.21) imply that  k(x − y) dy Ω\Bn (x,)   = k(x − y) dy + k(x − y) dy Ω\Bn (x,d(x)) Bn (x,d(x))\Bn (x,)   = k(x − y) dy + k dσ log(d(x)/) Ω\Bn (x,d(x)) ∂Bn (0,1)  = k(x − y) dy + 0 ∀ ∈]0, d(x)] . Ω\Bn (x,d(x))

If instead x ∈ Rn \ Ω, then the assumption that Ω has finite measure and the inequality

234

7 Poisson Equation and Volume Potentials

 |k(x − y)| ≤

 sup |k| |x − y| ∂Bn (0,1)

 −n

 sup |k| d(x)−n

≤

∀y ∈ Ω ,

∂Bn (0,1)

ensure that k(x − y) is Lebesgue integrable in the variable y ∈ Ω. Hence, the proof of statement (i) is complete. Statement (ii) is an immediate consequence of statement (i).   Next we set   0,1  ≡ k ∈ Cloc (Rn \ {0}) : k is positively homogeneous of degree − n , K0,1 and

  kK0,1 ≡ kC 0,1 (∂Bn (0,1)) ∀k ∈ K0,1 .     We can easily verify that K0,1 is a Banach space and we consider the , kK0,1 subspace ! )   ≡ K0,1;e,0

 k ∈ K0,1 : k is even,

k dσ = 0

(7.22)

∂Bn (0,1)  . Then we introduce the following known lemma (cf. Majda and Bertozzi of K0,1 [196, Prop. 8.12, pp. 348–350]), which we exploit later in the analysis of volume potentials in H¨older spaces. For the sake of completeness, we include a proof that follows the lines of that of Mateu, Orobitg, Verdera [198, estimate of (IV )δ , p. 408].

Lemma 7.11. Let Ω be a bounded open subset of Rn of class C 1,α . Then there exists c∗Ω ∈]0, +∞[ such that        k(x − y) dy  ≤ c∗Ω kK ∀k ∈ K0,1;e,0 . (7.23) sup  0,1   n (x,)∈R ×]0,+∞[ Ω\Bn (x,) Proof. Since Ω is bounded and of class C 1,α , Lemma 2.63 of the uniform cylinders implies that there exist 0 < r < δ < δ∗ ≡ min{1, diam(Ω)} with the property that for all p ∈ ∂Ω there exists Rp ∈ On (R) such that C(p, Rp , r, δ) is a coordinate cylinder for Ω around p and the corresponding function γp that represents ∂Ω in C(p, Rp , r, δ) satisfies the condition ∇γp (0) = 0 and the inequality (7.24) a ≡ sup γp C 1,α (Bn−1 (0,r)) < +∞ . p∈∂Ω

Next we note that if  ∈ [r, +∞[, then

7.2 Volume Potentials

235

       k(x − y) dy  ≤ |k(x − y)| dy   Ω\Bn (x,)  Ω\Bn (x,)  dy  ≤ r−n mn (Ω)kK0,1 ≤ sup |k| n |x − y| ∂Bn (0,1) Ω\Bn (x,)  for all x ∈ Rn and k ∈ K0,1;e,0 . On the other hand, we have

Ω \ Bn (x, ) = (Ω \ Bn (x, r)) ∪ [Ω ∩ (Bn (x, r) \ Bn (x, ))]

∀ ∈]0, r[

and accordingly       k(x − y) dy    Ω\Bn (x,)              ≤ k(x − y) dy  +  k(x − y) dy   Ω\Bn (x,r)   Ω∩(Bn (x,r)\Bn (x,))         ≤ r−n mn (Ω)kK0,1 + k(x − y) dy  ∀ ∈]0, r[  Ω∩(Bn (x,r)\Bn (x,))   for all x ∈ Rn and k ∈ K0,1;e,0 . Thus it suffices to show that there exists c∗0,Ω in ]0, +∞[ such that       (7.25) k(x − y) dy  sup    (x,)∈Rn ×]0,r[ Ω∩(Bn (x,r)\Bn (x,))

≤ c∗0,Ω kK

0,1

 ∀k ∈ K0,1;e,0 .

To shorten our notation, we set An (x, , r) ≡ Bn (x, r) \ Bn (x, )

∀x ∈ Rn ,  ∈]0, r[ .

We first estimate the integral in (7.25) in case x ∈ ∂Ω. Since r < δ, we have Bn (0, r) ⊆ Bn−1 (0, r)×] − δ, δ[ and accordingly Bn (x, r) = x + Rxt (Bn (0, r)) ⊆ C(x, Rx , r, δ) . Then, by exploiting the change of variables y = x + Rxt (η, ζ)t , we obtain  k(x − y) dy Ω∩An (x,,r)  = k(−Rxt (η, ζ)t ) dηdζ [Rx (Ω−x)]∩An (0,,r)

(7.26)

236

7 Poisson Equation and Volume Potentials

 k˜x (η, ζ) dηdζ

= 

[Rx (Ω−x)]∩[Bn−1 (0,r)×]−δ,δ[]∩An (0,,r)

k˜x (η, ζ) dηdζ

=

∀ ∈]0, r[ ,

hypographs (γx )∩An (0,,r)

where k˜x (η, ζ) ≡ k(−Rxt (η, ζ)t ) and

∀(η, ζ) ∈ (Rn−1 × R) \ {(0, 0)} ,

 k˜x (η, ζ) dηdζ

(7.27)

hypographs (γx )∩An (0,,r)



= + hypographs (γx )∩(B+ n (0,r)\Bn (0,))

k˜x (η, ζ) dηdζ



+ 

− hypographs (γx )∩(B− n (0,r)\Bn (0,))

= + hypographs (γx )∩(B+ n (0,r)\Bn (0,))

k˜x (η, ζ) dηdζ

k˜x (η, ζ) dηdζ



+  −

− B− n (0,r)\Bn (0,)

k˜x (η, ζ) dηdζ

− {(η,ζ)∈B− n (0,r)\Bn (0,): γx (η)≤ζ 0 , ∂Ω o

for all ρ ∈]0, ρ∗ ]. Then we set i o (θ,ρ , θ,ρ , c,ρ ) i o u [θ , θ , c , λ[]]˜ u dσ ∂Ω o  ,ρ ,ρ ,ρ

∗i ∗o (θ,ρ , θ,ρ , c∗,ρ ) ≡ 

for all ρ ∈]0, ρ∗ ]. Clearly  ∗i ∗o u [θ,ρ , θ,ρ , c∗,ρ , λ[]]˜ u dσ = 1 ∂Ω o

∀ρ ∈]0, ρ∗ ]

and accordingly ˜ δ2,n  log , θ∗i , θ∗o , c∗ , λ[]] = 0 Q[, ,ρ ,ρ ,ρ Since

∗i ∗o lim (θ,ρ , θ,ρ , c∗,ρ ) = (Θi [], Θo [], C[])

ρ→0

∀ρ ∈]0, ρ∗ ] . in X0,α × R ,

(9.82)

9.4 A Steklov Eigenvalue Problem

369

there exists ρ ∈]0, ρ∗ ] such that ∗i ∗o (θ,ρ , θ,ρ , c∗,ρ ) ∈ V˜

∀ρ ∈]0, ρ ] .

Then Theorem 9.20 implies that ∗i ∗o (θ,ρ , θ,ρ , c∗,ρ ) = (Θi [], Θo [], C[])

∀ρ ∈]0, ρ ] .

Then for any such ρ equalities (9.81) and (9.82) imply that (θi , θo , c ) is a multiple of (Θi [], Θo [], C[]). Hence, v is a constant multiple of u[] and the eigenspace corresponding to the eigenvalue λ[] has dimension one.   ˜ of Theorem 9.22 is l, i.e., if λ ˜ = λl [Ω o ], then If the index of the eigenvalue λ one can exploit the continuity result (9.32) of Nazarov [237], the limiting relation ˜ = λl [Ω o ], and (9.77) to show that, possibly shrinking ˆ δ2,n  log ] = λ lim→0 λ[, ∗ ˆ  , we have λ[, δ2,n  log ] = λl [Ω()] for all  ∈]0, ∗ [. We now consider the macroscopic behavior of the eigenfunction u[] away from boundary of the hole. ˜ u Theorem 9.23. Let λ, ˜, ∗ , and ∗1 be as in Theorem 9.22. Let ΩM be an open o / ΩM . Then there exist M ∈]0, ∗ [ and a real analytic subset of Ω such that 0 ∈   map UM from ] − M , M [×] − ∗1 , ∗1 [ to C 1,α (ΩM ) such that ΩM ⊆ Ω() ,

(9.83)

u[]|ΩM = UM [, δ2,n  log ] , for all  ∈]0, ΩM [. Moreover, UM [0, 0] = u ˜|ΩM . Proof. Let M ∈]0, ∗ [ be such that ΩM ⊆ Ω() for all  ∈ [−M , M ]. Then we set  UΩ(∗M ) [, 1 ](x) ≡ n−1 Sn (x − η)Θi [, 1 ](η) dση (9.84) i ∂Ω  Sn (x − y)Θo [, 1 ](y) dσy + C[, 1 ] ∀x ∈ ΩM + ∂Ω o

for all (, 1 ) ∈]−M , M [×]−∗1 , ∗1 [. Then the equality in (9.83) holds by definition of UΩ(∗M ) (see also (9.79)). Also, Proposition 9.18 (ii) and equality (9.78) imply the ˜|ΩM . validity of equality UM [0, 0] = u + We now prove the analyticity of UM . By Theorem 4.25 (i), vΩ o [·] is linear and 0,α o 1,α continuous from C (∂Ω ) to C (Ω o ), and the restriction map from C 1,α (Ω o ) to C 1,α (ΩM ) is linear and continuous, and the function Θo is real analytic from the set ] − M , M [×] − ∗1 , ∗1 [ to C 0,α (∂Ω o ). Then we conclude that the second summand in the right-hand side of (9.84) defines a real analytic map from ]−M , M [×]−∗1 , ∗1 [ to C 1,α (ΩM ). We now turn to consider the first summand. By the analyticity of nonlinear integral operators with analytic kernel of Theorem A.28 (i), we deduce that the map from ] − ΩM , ΩM [×L1 (∂Ω i ) to C 1,α (ΩM ) that takes

370

9 Other Problems with Linear Boundary Conditions in a Domain with a Small Hole

 (, f ) to the function ∂Ω i Sn (x − η)f (η) dση of the variable x ∈ ΩM is real analytic. Since Θi is real analytic from ] − M , M [×] − ∗1 , ∗1 [ to C 0,α (∂Ω i ), and C 0,α (∂Ω i ) is continuously embedded into L1 (∂Ω i ), we conclude that the function from ] − M , M [×] − ∗1 , ∗1 [ to C 1,α (ΩM ) that takes (, 1 ) to the first summand in the right-hand side of (9.84) is real analytic. Since C[·, ·] is real analytic, the proof is complete.   Next we turn to study the microscopic behavior of u[] close to the boundary of the hole, that is, the behavior of a suitable restriction of the rescaled function u[](·). We do so in the following theorem. ˜ u Theorem 9.24. Let λ, ˜, ∗ , and ∗1 be as in Theorem 9.22. Let Ωm be a bounded n open subset of R \ Ω i . Then there exist m,1 ∈]0, ∗ [ and a real analytic function Um from ] − m,1 , m,1 [×] − ∗1 , ∗1 [ to C 1,α (Ωm ) such that Ωm ⊆ Ω o , u[](ξ) = Um [, δ2,n  log ](ξ)

(9.85) ∀ξ ∈ Ωm ,

for all  ∈]0, m,1 [. Moreover, Um [0, 0] = u ˜(0) .

(9.86)

Proof. Let m,1 ∈]0, ∗ [ be such that Ωm ⊆ 1 Ω o for all  ∈ [−m,1 , m,1 ] \ {0}. Clearly, if  ∈]0, m,1 [ and ξ ∈ Ωm , then we have ξ ∈ Ω(). Then the equality Sn (ξ) = 2−n Sn (ξ) +

δ2,n log  2π

∀ ∈]0, +∞[

and formula (9.79) imply that  u[](ξ) =  Sn (ξ − η)Θi [, δ2,n  log ](η) dση (9.87) i ∂Ω  1 Θi [, δ2,n  log ](η) dση +δ2,n ( log ) 2π ∂Ω i  + Sn (ξ − y)Θo [, δ2,n  log ](y) dσy + C[, δ2,n  log ] ∂Ω o

for all ξ ∈ Ωm and all  ∈]0, m,1 ]. Since Θi is analytic, the function Um,1 from ] − ∗ , ∗ [×] − ∗1 , ∗1 [ to R defined by  1 U1 [, 1 ] ≡ Θi [, 1 ] dσ ∀(, 1 ) ∈] − ∗ , ∗ [×] − ∗1 , ∗1 [ (9.88) 2π ∂Ω i is real analytic. Then we set  Um,2 [, 1 ](ξ) ≡  ∂Ω i

Sn (ξ − η)Θi [, 1 ](η) dση

(9.89)

9.4 A Steklov Eigenvalue Problem

371

 Sn (ξ − y)Θo [, 1 ](y) dσy + C[, 1 ]

+

∀ξ ∈ Ωm ,

∂Ω o

for all (, 1 ) ∈] − m,1 , m,1 [×] − ∗1 , ∗1 [. The analyticity of the first summand in the right-hand side of (9.89) follows by the analyticity of Θi and by the linearity − and continuity of vΩ i [·] (see Theorem 4.25 (ii)). Next we consider the second summand in the right-hand side of (9.89). By the analyticity of nonlinear integral operators with analytic kernel of Theorem A.28 (i), we deduce that the map from ] − m,1 , m,1 [×L1 (∂Ω o ) to C 1,α (Ωm ) that takes  (, f ) to the function ∂Ω o Sn (ξ − y)f (y) dσy of the variable ξ ∈ Ωm is real analytic. Since Θo is real analytic from ] − m,1 , m,1 [×] − ∗1 , ∗1 [ to C 0,α (∂Ω o ), and C 0,α (∂Ω o ) is continuously embedded into L1 (∂Ω o ), we conclude that the function from ] − m,1 , m,1 [×] − ∗1 , ∗1 [ to C 1,α (Ωm ), that takes (, 1 ) to the second summand in the right-hand side of (9.89) is real analytic. Since C is analytic, we conclude that Um,2 is real analytic. Hence the map Um [, 1 ] ≡ 1 U1 [, 1 ] + Um,2 [, 1 ]

∀(, 1 ) ∈] − m,1 , m,1 [×] − ∗1 , ∗1 [

is real analytic from ]−m,1 , m,1 [×]−∗1 , ∗1 [ to C 1,α (Ωm ) and satisfies the equality u[](ξ) = Um [, δ2,n  log ](ξ)

∀ξ ∈ Ωm ,

for all  ∈]0, m,1 [, i.e. Um satisfies equality (9.85). Finally, equality (9.78) and ˜ = v +o [θ˜o ] + c˜ imply that equality u ˜=u ˜o [θ˜i , θ˜o , c˜, λ] Ω Um [0, 0] = u ˜(0) (see Proposition 9.18 (ii)). Hence, (9.86) follows and the proof is complete.

 

Finally, we turn to analyze the behavior of the energy integral of the eigenfunction u[] of (9.79). ˜ u Theorem 9.25. Let λ, ˜, ∗ and ∗1 be as in Theorem 9.22. Then there exist e in ]0, ∗ [ and a real analytic function E from ] − e , e [×] − ∗1 , ∗1 [ to R such that  |∇u[](x)|2 dx = E[, δ2,n  log ] ∀ ∈]0, e [ . (9.90) Ω()



Moreover, E[0, 0] =

|∇˜ u|2 dx . Ωo

Proof. By the first Green Identity of Theorem 4.2, we have   ∂ |∇u[](x)|2 dx = u[] u[] dσ ∂νΩ() Ω() ∂Ω()

(9.91)

372

9 Other Problems with Linear Boundary Conditions in a Domain with a Small Hole

 ˆ δ2,n  log ] = λ[,

u[]2 dσ ∂Ω()



  u[]2 dσ + u[]2 dσ i ∂Ω o  ∂Ω    ˆ δ2,n  log ] n−1 = λ[, u[]2 (ξ) dσξ + u[]2 dσ .

ˆ δ2,n  log ] = λ[,

∂Ω i

∂Ω o

Then we take ΩM as in Theorem 9.23 and Ωm as in Theorem 9.24 such that ∂Ω o ⊆ ΩM ,

∂Ω i ⊆ Ωm .

If UM and Um are the corresponding real analytic maps, then we have  |∇u[]|2 dx Ω()

  ˆ δ2,n  log ] n−1 = λ[,

Um [, δ2,n  log ](ξ) 2

UM [, δ2,n  log ]

+

2

 ∂Ω i





(9.92) dσξ

 dσ

∂Ω o

for all  ∈]0, min{M , m,1 }[. Then it suffices to set e ≡ min{M , m,1 } , and to define E as the function from ] − e , e [×] − ∗1 , ∗1 [ to R given by   ˆ 1 ] n−1 E[, 1 ] ≡ λ[,

2

 Um [, 1 ](ξ) ∂Ω i

dσξ 2





UM [, 1 ]

+

 dσ

∂Ω o

for all (, 1 ) ∈] − e , e [×] − ∗1 , ∗1 [. Then Equation (9.90) holds by the definition ˆ 0] = λ, ˜ UM [0, 0] = u ˜|ΩM and of E and by equality (9.92). Moreover, equalities λ[0, the first Green Identity of Theorem 4.2 imply that    ∂u ˜ 2 ˜ E[0, 0] = λ u ˜ dσ = u ˜ dσ = |∇˜ u|2 dx . ∂Ω o ∂Ω o ∂νΩ o Ωo Hence, equality (9.91) holds true and the theorem is proven.

 

Chapter 10

A Dirichlet Problem in a Domain with Two Small Holes

Abstract In this chapter we see an application of the Functional Analytic Approach to a domain perturbation of a somewhat different nature: instead of a domain with a single shrinking hole we consider a domain that contains two holes that collide into one another while shrinking in size. We confine ourselves to a Dirichlet problem for the Laplace equation. Different boundary value problems can also be analyzed and some examples can be found in Dalla Riva and Musolino (Comm Partial Differ Equ 41(5):812–837, 2016; Math Methods Appl Sci 41(3):986–993, 2018). Yet as occurs for the single hole problem of Chapter 8, the Dirichlet problem presents certain interesting features related to the appearance of a logarithmic behavior in the twodimensional case. For this reason we prefer to focus on this problem and investigate it thoroughly. Further applications of the Functional Analytic Approach in problems with different domain perturbations can be found in the papers [29] and [30] with Bonnaillie-No¨el and Dambrine, which concern the case of a small hole approaching the outer boundary of a domain, and in the paper [60] with Costabel and Dauge, which deals with perturbations of a plane sector domain. The mathematics of this chapter has been in part presented also in [79].

10.1 The Geometric Setting We begin by introducing the geometric setting of our problem. We fix a real number α ∈]0, 1[ and a natural number n ≥ 2. Then we take three sets Ω o , Ω1 , and Ω2 that satisfy the following condition: Ω o , Ω1 and Ω2 are open bounded connected subsets of Rn of class C 1,α , they contain the origin 0 of Rn and the exterior domains Ω o− , Ω1− , and Ω2− are connected.

© Springer Nature Switzerland AG 2021 M. Dalla Riva et al., Singularly Perturbed Boundary Value Problems, https://doi.org/10.1007/978-3-030-76259-9 10

373

374

10 A Dirichlet Problem in a Domain with Two Small Holes

As usual the letter ‘o’ stands for ‘outer domain’ and Ω o will play the role of the unperturbed outer domain in which we make two holes. To do so, we take two points p1 , p2 ∈ Rn , p1 = p2 and we choose δ2 > 0 such that (p1 + 2 Ω1 ) ∩ (p2 + 2 Ω2 ) = ∅

∀2 ∈ [−δ2 , δ2 ] .

(10.1)

Then we define the rescaled sets Ω1 (1 , 2 ) ≡ 1 p1 + 1 2 Ω1 ,

Ω2 (1 , 2 ) ≡ 1 p2 + 1 2 Ω2 ,

∀1 , 2 ∈ R ,

which will play the role of the holes. We observe that, for 1 , 2 ∈ R \ {0} and i ∈ {1, 2}, each Ωi (1 , 2 ) is an open bounded subset of R2 and contains the point 1 pi . Instead, when 1 = 0 or 2 = 0, Ωi (1 , 2 ) collapses to a point and we have Ωi (0, 2 ) = {0} and Ωi (1 , 0) = {1 pi }. In addition, condition (10.1) implies that Ω1 (1 , 2 ) ∩ Ω2 (1 , 2 ) = ∅

∀1 ∈ R \ {0} , 2 ∈ [−δ2 , δ2 ] .

Then we note that the mutual distance between Ω1 (1 , 2 ) and Ω2 (1 , 2 ) is controlled by |1 |, while their size is proportional to |1 2 |. As a consequence, when both 1 and 2 approach zero, the size tends to zero at a faster rate than the mutual distance. When this happens, we say that the holes are ‘moderately close’ (here we follow the terminology of Bonnaillie-No¨el et al. [31]). We will also consider the case when the size and the distance are comparable, i.e. when 1 tends to zero and 2 stays away from zero. Since we want the holes to be contained in Ω o , we have to restrict the set of the ‘admissible’ parameters 1 for which we define the perforated domain. Then we take δ1 > 0 such that Ω1 (1 , 2 ) ∪ Ω2 (1 , 2 ) ⊆ Ω o

∀(1 , 2 ) ∈ [−δ1 , δ1 ] × [−δ2 , δ2 ]

(10.2)

and we regard as admissible parameters the pairs (1 , 2 ) in the rectangular domain [−δ1 , δ1 ] × [−δ2 , δ2 ]. For these (1 , 2 ) we define the perforated domain   Ω(1 , 2 ) ≡ Ω o \ Ω1 (1 , 2 ) ∪ Ω2 (1 , 2 ) (see Figure 10.1). We observe that for 1 ∈ [−δ1 , δ1 ] \ {0} and 2 ∈ [−δ2 , δ2 ] \ {0}, the domain Ω(1 , 2 ) is an open bounded connected subset of Rn of class C 1,α and the boundary of Ω(1 , 2 ) consists of the disjoint union of ∂Ω o , ∂Ω1 (1 , 2 ), and ∂Ω2 (1 , 2 ) (cf. Section 2.13 for the definition of class C 1,α , see also Theorem A.10 for a proof that Ω(1 , 2 ) is connected). For 1 = 0 the set Ω(0, 2 ) equals Ω o \ {0}

10.2 A Dirichlet Problem in Ω(1 , 2 )

375

Fig. 10.1 The perforated domain Ω(1 , 2 ) (n = 2) 1 p1 + 1 2 Ω1

∂Ω o

Ω(1 , 2 ) 1 p2 + 1 2 Ω2

1 p1 2 p2

and for 2 = 0 we have Ω(1 , 0) = Ω o \ ({1 p1 } ∪ {1 p2 }). Thus Ω(0, 2 ) and Ω(1 , 0) are not local strict hypographs of class C 0 in the sense of Definition 2.37. We also find convenient to denote by Ω (2 ) the set Ω (2 ) ≡ Ω1 (1, 2 ) ∪ Ω2 (1, 2 )

∀2 ∈ [−δ2 , δ2 ] .

(10.3)

We note that the set Ω (2 ) is not necessarily a subset of Ω o .

10.2 A Dirichlet Problem in Ω(1 , 2 ) We now introduce a Dirichlet boundary value problem in the domain Ω(1 , 2 ). To define the boundary conditions, we fix three functions f o ∈ C 1,α (∂Ω o ) , f1 ∈ C 1,α (∂Ω1 ) , and f2 ∈ C 1,α (∂Ω2 ) . Then, for 1 ∈]0, δ1 [ and 2 ∈]0, δ2 [, we consider the Dirichlet boundary value problem ⎧ Δu = 0 in Ω(1 , 2 ), ⎪ ⎪ ⎨ on ∂Ω o , u = fo (10.4) 1 u(x) = f1 ((x − 1 p )/(1 2 )) ∀x ∈ ∂Ω1 (1 , 2 ) , ⎪ ⎪ ⎩ 2 u(x) = f2 ((x − 1 p )/(1 2 )) ∀x ∈ ∂Ω2 (1 , 2 ) for a function u ∈ C 1,α (Ω(1 , 2 )). The existence and uniqueness Theorem 6.27 implies that the solution of problem (10.4) exists and is unique (see also Theorem 3.6). Clearly, such solution depends on the pair (1 , 2 ) and for this reason we denote it by u1 ,2 . Then we can consider the map that takes (1 , 2 ) to (a suitable restriction of) u1 ,2 . We will analyze separately case n ≥ 3 and n = 2. For n ≥ 3 we will study both the cases of ‘moderately close’ holes, where the pair (1 , 2 ) approaches (0, 0), and the case of ‘close’ holes, where (1 , 2 ) approaches a singular pair (0, ˜2 ) with ˜2 > 0 (cf. Section 10.1 for the notion of ‘close’ and ‘moderately close’ holes). For n = 2 instead, we will focus on the case of moderately close holes. The reader can find the analysis of the two-dimensional problem with close

376

10 A Dirichlet Problem in a Domain with Two Small Holes

holes in the paper [79]. Both for n ≥ 3 and n = 2, our aim is to represent the map that takes (1 , 2 ) to a suitable restriction of u1 ,2 in terms of two ingredients: • Real analytic maps of (1 , 2 ) defined in a neighborhood of a singular pair. • Completely known elementary functions of 1 and 2 . Similarly we wish to do for the maps that take (1 , 2 ) to suitable restrictions of the rescaled functions u1 ,2 (1 p1 + 1 2 ·) and u1 ,2 (1 p2 + 1 2 ·) and to the energy integral of u1 ,2 . We observe that here the functions u1 ,2 (1 pi + 1 2 ·) with i ∈ {1, 2} take care of the behavior of u1 ,2 close to the boundary of the holes. As we did in the previous chapters, we will use the capital subscript ‘M ’ to indicate sets or maps that we use to describe the ‘macroscopic’ behavior of u1 ,2 away from the holes and the small ‘m’ for analogous objects that concern the ‘microscopic’ behavior of u1 ,2 near the boundary of the holes. Both for n ≥ 3 and n = 2 the analysis of this chapter follows the guidelines of the Functional Analytic Approach introduced in Chapter 8 for the Dirichlet problem in a domain with a single small hole, with some additional complications due to the additional hole and perturbation parameter. When n ≥ 3, also the results that we obtain are similar to those of Section 8.3 for the corresponding problem with a single hole. For example, we show that a suitable restriction of u1 ,2 can be continued in a neighborhood of (0, 0) (or (0, ˜2 ) with ˜2 > 0) by a real analytic function of the / ΩM , then pair (1 , 2 ). More precisely, if ΩM is an open subset of Ω o with 0 ∈ we prove in Theorem 10.7 that there exists a real analytic map UM from a suitably small square domain ] − δM , δM [2 to C 1,α (ΩM ) such that u1 ,2 |ΩM = UM [1 , 2 ]

∀(1 , 2 ) ∈]0, δM [2 .

Then, for a possibly smaller δM > 0, there exists a family {u(j,k) }(j,k)∈N2 of functions of C 1,α (ΩM ) such that  u(j,k) j1 k2 ∀(1 , 2 ) ∈]0, δM [2 , u1 ,2 |ΩM = (j,k)∈N2

where the series converges normally in C 1,α (ΩM ). We could also compute the coefficients {u(j,k) }(j,k)∈N2 , either using the expansion methods of Asymptotic Analysis or adapting the technique based on the Functional Analytic Approach that we have introduced in Section 8.5. (We will not present such computation, which is however quite straightforward. The reader can find a similar computation in [30].) The case of dimension n = 2 is instead more involved: Not only we have to deal with a logarithmic behavior, which is expected for the Dirichlet problem in a planar perforated domain, but, due to the presence of two moderately close small holes, a new feature appears that was not present when dealing with a single hole. Specifically, to provide explicit expressions of the first coefficient of an expansion it is necessary to introduce some relation between the perturbation parameters 1 and 2 , that is to say, between the size and the mutual distance of the holes (cf. Section 10.4).

10.3 Close and Moderately Close Holes in Dimension n ≥ 3

377

10.3 Close and Moderately Close Holes in Dimension n ≥ 3 As we have done for the Dirichlet problem in a domain with a single shrinking hole of Chapter 8, we split the analysis for n ≥ 3 and n = 2. In particular, for n ≥ 3 we can represent the solution in terms of a single layer potential. Indeed, Theorem 6.48 (ii) implies that there exists a density function θ ∈ C 0,α (∂Ω(1 , 2 )) such that + vΩ( [θ] 1 ,2 ) + [θ] is the solution of (10.4) is the solutions of (10.4). Then we observe that vΩ( 1 ,2 ) if and only if the following boundary conditions are satisfied:

vΩ(1 ,2 ) [θ](x) = f1 ((x − 1 p1 )/(1 2 ))

for x ∈ ∂Ω1 (1 , 2 ) ,

vΩ(1 ,2 ) [θ](x) = f2 ((x − 1 p )/(1 2 )) vΩ(1 ,2 ) [θ](x) = f o (x)

for x ∈ ∂Ω2 (1 , 2 ) , for x ∈ ∂Ω o .

2

(10.5)

Our next step is to transform the equations on the (1 , 2 )-dependent boundaries ∂Ω1 (1 , 2 ) and ∂Ω2 (1 , 2 ) into equations on the fixed boundaries ∂Ω1 and ∂Ω2 . To do so, we introduce the rescaled densities θ1 (ξ) ≡ 1 2 θ(1 p1 + 1 2 ξ)

∀ξ ∈ ∂Ω1 ,

θ2 (ξ) ≡ 1 2 θ(1 p + 1 2 ξ)

∀ξ ∈ ∂Ω2 ,

2

(10.6)

and we set θo (x) = θ(x) for all x ∈ ∂Ω o . We also observe that Sn (1 p1 + 1 2 ξ − 1 p1 + 1 2 η) = 2−n 2−n Sn (ξ − η) 1 2 for all 1 , 2 > 0 and all ξ, η ∈ Rn with ξ = η and Sn (1 p1 + 1 2 ξ − 1 p2 + 1 2 η) = 2−n Sn (p1 − p2 + 2 (ξ − η)) 1 for all 1 , 2 > 0 and all ξ, η ∈ Rn such that p1 − p2 + 2 (ξ − η) = 0. Then, by Definition 4.18 of single layer potential and by the rule of change of variables in integrals, we obtain a new set of conditions, equivalent to (10.5) and defined on the fixed boundaries ∂Ω1 , ∂Ω2 , and ∂Ω o :     n−2 Sn (ξ − η)θ1 (η) dση + 2 Sn p1 − p2 + 2 (ξ − η) θ2 (η) dση ∂Ω1 ∂Ω2  + Sn (1 p1 + 1 2 ξ − y)θo (y) dσy = f1 (ξ) ∀ξ ∈ ∂Ω1 , ∂Ω o



(10.7)





Sn (ξ − η)θ2 (η) dση + n−2 2 ∂Ω2



Sn p2 − p1 + 2 (ξ − η) θ1 (η) dση ∂Ω1

378

10 A Dirichlet Problem in a Domain with Two Small Holes

 Sn (1 p2 + 1 2 ξ − y)θo (y) dσy = f2 (ξ)

+

∀ξ ∈ ∂Ω2 ,

∂Ω o

(10.8)

 n−2 n−2 1 2

Sn (x − 1 p1 + 1 2 η)θ1 (η) dση  n−2 n−2 + 1 2 Sn (x − 1 p2 + 1 2 η)θ2 (η) dση ∂Ω2  + Sn (x − y)θo (y) dσy = f o (x) ∂Ω1

∀x ∈ ∂Ω o .

∂Ω o

(10.9) In view of (10.7)–(10.9), we introduce for all (1 , 2 ) ∈] − δ1 , δ1 [×] − δ2 , δ2 [ the bounded linear operator V(1 , 2 ) ≡ (V1 (1 , 2 ), V2 (1 , 2 ), V o (1 , 2 )) from C 0,α (∂Ω1 )×C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) to C 1,α (∂Ω1 )×C 1,α (∂Ω2 ) × C 1,α (∂Ω o ) that takes a triple (θ1 , θ2 , θo ) to  V1 (1 , 2 )[θ1 , θ2 , θo ](ξ) ≡ Sn (ξ − η)θ1 (η) dση (10.10) ∂Ω1    Sn p1 − p2 + 2 (ξ − η) θ2 (η) dση + n−2 2 ∂Ω2  + Sn (1 p1 + 1 2 ξ − y)θo (y) dσy ∂Ω o

∀ξ ∈ ∂Ω1 ,

 V2 (1 , 2 )[θ1 , θ2 , θo ](ξ) ≡

Sn (ξ − η)θ2 (η) dση (10.11)    Sn p2 − p1 + 2 (ξ − η) θ1 (η) dση + n−2 2 ∂Ω1  + Sn (1 p2 + 1 2 ξ − y)θo (y) dσy ∂Ω2

∂Ω o

∀ξ ∈ ∂Ω2 ,

 V o (1 , 2 )[θ1 , θ2 , θo ](x) ≡ n−2 n−2 1 2

Sn (x − 1 p1 + 1 2 η)θ1 (η) dση ∂Ω1



(10.12)

+ n−2 n−2 Sn (x − 1 p2 + 1 2 η)θ2 (η) dση 1 2 ∂Ω2  + Sn (x − y)θo (y) dσy ∀x ∈ ∂Ω o . ∂Ω o

Then the three equations in (10.7)–(10.9) are equivalent to

10.3 Close and Moderately Close Holes in Dimension n ≥ 3

V(1 , 2 )[θ1 , θ2 , θo ] = (f1 , f2 , f o ) .

379

(10.13)

Accordingly, Equation (10.13) is equivalent to (10.5) via (10.6). Moreover, for n ≥ 3 Theorem 6.46 implies that the single layer operator VΩ(1 ,2 ) is an isomorphism (i.e., a linear homeomorphism) from C 0,α (∂Ω(1 , 2 )) to C 1,α (∂Ω(1 , 2 )). Then the system of equations in (10.5) have a unique solution θ ∈ C 0,α (∂Ω(1 , 2 )) and we deduce the validity of the following. Proposition 10.1. Let 1 ∈]0, δ1 [ and 2 ∈]0, δ2 [. There exists one and only one triple of functions (θ1 , θ2 , θo ) ∈ C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) such that (10.13) holds true. Moreover, the function u defined by  n−2  Sn (x − 1 p1 + 1 2 η)θ1 (η) dση u(x) ≡ n−2 1 2 ∂Ω1  n−2 + n−2  Sn (x − 1 p2 + 1 2 η)θ2 (η) dση (10.14) 1 2 ∂Ω2  + Sn (x − y)θo (y) dσy ∀x ∈ Ω(1 , 2 ) ∂Ω o

coincides with the unique solution u1 ,2 in C 1,α (Ω(1 , 2 )) of the boundary value problem (10.4).

10.3.1 Moderately Close Holes in Dimension n ≥ 3 In this subsection we study the solution u1 ,2 when both 1 and 2 approach 0. That is, in the case of moderately close holes (cf. Section 10.1). By Proposition 10.1, we can consider the map from ]0, δ1 [×]0, δ2 [ to the space C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) that takes (1 , 2 ) to the unique solution (θ1 (1 , 2 ), θ2 (1 , 2 ), θo (1 , 2 ))

(10.15)

of (10.13). We will show that such map has a real analytic continuation in a open neighborhood of (1 , 2 ) = (0, 0). To do so, we first prove that the map that takes (1 , 2 ) to V(1 , 2 ) is real analytic and then we verify that for (1 , 2 ) = (0, 0) the operator V(0, 0) is an isomorphism of Banach spaces. We deduce that V(1 , 2 ) is invertible in a neighborhood of (0, 0) and that the map that takes (1 , 2 ) to its inverse V(1 , 2 )(−1) is real analytic. Then the real analytic continuation of the map (1 , 2 ) → (θ1 (1 , 2 ), θ2 (1 , 2 ), θo (1 , 2 )) follows by the equality (θ1 (1 , 2 ), θ2 (1 , 2 ), θo (1 , 2 )) = V(1 , 2 )(−1) [f1 , f2 , f o ] .

380

10 A Dirichlet Problem in a Domain with Two Small Holes

So, our first lemma states that the map (1 , 2 ) → V(1 , 2 ) is real analytic. Lemma 10.2. The map from ] − δ1 , δ1 [×] − δ2 , δ2 [ to  L C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) , C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) × C 1,α (∂Ω o )



that takes (1 , 2 ) to V(1 , 2 ) is real analytic. Proof. The statement is a consequence of definition (10.10)–(10.12) of V, of the properties of integral operators with real analytic kernel of Corollary A.29 of the Appendix, and of the mapping properties of the single layer potential of Theorem 6.4 (see also the Definition 4.18 of single layer potential).   In the following Lemmas 10.3 we consider Equation (10.13) for (1 , 2 ) = (0, 0). Lemma 10.3. For all (g1 , g2 , g o ) ∈ C 1,α (∂Ω1 ) × C 1,α (∂Ω1 ) × C 1,α (∂Ω o ) and all (θ1 , θ2 , θo ) ∈ C 0,α (∂Ω1 ) × C 0,α (∂Ω1 ) × C 0,α (∂Ω o ) we have V(0, 0)[θ1 , θ2 , θo ] = (g1 , g2 , g o ) if and only if

⎧ o ⎪ ⎨VΩ1 [θ1 ] + vΩ o [θ ](0) = g1 , VΩ2 [θ2 ] + vΩ o [θo ](0) = g2 , ⎪ ⎩ o o VΩ [θ ] = g o .

(10.16)

(10.17)

Moreover, for all (g1 , g2 , g o ) ∈ C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) × C 1,α (∂Ω o ) there exists one and only one triple (θ1 , θ2 , θo ) ∈ C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) for which (10.16) (and, as a consequence, (10.17)) holds true. Proof. The equivalence of (10.16) and (10.17) follows by the definition of V in (10.10)–(10.12). To prove the last statement of the lemma we observe that Theorem 6.46 implies that the third equation in (10.17) has a unique solution θo ∈ C 0,α (∂Ω o ). Then, again Theorem 6.46 implies that also the first and second equations in (10.17) have unique solutions θ1 ∈ C 0,α (∂Ω1 ) and θ2 ∈ C 0,α (∂Ω2 ), respectively.   Then we verify that V(0, 0) is an isomorphism of Banach spaces. Lemma 10.4. The operator V(0, 0) is an isomorphism from C 0,α (∂Ω1 )×C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) to C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) × C 1,α (∂Ω o ). Proof. The previous Lemma 10.2 implies that V(0, 0) is linear and bounded. Then, by the Open Mapping Theorem 2.2, it is an a homeomorphism provided that it is bijective. Hence, we can conclude the proof by Lemma 10.3.   Since the set of invertible operators is open in the space of bounded operators and since the map that takes an invertible operator to its inverse is real analytic, we now deduce the validity of following lemma (cf. Proposition A.15 in the Appendix).

10.3 Close and Moderately Close Holes in Dimension n ≥ 3

381

Lemma 10.5. There exists 0 < δ∗ < min{δ1 , δ2 } such that V(1 , 2 ) is invertible for all (1 , 2 ) ∈] − δ∗ , δ∗ [2 and the map from ] − δ∗ , δ∗ [2 to  L C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) × C 1,α (∂Ω o ) ,  C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) that takes (1 , 2 ) to V(1 , 2 )(−1) is real analytic.  

Proof. The statement follows by Lemmas 10.2 and 10.4.

Then, by Lemma 10.5 can show that there exists a real analytic continuation of the map (1 , 2 ) → (θ1 (1 , 2 ), θ1 (1 , 2 ), θo (1 , 2 )). Proposition 10.6. There exists a real analytic map Θ ≡ (Θ1 , Θ2 , Θo ) : ] − δ∗ , δ∗ [2 → C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) such that V(1 , 2 )[Θ1 [1 , 2 ], Θ2 [1 , 2 ], Θo [1 , 2 ]] = (f1 , f2 , f o )

∀(1 , 2 ) ∈] − δ∗ , δ∗ [2 .

In particular, (θ1 (1 , 2 ), θ1 (1 , 2 ), θo (1 , 2 )) = Θ[1 , 2 ]

∀(1 , 2 ) ∈]0, δ∗ [2 .

Proof. We set Θ[1 , 2 ] ≡ V(1 , 2 )(−1) [f1 , f2 , f o ]

∀ ∈] − δ∗ , δ∗ [2

(with δ∗ as in Lemma 10.5). Then the proposition follows by Proposition 10.1 and Lemma 10.5. Indeed, the map that takes an operator  A ∈ L C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) × C 1,α (∂Ω o ) ,  C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) to A[f1 , f2 , f o ] ∈ C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) is linear and continuous, and therefore real analytic.   We can now exploit Proposition 10.6 to deduce some real analytic continuation results for the solution u1 ,2 of problem (10.4). In the next Theorem 10.7 we will examine the ‘macroscopic’ behavior of u1 ,2 away from the holes Ω1 (1 , 2 ) and Ω2 (1 , 2 ). Theorem 10.7. Let ΩM be an open subset of Ω o such that 0 ∈ / ΩM . Let δM ∈]0, δ∗ [ be such that ΩM ∩ Ω1 (1 , 2 ) = ∅

and

ΩM ∩ Ω2 (1 , 2 ) = ∅ ∀(1 , 2 ) ∈] − δM , δM [2 .

Then, there exists a real analytic map

382

10 A Dirichlet Problem in a Domain with Two Small Holes

UM : ] − δM , δM [2 → C 1,α (ΩM ) such that u1 ,2 |ΩM = UM [1 , 2 ]

∀(1 , 2 ) ∈]0, δM [2 .

(10.18)

Moreover, UM [0, 0] = uo|ΩM

(10.19)

where uo ∈ C 1,α (Ω o ) is the unique solution of the unperturbed problem ! Δuo = 0 in Ω o , uo = f o on ∂Ω o .

(10.20)

Proof. In view of formula (10.14) for u1 ,2 , we set  n−2  Sn (x − 1 p1 + 1 2 η) Θ1 [1 , 2 ](η) dση UM [1 , 2 ](x) ≡ n−2 1 2 ∂Ω1  n−2 n−2 + 1 2 Sn (x − 1 p2 + 1 2 η) Θ2 [1 , 2 ](η) dση + vΩ o [Θo [1 , 2 ]](x) ∂Ω2

for all x ∈ ΩM and for all (1 , 2 ) ∈] − δM , δM [2 . To show that (1 , 2 ) → UM [1 , 2 ] is real analytic we first observe that, by Theorem 4.25 (i) and Proposition 10.6, the map from ]−δM , δM [2 to C 1,α (ΩM ) that takes (1 , 2 ) to vΩ o [Θo [1 , 2 ]]|ΩM is real analytic. Thus, it remains to prove that the maps from ] − δM , δM [2 to C 1,α (ΩM ) that take (1 , 2 ) to the functions  n−2 n−2  Sn (x − 1 p1 + 1 2 η) Θ1 [1 , 2 ](η) dση ∀x ∈ ΩM 1 2 ∂Ω1

and

 Sn (x − 1 p2 + 1 2 η) Θ2 [1 , 2 ](η) dση

n−2 n−2 1 2

∀x ∈ ΩM ,

∂Ω2

are also real analytic, a result that follows by the analyticity of Θ of Proposition 10.6 and by Theorem A.28 (i) on the analyticity of integral operators with a real analytic kernel. Equality (10.18) is a consequence of Propositions 10.1 and 10.6. To prove equality (10.19), we observe that UM [0, 0](x) = vΩ o [Θo [0, 0]](x) for all x ∈ ΩM and that vΩ o [Θo [0, 0]] is the only solution in C 1,α (Ω o ) of the unperturbed problem (10.20). Indeed, Lemma 10.3 and Proposition 10.6 imply that VΩ o [Θo [0, 0]] = f o .  

10.3 Close and Moderately Close Holes in Dimension n ≥ 3

383

We now analyze the behavior of u1 ,2 near the boundary of the holes Ω1 (1 , 2 ) and Ω2 (1 , 2 ). We will avoid to write two very similar statements, one for Ω1 (1 , 2 ) and one for Ω2 (1 , 2 ). Instead, we will consider a hole Ωi (1 , 2 ) with i ∈ {1, 2}. Then, for the rescaled function u1 ,2 (1 pi + 1 2 ξ) of the variable ξ ∈ [(Ω o − 1 pi )/(1 2 )] \ Ωi , we have the following. Theorem 10.8. Let i ∈ {1, 2} be fixed. Let Ωi,m be an open bounded subset of (i) Rn \ Ωi . Let δm ∈]0, δ∗ [ be such that 1 pi + 1 2 Ωi,m ⊆ Ω o and (1 pi + 1 2 Ωi,m ) ∩ Ωj (1 , 2 ) = ∅ (i)

(i)

for all (1 , 2 ) ∈] − δm , δm [2 , where j = 1 if i = 2 and j = 2 if i = 1. Then, there exists a real analytic map (i) (i) 2 Ui,m : ] − δm , δm [ → C 1,α (Ωi,m )

such that u1 ,2 (1 pi + 1 2 ξ) = Ui,m [1 , 2 ](ξ)

(i) 2 ∀ξ ∈ Ωi,m , (1 , 2 ) ∈]0, δm [ . (10.21)

Moreover, i Ui,m [0, 0] = v|Ω i,m 1,α where v i ∈ Cloc (Rn \ Ωi ) is the unique solution of ⎧ i ⎪ in Rn \ Ωi , ⎨Δv = 0 on ∂Ωi , vi = f i ⎪ ⎩ limξ→∞ v i (ξ) = uo (0)

(10.22)

(10.23)

and uo is the unique solution of (10.20). Proof. We take j ≡ 1 if i = 2 and j ≡ 2 if i = 1. Since u1 ,2 is delivered by the right-hand side of (10.14), we find natural to set Ui,m [1 , 2 ](ξ) ≡ vΩi [Θi [1 , 2 ]](ξ)    n−2 + 2 Sn pi − pj + 2 (ξ − η) Θj [1 , 2 ](η) dση ∂Ωj



  Sn 1 p1 + 1 2 ξ − y Θo [1 , 2 ](y) dσy

+ ∂Ω o

(i)

(i)

for all ξ ∈ Ωi,m and (1 , 2 ) ∈] − δm , δm [2 . To show that (1 , 2 ) → Ui,m [1 , 2 ]

384

10 A Dirichlet Problem in a Domain with Two Small Holes

is real analytic we first observe that, by Theorem 4.25 (ii) and Proposition 10.6, the (i) (i) map from ] − δm , δm [2 to C 1,α (Ωi,m ) that takes (1 , 2 ) to vΩi [Θi [1 , 2 ]] is real (i) (i) analytic. Thus, it remains to prove that the map from ] − δm , δm [2 to C 1,α (Ωi,m ) that takes (1 , 2 ) to the function    n−2 2 Sn pi − pj + 2 (ξ − η) Θj [1 , 2 ](η) dση ∂Ωj



  Sn 1 p1 + 1 2 ξ − y Θo [1 , 2 ](y) dσy

+ ∂Ω o

of ξ ∈ Ωi,m is also real analytic, a fact that follows by the analyticity of Θ in Proposition 10.6 and by Theorem A.28 (i) for the integral operators with real analytic kernels. Equality (10.21) holds by the definition of Ui,m . To prove equality (10.22), we observe that Ui,m [0, 0](ξ) = vΩi [Θi [0, 0]](ξ) + vΩ o [Θo [0, 0]](0)

∀ξ ∈ Ωi,m

and that the function vΩi [Θi [0, 0]](·) + vΩ o [Θo [0, 0]](0) is the only solution in 1,α (Rn \ Ωi ) of problem (10.23). Indeed, Lemma 10.3 and Proposition 10.6 imply Cloc that on ∂Ωi VΩi [Θi [0, 0]] + vΩ o [Θo [0, 0]](0) = f i and Theorem 4.23 (i) implies that vΩi [Θi [0, 0]] is harmonic at infinity (see also Proposition 6.1 on the uniqueness of the solution of the exterior Dirichlet problem).   Theorems 10.7 and 10.8 can be used to study functionals related to the solution u1 ,2 . For example, in the following Theorem 10.9 we consider the energy integral. Theorem 10.9. There exist δE ∈]0, δ∗ [ and a real analytic function E : ] − δ E , δE [2 → R such that

 2

|∇u1 ,2 | dx = E(1 , 2 )

∀(1 , 2 ) ∈]0, δE [2 .

(10.24)

Ω()



Moreover,

2

E(0, 0) =

|∇uo | dx ,

(10.25)

Ωo

where uo is the unique solution of the unperturbed problem (10.20). Proof. By the first Green Identity of Theorem 4.2 and by the rule of change of variable in integrals we have

10.3 Close and Moderately Close Holes in Dimension n ≥ 3

385



2

|∇u1 ,2 | dx Ω(1 ,2 )



u1 ,2 νΩ o · ∇u1 ,2 dσ −

= 

∂Ω o

  i=1,2

u1 ,2 νΩi (1 ,2 ) · ∇u1 ,2 dσ ∂Ωi (1 ,2 )

u1 ,2 νΩ o · ∇u1 ,2 dσ     n−2 u1 ,2 (1 pi +1 2 η) νΩi (η) · ∇η u1 ,2 (1 pi +1 2 η) dση −(1 2 ) =

∂Ω o

i=1,2

∂Ωi

for all (1 , 2 ) ∈]0, δ1 [×]0, δ2 [. Then, recalling that u1 ,2 is a solution of the boundary value problem (10.4), we deduce that   2 |∇u1 ,2 | dx = f o νΩ o · ∇u1 ,2 dσ Ω(1 ,2 ) ∂Ω o     n−2 − (1 2 ) fi (η) νΩi (η) · ∇η u1 ,2 (1 pi + 1 2 η) dση i=1,2

∂Ωi

for all (1 , 2 ) ∈]0, δ1 [×]0, δ2 [. We now take ΩM as in Theorem 10.7 and Ωi,m with i ∈ {1, 2} as in Theorem 10.8. In addition, we assume that ∂Ω o ⊆ ΩM

and

∂Ωi ⊆ Ωi,m

for i ∈ {1, 2} .

By exploiting the corresponding maps UM and Ui,m , i ∈ {1, 2}, we can write  2 |∇u1 ,2 | dx Ω(1 ,2 )    o n−2 = f νΩ o · ∇UM [1 , 2 ] dσ − (1 2 ) fi νΩi · ∇Ui,m [1 , 2 ] dσ ∂Ω o

∂Ωi

i=1,2 (1)

(2)

for all (1 , 2 ) ∈]0, δE [2 , with δE ≡ min{δM , δm , δm }. Then (10.24) holds with E defined by E(1 , 2 )    ≡ f o νΩ o · ∇UM [1 , 2 ] dσ − (1 2 )n−2 ∂Ω o

i=1,2

fi νΩi · ∇Ui,m [1 , 2 ] dσ ∂Ωi

for all (1 , 2 ) ∈]0, δE [2 . Moreover, equality (10.19) implies that  E(0, 0) = f o νΩ o · ∇uo dσ ∂Ω o

and the validity of (10.25) follows by a computation based on the Divergence Theorem 4.1 and on the fact that uo is the solution of the unperturbed problem (10.20).

386

10 A Dirichlet Problem in a Domain with Two Small Holes

To verify that E is real analytic we note that the map UM is real analytic from ] − δE , δE [2 to C 1,α (ΩM ) by Theorem 10.7. Since the map from C 1,α (ΩM ) to C 0,α (ΩM )n that takes a function to its gradient is linear and continuous, we deduce that the map (1 , 2 ) → ∇UM [1 , 2 ] is real analytic from ] − δE , δE [2 to C 0,α (ΩM )n . Similarly, Theorems 10.8 implies that (1 , 2 ) → ∇Ui,m [1 , 2 ] is real analytic from ]−δE , δE [2 to C 0,α (Ωi,m )n for all i ∈ {1, 2}. Then, by the continuity of the linear map that takes a function of C 0,α (∂Ω o ) to its integral over ∂Ω o and of the linear map that takes a function of C 0,α (∂Ωi ) to its integral over ∂Ωi , with i ∈ {1, 2}, the map E is real analytic.  

10.3.2 Close Holes in Dimension n ≥ 3 If 1 approaches 0 and 2 remains away from 0, then the holes shrink to one another while reducing their size at a comparable speed. In this case we say that the holes are ‘close,’ rather than ‘moderately close.’ To analyze the behavior of the solution u1 ,2 under such assumption we consider the operator V(0, ˜2 ) of (10.10)–(10.12) for a fixed ˜2 in ]0, δ2 [. For the equation V(0, ˜2 )[θ1 , θ2 , θo ] = (g1 , g2 , g o ) we have the following. Lemma 10.10. If ˜2 ∈]0, δ2 [ and (g1 , g2 , g o ) is a triple of C 1,α (∂Ω1 )×C 1,α (∂Ω1 )× C 1,α (∂Ω o ), then equation V(0, ˜2 )[θ1 , θ2 , θo ] = (g1 , g2 , g o ). is equivalent to

! VΩ (˜2 ) [θ ] + vΩ o [θo ](0) = g , VΩ o [θo ] = g o ,

(10.26)

(10.27)

with θ ∈ C 0,α (∂Ω (˜ 2 )) and g ∈ C 1,α (∂Ω (˜ 2 )) given by θ (x) ≡ ˜−1 2 θi

 x − pi  ˜2

, g (x) ≡ gi

 x − pi  ˜2

,

∀i ∈ {1, 2} , x ∈ ∂Ωi (1, ˜2 ) ,

(see (10.3) for the definition of Ω (˜ 2 )). Moreover, there exists one and only one triple (θ1 , θ2 , θo ) in C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) for which (10.26) (and (10.27)) holds true. Proof. The equivalence of (10.26) and (10.27) follows by the definition of V in (10.10)–(10.12) and by a computation based on the rule of change of variable in integrals. To prove the last statement of the lemma we observe that, by Theorem 6.46, the third equation in (10.27) has a unique solution θo ∈ C 0,α (∂Ω o ), then, again by Theorem 6.46, also the first equation in (10.27) has a unique solu2 )).   tions θ ∈ C 0,α (∂Ω (˜ Then we verify that the operator V(0, ˜2 ) is an isomorphism of Banach spaces.

10.3 Close and Moderately Close Holes in Dimension n ≥ 3

387

Lemma 10.11. Let ˜2 ∈]0, δ2 [. Then the operator V(0, ˜2 ) is an isomorphism from C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) to C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) × C 1,α (∂Ω o ). Proof. By Lemma 10.2 we know that V(0, ˜2 ) is a linear bounded operator. Then, to prove that it is a homeomorphism, it suffices to show that it is a bijection and invoke the Open Mapping Theorem 2.2. Hence the statement follows by Lemma 10.10.   At this point, we have the following. Lemma 10.12. Let ˜2 ∈]0, δ2 [. Then there exists 0 < δ∗ < min{δ1 , δ2 − ˜2 } such 2 − δ∗ , ˜2 + δ∗ [ and the map that V(1 , 2 ) is invertible for all (1 , 2 ) ∈] − δ∗ , δ∗ [×]˜ 2 − δ∗ , ˜2 + δ∗ [ to from ] − δ∗ , δ∗ [×]˜  L C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) × C 1,α (∂Ω o ) ,  C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) that takes (1 , 2 ) to V(1 , 2 )(−1) is real analytic. Proof. Since the set of invertible operators is open in the space of bounded operators and since the map that takes an invertible operator to its inverse is real analytic the lemma follows by Lemmas 10.2 and 10.11 (cf. Proposition A.15 in the Appendix).   Then by Lemma 10.12 we deduce the following. Proposition 10.13. Let ˜2 ∈]0, δ2 [. Let 0 < δ∗ < min{δ1 , δ2 − ˜2 } be as in Lemma 10.12. Then there exists a real analytic map Θ ≡ (Θ1 , Θ2 , Θo ) from the square domain 2 − δ∗ , ˜2 + δ∗ [ ] − δ∗ , δ∗ [×]˜ to C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) such that V(1 , 2 )[Θ1 [1 , 2 ], Θ2 [1 , 2 ], Θo [1 , 2 ]] = (f1 , f2 , f o ) for all (1 , 2 ) ∈] − δ∗ , δ∗ [×]˜ 2 − δ∗ , ˜2 + δ∗ [. In particular, (θ1 (1 , 2 ), θ1 (1 , 2 ), θo (1 , 2 )) = Θ[1 , 2 ] for all 1 ∈]0, δ∗ [ and 2 > 0 in the interval ]˜ 2 − δ∗ , ˜2 + δ∗ [. Proof. We take Θ[1 , 2 ] ≡ V(1 , 2 )(−1) [f1 , f2 , f o ]

∀ ∈] − δ∗ , δ∗ [×]˜ 2 − δ∗ , ˜2 + δ∗ [ .

Then the proposition follows by Proposition 10.1 and Lemma 10.12 (see also the argument in the proof of Proposition 10.6).  

388

10 A Dirichlet Problem in a Domain with Two Small Holes

We observe that a priori the scalar δ∗ and the map Θ do not necessarily coincide with those introduced in Proposition 10.6. We can, however, justify our notation by noting that we can choose the same δ∗ in Propositions 10.6 and 10.13 and that the two corresponding maps Θ coincide in the intersection of their domains by virtue of the Identity Principle for real analytic functions of Proposition A.14. We adopt a similar (and similarly justified) abuse of notation also in the following Theorems 10.14, 10.15, and 10.16, where we derive some consequences of Proposition 10.6 and of the integral representation (10.14). We begin with Theorem 10.14, where we study the ‘macroscopic’ behavior of u1 ,2 away from the holes Ω1 (1 , 2 ) and Ω2 (1 , 2 ). Theorem 10.14. Let ˜2 ∈]0, δ2 [. Let δ∗ be as in Lemma 10.12. Let ΩM be an open / ΩM . Let δM ∈]0, δ∗ [ be such that subset of Ω o such that 0 ∈ ΩM ∩ Ω1 (1 , 2 ) = ∅ and

ΩM ∩ Ω2 (1 , 2 ) = ∅

for all (1 , 2 ) ∈] − δM , δM [×]˜ 2 − δM , ˜2 + δM [. Then, there exists a real analytic map 2 − δM , ˜2 + δM [→ C 1,α (ΩM ) UM : ] − δM , δM [×]˜ such that u1 ,2 |ΩM = UM [1 , 2 ]

(10.28)

for all 1 ∈]0, δM [ and 2 > 0 in the interval ]˜ 2 − δM , ˜2 + δM [. Moreover, UM [0, ˜2 ] = uo|ΩM

(10.29)

where uo ∈ C 1,α (Ω o ) is the unique solution of the unperturbed problem (10.20). Proof. The proof is very similar to that of Theorem 10.7. First we define  n−2 UM [1 , 2 ](x) ≡ n−2  Sn (x − 1 p1 + 1 2 η) Θ1 [1 , 2 ](η) dση 1 2 ∂Ω1  n−2 + n−2  Sn (x − 1 p2 + 1 2 η) Θ2 [1 , 2 ](η) dση + vΩ o [Θo [1 , 2 ]](x) 1 2 ∂Ω2

for all x ∈ ΩM and (1 , 2 ) ∈] − δM , δM [×]˜ 2 − δM , ˜2 + δM [. Then, applying the argument in the proof of Theorem 10.7 around the point (0, ˜2 ) instead of (0, 0), we show that (1 , 2 ) → UM [1 , 2 ] is real analytic. Equality (10.28) is a consequence of Propositions 10.1 and 10.13. Finally, to prove (10.29) we observe that UM [0, ˜2 ](x) = vΩ o [Θo [0, ˜2 ]](x) for all x ∈ ΩM and that vΩ o [Θo [0, ˜2 ]] is the only solution in C 1,α (Ω o ) of the unperturbed problem (10.20). Indeed, Lemma 10.10 and Proposition 10.13 imply that VΩ o [Θo [0, ˜2 ]] = f o .

(10.30)  

10.3 Close and Moderately Close Holes in Dimension n ≥ 3

389

Then, for the ‘microscopic’ behavior of u1 ,2 close to the boundary of the holes Ω1 (1 , 2 ) and Ω2 (1 , 2 ) we have the following. Theorem 10.15. Let ˜2 ∈]0, δ2 [. Let δ∗ be as in Lemma 10.12. Let i ∈ {1, 2}. Let (i) Ωi,m be an open bounded subset of Rn \ Ωi . Let δm ∈]0, δ∗ [ be such that 1 pi + 1 2 Ωi,m ⊆ Ω o and (1 pi + 1 2 Ωi,m ) ∩ Ωj (1 , 2 ) = ∅ (i)

(i)

(i)

(i)

2 − δm , ˜2 + δm [, where j = 1 if i = 2 and j = 2 for all (1 , 2 ) ∈] − δm , δm [×]˜ if i = 1. Then, there exists a real analytic map (i) (i) (i) (i) Ui,m : ] − δm , δm [×]˜  2 − δm , ˜2 + δm [→ C 1,α (Ωi,m )

such that u1 ,2 (1 pi + 1 2 ξ) = Ui,m [1 , 2 ](ξ) for all

(i) 1 ∈]0, δm [

and 2 > 0 in the interval ]˜ 2 −

∀ξ ∈ Ωi,m

(i) δm , ˜2

Ui,m [0, ˜2 ](ξ) = v (pi + ˜2 ξ)

+

(i) δm [.

(10.31)

Moreover,

∀ξ ∈ Ωi,m

1,α (Rn \ Ω (˜ 2 )) is the unique solution of where v ∈ Cloc ⎧ ⎪ in Rn \ Ω (˜ 2 ) , ⎨Δv = 0 on ∂Ω (˜ 2 ) , v  = f ⎪ ⎩ limξ→∞ v (ξ) = uo (0) ,

(10.32)

(10.33)

where uo ∈ C 1,α (Ω o ) is the unique solution of the unperturbed problem (10.20), and where f is defined by f (x) ≡ fk

 x − pk  ˜2

,

∀k ∈ {1, 2} , x ∈ ∂Ωk (1, ˜2 ) .

Proof. Let j = 1 if i = 2 and j = 2 otherwise. We define Ui,m [1 , 2 ](ξ) ≡ vΩi [Θi [1 , 2 ]](ξ)    + n−2 Sn pi − pj + 2 (ξ − η) Θj [1 , 2 ](η) dση 2 ∂Ωj



  Sn 1 pi + 1 2 ξ − y Θo [1 , 2 ](y) dσy

+ ∂Ω o

(i)

(i)

(i)

(i)

for all ξ ∈ Ωi,m and (1 , 2 ) ∈] − δm , δm [×]˜ 2 − δm , ˜2 + δm [. To show that (1 , 2 ) → Ui,m [1 , 2 ]

390

10 A Dirichlet Problem in a Domain with Two Small Holes

is real analytic, we argue as in the proof of Theorem 10.8. Equality (10.31) can be verified by a change of variable in the integral representation (10.14). Then to prove the statement it remains to verify (10.32). By a change of variables in the integrals over ∂Ωi and ∂Ωj we can observe that Ui,m [0, ˜2 ](ξ) = vΩ (˜2 ) [θ ](pi + ˜2 ξ) + vΩ o [Θo [0, ˜2 ]](0)

∀ξ ∈ Ωi,m

with θ ∈ C 0,α (∂Ω (˜ 2 )) given by θ (y) ≡ ˜−1 2 Θk [0, 2 ]

 y − pk 

∀y ∈ ∂Ωk (1, ˜2 ) , k ∈ {1, 2} .

˜2

Then we note that vΩ (˜2 ) [θ ] + vΩ o [Θo [0, ˜2 ]](0) 1,α is the unique solution in Cloc (Rn \Ω (˜ 2 )) of problem (10.33). Indeed, Lemma 10.10 and Proposition 10.13 imply that

VΩ (˜2 ) [θ ] + vΩ o [Θo [0, ˜2 ]](0) = f

on ∂Ω (˜ 2 ) ,

+ o ˜2 ]] = uo with uo as in (10.20). Hence, the and by (10.30) we have vΩ o [Θ [0,  validity of equality (10.32) follows.  

We observe that the limiting problem (10.33) is defined in the exterior of Ω (˜ 2 ) and contains boundary conditions both on ∂Ω1 (1, ˜2 ) and on ∂Ω2 (1, ˜2 ). Thus, both the sets Ω1 and Ω2 and the corresponding functions f1 and f2 take a part in it. Instead, the corresponding problem (10.23) obtained in the case of ‘moderately close’ holes is defined in the exterior of Ωi and has only a condition on the boundary ∂Ωi . In a sense, we can say that the ‘microscopic’ limit of the rescaled solution u1 ,2 (1 pi + 1 2 ·) ‘continues to see’ both the holes if these are ‘close’ and ‘sees’ only one of the holes when these are ‘moderately close’ to one another. Finally, for the energy integral of u1 ,2 we have the following Theorem 10.16. The proof can be effected following the footsteps of the proof of Theorem 10.9. We just need to use Theorem 10.14 instead of Theorem 10.7 and Theorem 10.15 instead of Theorem 10.8. Theorem 10.16. Let ˜2 ∈]0, δ2 [. Let δ∗ be as in Lemma 10.12. Then there exist δE ∈]0, δ∗ [ and a real analytic function E : ] − δE , δE [×]˜ 2 − δE , ˜2 + δE [→ R such that

 2

|∇u1 ,2 | dx = E(1 , 2 ) Ω()

2 − δE , ˜2 + δE [. Moreover, for all 1 ∈]0, δE [ and 2 > 0 in the interval ]˜  2 E(0, 0) = |∇uo | dx , Ωo

10.4 Moderately Close Holes in Dimension n = 2

391

where uo is the unique solution of the unperturbed problem (10.20).

10.4 Moderately Close Holes in Dimension n = 2 We now turn to the case of dimension n = 2. In contrast with what we have done for n ≥ 3, we consider only the case where the pair (1 , 2 ) approaches (0, 0) and the holes are ‘moderately close’ to one another. As we have explained in Section 10.1, this means that the holes are colliding into one another while shrinking their size at a faster speed. Instead, we do not treat the case where 1 tends to zero and 2 stays away from zero. That is, the case of ‘close’ holes. The interested reader can find the analysis of this last case in the the paper [79]. Before entering into mathematical details, we would like to comment some of the results that we are going to prove. In particular, in Theorem 10.35 we will analyze the behavior of the solution u1 ,2 away from the boundary of the holes ∂Ω1 (1 , 2 ) and ∂Ω2 (1 , 2 ). As usual, we will consider the restriction of u1 ,2 to an open subset of Ω o that does not contain the origin in its closure. If we content ourselves to pick a single point x ∈ Ω o \ {0} and look at the evaluation of u1 ,2 at x, then Theorem 10.35 implies that, possibly shrinking δ1 and δ2 , we have u1 ,2 (x) = uo (x) + 1 2 Ux [1 , 2 ] + F(1 , 2 )t Λ(1 , 2 )−1 Vx [1 , 2 ] (10.34) for all 1 ∈]0, δ1 [ and all 2 ∈]0, δ2 [. In the equality above uo is the solution of the unperturbed Dirichlet problem in Ω o with boundary datum f o , the function Ux is real analytic from ] − δ1 , δ1 [×] − δ2 , δ2 [ to R, the functions F and Vx are real analytic from ] − δ1 , δ1 [×] − δ2 , δ2 [ to R2 , and Λ(1 , 2 ) is a 2 × 2 matrix such that ⎞ ⎛ log(1 2 ) log 1 1 ⎝ ⎠ = R(1 , 2 ) Λ(1 , 2 ) − 2π log 1 log(1 2 ) where R is a real analytic matrix-valued function on ] − δ1 , δ1 [×] − δ2 , δ2 [. We shall see that Λ(1 , 2 ) is invertible for every pair (1 , 2 ) of positive real numbers in ]0, δ1 [×]0, δ2 [. Accordingly, the inverse matrix Λ(1 , 2 )−1 in the right-hand side of (10.34) makes sense. We also observe that the matrix Λ(1 , 2 ) takes care of the logarithmic behavior which is expected for the Dirichlet problems in a perforated domain in dimension two. Moreover, equality (10.34) fulfills the goal stated in Section 10.2. That is, it represents the map that takes (1 , 2 ) to u1 ,2 (x) in terms of the following two ingredients. • Real analytic functions of the perturbation parameter, in this case the pair (1 , 2 ), defined in a open neighborhood of the singular point (0, 0) in R2 . • Singular but completely known functions of the perturbation parameter, in this case log(1 2 ) and log 1 .

392

10 A Dirichlet Problem in a Domain with Two Small Holes

However, the presence of the matrix Λ(1 , 2 )−1 in (10.34) produces a specific problem that we did not meet when dealing with a single hole in Section 8.4 or with two holes in dimension n ≥ 3 in Section 10.3. Indeed, if we want to use (10.34) to deduce an asymptotic approximation of u1 ,2 (x), as we did in Section 8.5 for the one hole problem, then we have to compute the inverse matrix Λ(1 , 2 )−1 . Doing so, we obtain an expression which involves the quotient log 1 log(1 2 )

(10.35)

(cf. Proposition 10.37 below). Now, the problem with (10.35) is that its limit as (1 , 2 ) → (0, 0) does not exist. This is a new situation that we did not foresee when we have introduced the problem in Section 10.2. One way to overcome this difficulty is to introduce a relation between the parameters 1 and 2 . For example, we can replace 1 with a positive real parameter t and set 2 = γ(t), with γ a function from the interval ]0, 1[ to itself such that lim γ(t) = 0

t→0+

and such that the limit λ0 ≡ lim+ t→0

log t log(tγ(t))

(10.36)

exists finite in [0, +∞[. In this way, we switch from a two parameter problem to a one parameter problem and, thanks to the existence of the limit (10.36), we can compute the an asymptotic approximation of the map t → ut,γ(t) for t > 0 that tends to zero. In particular, if we stop our computation to the first two terms we obtain ut,γ(t) (x) = uo (x)

 o 2π  1 lim u1 (η) + lim u2 (η) − 2uo (0) GΩ (x, 0) η→∞ log(tγ(t)) 1 + λ0 η→∞   1 +o log(tγ(t)) (10.37)

+

as t > 0 tends to zero (see Proposition 10.39). In (10.37), u1 and u2 denote the harmonic solution of the exterior Dirichlet problem in R2 \ Ω1 and R2 \ Ω2 with o boundary datum f1 and f2 , respectively, and GΩ is the Dirichlet Green’s function of Ω o . A particular feature of (10.37) is that the limit value λ0 appears explicitly in the second asymptotic term of the expansion. This seems to indicate that no asymptotic approximation with at least two terms could be obtained without prescribing a limiting value for the quotient (10.35). In this sense, the necessity of introducing a specific relation between the size of the holes and their mutual distance is intrinsic

10.4 Moderately Close Holes in Dimension n = 2

393

in the nature of the problem and could not be avoided, no matter what method we adopt to analyze it. It is thus no surprise that some kind of condition on the size and the distance of the holes has appeared in previous research papers dealing with the Dirichlet problem in a domain with moderately close holes, also when the adopted techniques are very different from those of the Functional Analytic Approach of the present book. For example, in [31], Bonnaillie-No¨el, Dambrine, and Lacave use the Multi-Scale Approximation Method to study the Poisson problem with Dirichlet conditions in a domain with two moderately close holes. To carry on their computations they assume that the distance behaves like the size to some power β ∈]0, 1[, a condition that corresponds, with our notation, to the case where γ(t) = t(1−β)/β and the quotient (10.35) is constant and equal to 1 − β. Somehow related to this problem is also the work of Maz’ya, Movchan, and Nieves on the Meso-Scale Approximation Method, which aims at studying boundary value problems in domains with a large (but fixed) number of small close holes, a so-called ‘cloud of voids’ (cf. [201]). Most of their papers deal with the threedimensional case, but a relation between the size and the distance of the holes still plays a role in the analysis. We mention, for example, the paper [199] of Maz’ya and Movchan, on a three-dimensional Poisson problem with Dirichlet conditions, and the paper [202] of Maz’ya, Movchan, and Nieves, on a two-dimensional scattering problem for an elastic membrane.

10.4.1 Integral Representation of the Solution We now begin our analysis of the Dirichlet problem (10.4) in dimension n = 2. As for the case of the one hole problem of Section 8.4, we exploit the integral representation (6.19) for the solution u1 ,2 . Then, by writing (6.19) for the specific case when the domain Ω = Ω(1 , 2 ) has two holes, we obtain + [μ1 ,2 ] u1 ,2 = wΩ( 1 ,2 )    2  + (i) (j) f1 ,2 τ1 ,2 dσ (Λ−1 + Ω(1 ,2 )− )i,j vΩ(1 ,2 ) [τ1 ,2 ] i,j=1

∂Ω(1 ,2 )

with f1 ,2 ∈ C 1,α (∂Ω(1 , 2 )) defined by  ⎧  x −  1 p1 ⎪ ⎪ f 1 ⎪ ⎪ ⎪ ⎨  1 2  x −  1 p2 f1 ,2 (x) ≡ f2 ⎪ ⎪ 1 2 ⎪ ⎪ ⎪ ⎩ o f (x)

if x ∈ ∂Ω1 (1 , 2 ) , if x ∈ ∂Ω2 (1 , 2 ) , if x ∈ ∂Ω o ,

(10.38)

394

10 A Dirichlet Problem in a Domain with Two Small Holes (1)

(2)

and where τ1 ,2 , τ1 ,2 ∈ C 0,α (∂Ω(1 , 2 )) and μ1 ,2 ∈ C 1,α (∂Ω(1 , 2 )) are the unique solutions of the following systems of integral equations  ⎧ 1 t (j) ⎪ ⎪ ⎨ 2 I + WΩ(1 ,2 ) [τ1 ,2 ] = 0 ,  (10.39) ⎪ ⎪ τ(j) dσ = δ ∀i ∈ {1, 2} ⎩ i,j 1 ,2 ∂Ωi (1 ,2 )

for j ∈ {1, 2}, and ⎧ 1  ⎪ ⎨ 2 I + WΩ(1 ,2 )[μ1 ,2 ] (j) 2 = f1 ,2 − j=1 ∂Ω(1 ,2 ) f1 ,2 τ1 ,2 dσ ζ∂Ωj (1 ,2 ) , ⎪  ⎩ μ dσ = 0 ∂Ωj (1 ,2 ) 1 ,2

(10.40)

(see (6.17) for the definition of ζ∂Ωj (1 ,2 ) ). Finally, the matrix ΛΩ(1 ,2 )− that appears in (10.38) is given by ΛΩ(1 ,2 )−  =

(1)

(1)



(1)



− ∂Ω1 (1 ,2 )

VΩ(1 ,2 ) [τ1 ,2 ] dσ

− ∂Ω2 (1 ,2 )

VΩ(1 ,2 ) [τ1 ,2 ] dσ



(2)

(2)



− ∂Ω1 (1 ,2 )

VΩ(1 ,2 ) [τ1 ,2 ] dσ

− ∂Ω2 (1 ,2 )

VΩ(1 ,2 ) [τ1 ,2 ] dσ (10.41)

(2)

with τ1 ,2 and τ1 ,2 as in (10.39). As we have observed in Lemma 6.30, ΛΩ(1 ,2 )− is invertible. Therefore, we are allowed to write Λ−1 Ω(1 ,2 )− in (10.38). To analyze problem (10.4) we proceed as follows. First we analyze system (1) (2) (10.39) and prove a prove a real analytic continuation result for τ1 ,2 and τ1 ,2 , then we turn to system (10.40) and prove a similar real analytic continuation result also for μ1 ,2 . At this point we will be ready to study the solution u1 ,2 by means of the representation formula (10.38). As usual our aim is to represent the maps that take (1 , 2 ) to suitable restrictions of u1 ,2 and of the rescaled functions u1 ,2 (1 pi + 1 2 ·) with i ∈ {1, 2} and to the energy Ω(1 ,2 ) |∇u1 ,2 |2 dx in terms of real analytic maps of (1 , 2 ) and of elementary functions. To describe the limiting values of such analytic maps, we will introduce in Section 10.4.4 certain auxiliary functions related to the Dirichlet Green’s functions of Ω o and of the exteriors of Ω1 and Ω2 . In Section 10.4.5, we state Theorem 10.35 on the representation of u1 ,2 and the u1 ,2 (1 pi + 1 2 ·)’s in terms of real analytic maps and known functions. Then in Theorem 10.36 we consider the energy of u1 ,2 . In the last Section 10.4.6 of this chapter, we examine the problem of computing an asymptotic expansion of the solution. We will need to introduce a specific relation between 1 and 2 to ensure the existence of the limit of the quotient (10.35) for (1 , 2 ) → (0, 0). We also verify that the second term in the asymptotic expansion of u1 ,2 depends explicitly on such limit.

10.4 Moderately Close Holes in Dimension n = 2

395

10.4.2 Analysis of System (10.39) In this subsection we study system (10.39) and prove a real analytic continuation (1) (2) result for the densities τ1 ,2 and τ1 ,2 . We exploit the same strategy that we have used in Section 8.4 to deal with the case of a single shrinking hole. Accordingly, we first split the first equation of (10.39) over ∂Ω1 (1 , 2 ), ∂Ω2 (1 , 2 ), and ∂Ω o , and then we perform a change of variable in the integrals over the (1 , 2 )-dependent boundaries ∂Ω1 (1 , 2 ), ∂Ω2 (1 , 2 ). We observe that ∇S2 (1 pi + 1 2 ξ − 1 pi − 1 2 η) = (1 2 )−1 ∇S2 (ξ − η)

(10.42)

i j ∇S2 (1 pi + 1 2 ξ − 1 pj − 1 2 η) = −1 1 ∇S2 (p − p + 2 (ξ − η)),

(10.43)

and

for all i, j ∈ {1, 2}, i = j, and all ξ, η ∈ ∂Ωi for which the functions in (10.42) and (10.43) are defined. Then, introducing the new densities φ1,(j) (ξ) ≡ 1 2 τ(j) (1 p1 + 1 2 ξ) 1 ,2 1 ,2 φ2,(j) (ξ) 1 ,2

≡

1 2 τ(j) (1 p2 1 ,2

φo,(j) (x) 1 ,2

≡

τ(j) (x) 1 ,2

+ 1 2 ξ)

∀ξ ∈ ∂Ω1 , ∀ξ ∈ ∂Ω2 ,

(10.44)

∀x ∈ ∂Ω , o

we find that system (10.39) for j ∈ {1, 2} is equivalent to the system of the following four equations: 1 1,(j) φ (ξ) − WΩt 1 [φ1,(j) ](ξ) 1 ,2 2 1 ,2  − 2 ∂Ω2

φ2,(j) (η) νΩ1 (ξ) · ∇S2 (p1 − p2 + 2 (ξ − η)) dση 1 ,2



− 1 2 ∂Ω o

φo,(j) (y) νΩ1 (ξ) · ∇S2 (1 p1 + 1 2 ξ − y) dσy = 0 1 ,2

∀ξ ∈ ∂Ω1 , (10.45)

1 2,(j) φ (ξ) − WΩt 2 [φ2,(j) ](ξ) 1 ,2 2 1 ,2  − 2 ∂Ω1

φ1,(j) (η) νΩ2 (ξ) · ∇S2 (p2 − p1 + 2 (ξ − η)) dση 1 ,2



− 1 2 ∂Ω o

φo,(j) (y) νΩ2 (ξ) · ∇S2 (1 p2 + 1 2 ξ − y) dσy = 0 1 ,2

∀ξ ∈ ∂Ω2 , (10.46)

1 o,(j) φ (x) + WΩt o [φo,(j) ](x) 1 ,2 2 1 ,2

396

10 A Dirichlet Problem in a Domain with Two Small Holes

+

2   h=1

∂Ωh

h φ(i,h) 1 ,2 (η) νΩ o (x) · ∇S2 (x − 1 p − 1 2 η) dση = 0

∀x ∈ ∂Ω o , (10.47)

 ∂Ωi

∀i ∈ {1, 2} .

φi,(j) 1 ,2 dσ = δi,j

(10.48) In order to cast the system of the four equations in (10.45)–(10.48) into an operator equation in Banach space, we introduce for each (1 , 2 ) in ] − δ1 , δ1 [×] − δ2 , δ2 [ an auxiliary operator. Namely, the operator   N (1 , 2 ) ≡ N 1 (1 , 2 ) , N 2 (1 , 2 ) , N o (1 , 2 ) , N s (1 , 2 ) (the superscript ‘s’ of N s stands for ‘scalars’) from C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) to C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) × R2 defined by 1 N 1 (1 , 2 )[φ1 , φ2 , φo ](ξ) ≡ φ1 (ξ) − WΩt 1 [φ1 ](ξ) 2  − 2 φ2 (η) νΩ1 (ξ) · ∇S2 (p1 − p2 + 2 (ξ − η)) dση ∂Ω2  − 1 2 φo (y) νΩ1 (ξ) · ∇S2 (1 p1 + 1 2 ξ − y) dσy

(10.49)

∀ξ ∈ ∂Ω1 ,

∂Ω o

1 N 2 (1 , 2 )[φ1 , φ2 , φo ](ξ) ≡ φ2 (ξ) − WΩt 2 [φ2 ](ξ) 2  − 2 φ1 (η) νΩ2 (ξ) · ∇S2 (p2 − p1 + 2 (ξ − η)) dση ∂Ω1  − 1 2 φo (y) νΩ2 (ξ) · ∇S2 (1 p2 + 1 2 ξ − y) dσy

(10.50)

∀ξ ∈ ∂Ω2 ,

∂Ω o

N o (1 , 2 )[φ1 , φ2 , φo ](x) ≡ +

2   h=1

1 o φ (x) + WΩt o [φo ](x) 2

(10.51)

φh (η) νΩ o (x) · ∇S2 (x − 1 ph − 1 2 η) dση ∂Ωh







N s (1 , 2 )[φ1 , φ2 , φo ] ≡

φ1 dσ , ∂Ω1

∀x ∈ ∂Ω o ,

φ2 dσ

(10.52)

∂Ω2

for all (1 , 2 ) ∈] − δ1 , δ1 [×] − δ2 , δ2 [ and all (φ1 , φ2 , φo ) ∈ C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ). It is apparent that the system of Equations (10.45)– (10.48) is equivalent to

10.4 Moderately Close Holes in Dimension n = 2

397

N (1 , 2 )[φ1,(j) , φ2,(j) , φo,(j) ] = (0, 0, 0, (δ1,j , δ2,j )) 1 ,2 1 ,2 1 ,2 (j)

1,(j)

(10.53) 2,(j)

o,(j)

and thus system (10.39) for τ1 ,2 is equivalent to (10.53) with (φ1 ,2 , φ1 ,2 , φ1 ,2 ) (j) and τ1 ,2 related by (10.44). Accordingly, Lemma 6.28 on the unique solvability of system (10.39) implies the validity of the following. Proposition 10.17. For all (1 , 2 ) ∈]0, δ1 [×]0, δ2 [ and j ∈ {1, 2}, there exists one and only one solution (φ1,(j) , φ2,(j) , φo,(j) ) ∈ C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) 1 ,2 1 ,2 1 ,2 of (10.53). Proposition 10.17 implies that, for each j ∈ {1, 2}, there is a map from ]0, δ1 [×]0, δ2 [ to C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) that takes a pair (1 , 2 ) to the triple , φ2,(j) , φo,(j) ). (10.54) (φ1,(j) 1 ,2 1 ,2 1 ,2 We now prove that such a map has a real analytic continuation in a neighborhood of the singular pair (0, 0). To do so, we begin with the following technical lemma. Lemma 10.18. For all (1 , 2 ) in ] − δ1 , δ1 [×] − δ2 , δ2 [ and all (φ1 , φ2 , φo ) in C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) the pair of functions  1  N (1 , 2 )[φ1 , φ2 , φo ] , N 2 (1 , 2 )[φ1 , φ2 , φo ] (10.55) belongs to C 0,α (∂Ω1 )0 × C 0,α (∂Ω2 )0 . Proof. The membership of the pair of (10.55) in C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) is a consequence of the mapping properties of the operator W t (cf. Theorem 6.7) and of the mapping properties of the integral operators with a real analytic kernel of Theorem A.28 (ii) in the Appendix. To prove the lemma it remains to show that  N 1 (1 , 2 )[φ1 , φ2 , φo ] dσ = 0 (10.56) ∂Ω1

and

 N 2 (1 , 2 )[φ1 , φ2 , φo ] dσ = 0.

(10.57)

∂Ω2

We confine ourselves to verify (10.56). The proof of (10.57) is indeed the same. To deduce (10.56) we exploit the definition of N 1 (1 , 2 )[φ1 , φ2 , φo ] in (10.49). We begin by observing that, by the jump formulas for the normal derivative of the single layer potential of Theorem 6.6 and by the first Green Identity of Theorem 4.2, we have     1 + t φ1 − WΩ1 [φ1 ] dσ = − νΩ1 · ∇vΩ [φ1 ] dσ = 0 . 1 2 ∂Ω1 ∂Ω1

398

10 A Dirichlet Problem in a Domain with Two Small Holes

If 2 = 0, the equality above is sufficient to conclude the proof. If instead 2 = 0, then we have also to deal with the second term in the right-hand side of (10.49), and if both 1 and 2 are not 0, then we have to consider also the third term. So, we now assume that 2 = 0 and we study the second term in the expression for N 1 (1 , 2 )[φ1 , φ2 , φo ] in (10.49). We observe that the classical theorem of differentiability for integrals depending on a parameter and the chain rule imply that  2 φ2 (η) νΩ1 (ξ) · ∇S2 (p1 − p2 + 2 (ξ − η)) dση ∂Ω2   1 2 = νΩ1 (ξ) · ∇ξ φ2 (η) S2 (p − p + 2 (ξ − η)) dση . ∂Ω2

Since the function that takes ξ to  φ2 (η) S2 (p1 − p2 + 2 (ξ − η)) dση ∂Ω2

is harmonic in a neighborhood of Ω1 for 2 ∈ [−δ2 , δ2 ] (cf. assumption (10.1)), we deduce by the first Green Identity of Theorem 4.2 that   2 φ2 (η) νΩ1 (ξ) · ∇S2 (p1 − p2 + 2 (ξ − η)) dση dσξ ∂Ω1 ∂Ω    2 1 2 = νΩ1 (ξ) · ∇ξ φ2 (η) S2 (p − p + 2 (ξ − η)) dση dσξ = 0. ∂Ω1

∂Ω2

Finally, we assume that both 1 and 2 are different from 0 and we consider the third term in the expression for N 1 (1 , 2 )[φ1 , φ2 , φo ] in (10.49). First we observe that the classical theorem of differentiability for integrals depending on a parameter and the chain rule imply that  φo (y) νΩ1 (ξ) · ∇S2 (1 p1 + 1 2 ξ − y) dσy 1 2 ∂Ω o   o 1 = νΩ1 (ξ) · ∇ξ φ (y) S2 (1 p + 1 2 ξ − y) dσy . ∂Ω o

Then we note that the function that takes ξ to  φo (y) S2 (1 p1 + 1 2 ξ − y) dσy ∂Ω o

is harmonic in a neighborhood of Ω1 for (1 , 2 ) ∈ [−δ1 , δ1 ] × [−δ2 , δ2 ] (cf. assumptions (10.1) and (10.2)). Then the first Green Identity of Theorem 4.2 implies that

10.4 Moderately Close Holes in Dimension n = 2



399



1 2 ∂Ω1



φo (y) νΩ1 (ξ) · ∇S2 (1 p1 + 1 2 ξ − y) dσy dσξ   νΩ1 (ξ) · ∇ξ φo (y) S2 (1 p1 + 1 2 ξ − y) dσy dσξ = 0.

∂Ω o

=

∂Ω o

∂Ω1

 

The validity of the lemma is now proved. We now show that the map (1 , 2 ) → N (1 , 2 ) is real analytic. Lemma 10.19. The map from ] − δ1 , δ1 [×] − δ2 , δ2 [ to L(C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) ,

C 0,α (∂Ω1 )0 × C 0,α (∂Ω2 )0 × C 0,α (∂Ω o ) × R2 ) that takes (1 , 2 ) to N (1 , 2 ) is real analytic. Proof. The analyticity of N follows by Corollary A.29 of the Appendix on the analyticity of integral operators with real analytic kernel, by the continuity of the linear boundary operators WΩt 1 , WΩt 2 , and WΩt o (see Theorem 6.7), and by the continuity of the linear map from C 0,α (∂Ωi ), i ∈ {1, 2}, to R that takes a function to its integral.   We now consider the operator N (0, 0). In the following Lemma 10.20 we investigate equation N (0, 0)[φ1 , φ2 , φo ] = (g1 , g2 , g o , (ρ1 , ρ2 )) with j ∈ {1, 2} and we show that it has a unique solution. Lemma 10.20. For all (g1 , g2 , g o , (ρ1 , ρ2 )) ∈ C 0,α (∂Ω1 )0 × C 0,α (∂Ω2 )0 × C 0,α (∂Ω o ) × R2 and all (φ1 , φ2 , φo ) ∈ C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) we have N (0, 0)[φ1 , φ2 , φo ] = (g1 , g2 , g o , (ρ1 , ρ2 )) if and only if ⎧ ⎪ − 12 φ1 + WΩt 1 [φ1 ] = −g1 , ⎪ ⎪ ⎪ 1 t ⎪ ⎪ ⎨− 2 φ2 + WΩ2 [φ2 ] = −g2 , 1 o t φ + WΩ o [φo ] = g o − (ρ1 + ρ2 ) νΩ o · (∇S2 )|∂Ω o , 2 ⎪ ⎪ ⎪ ⎪ ⎪∂Ω1 φ1 dσ = ρ1 , ⎪ ⎩ φ dσ = ρ2 . ∂Ω2 2

(10.58)

(10.59)

In particular, for all (g1 , g2 , g o , (ρ1 , ρ2 )) ∈ C 0,α (∂Ω1 )0 × C 0,α (∂Ω2 )0 × C 0,α (∂Ω o ) × R2 there exists one and only one (φ1 , φ2 , φo ) ∈ C 0,α (∂Ω1 )×C 0,α (∂Ω2 )×C 0,α (∂Ω o ) solution of Equation (10.58).

400

10 A Dirichlet Problem in a Domain with Two Small Holes

Proof. (The reader may notice that this proof follows the same steps of the proof of Lemma 8.17.) The equivalence of (10.58) and (10.59) is a consequence of the definition of N in (10.49)–(10.52). We now prove the last statement of the lemma. Since R2 \ Ω o is connected, Corollary 6.15 implies that the operator 12 I + WΩt o is an isomorphism from C 0,α (∂Ω o ) to itself. As a consequence, there exists one and only one solution φo of the third equation of (10.59) (note that ν o ∈ C 0,α (∂Ω o )). We now consider the system consisting of the first and fourth equation of (10.59). That is, ! − 1 φ + WΩt 1 [φ1 ] = −g1 ,  2 1 (10.60) φ dσ = ρ1 . ∂Ω1 1  By the membership of g1 in C 0,α (∂Ω1 )0 we have ∂Ω1 g1 dσ = 0. Since Ω1 is connected, it follows that g1 is orthogonal to the null space of − 12 I + WΩ1 , which consists of the constant functions on ∂Ω1 (cf. Lemma 6.21). Thus, by the Fredholm Alternative of Theorem 5.8 there exists φ˜1 ∈ C 0,α (∂Ω1 ) that satisfies the first equation of (10.60). Moreover, the Fredholm Alternative Theorem 5.8 also implies that the null space of − 12 I + WΩt 1 is one-dimensional andis accordingly spanned by a function φ ∈ C 0,α (∂Ω1 ). By Theorem 6.25, we have ∂Ω1 φ dσ = 0. Then we can verify that the function  ρ1 − ∂Ω1 φ˜1 dσ ˜  φ φ1 ≡ φ1 + φ dσ ∂Ω1 is a solution of (10.60). Finally, Theorem 6.25 implies that φ1 is unique. A similar argument shows that there is a unique solution φ2 of the second and last equation of (10.59).   Then, Lemma 10.20 and the Open Mapping Theorem 2.2 imply the validity of the following Lemma 10.21. Lemma 10.21. The operator N (0, 0) is an isomorphism (i.e., a linear homeomorphism) from C 0,α (∂Ω1 )×C 0,α (∂Ω2 )×C 0,α (∂Ω o ) to C 0,α (∂Ω1 )0 ×C 0,α (∂Ω2 )0 × C 0,α (∂Ω o ) × R2 . Proof. Lemma 10.19 implies that N (0, 0) is a linear bounded operator. Then by the Open Mapping Theorem 2.2, it is sufficient to verify that it is a bijection in order to prove that it is an isomorphism. Accordingly, the statement follows by Lemma 10.20.   Since the set of invertible operators is open in the space of bounded operators and since the map that takes an invertible operator to its inverse is real analytic (cf. Proposition A.15 of the Appendix), we deduce the following.

10.4 Moderately Close Holes in Dimension n = 2

401

Lemma 10.22. There exists 0 < δΦ < min{δ1 , δ2 } such that N (1 , 2 ) is invertible for all (1 , 2 ) ∈] − δΦ , δΦ [2 and the map from ] − δΦ , δΦ [2 to  L C 0,α (∂Ω1 )0 × C 0,α (∂Ω2 )0 × C 0,α (∂Ω o ) × R2 ,  C 0,α (∂Ω1 ) × C 0,α (∂Ω2 ) × C 0,α (∂Ω o ) that takes (1 , 2 ) to the inverse operator N (1 , 2 )(−1) is real analytic. Finally, by Lemma 10.22 we deduce that for j ∈ {1, 2} the map that takes (1 , 2 ) to the triple in (10.54) has a real analytic continuation in a neighborhood of (0, 0). Proposition 10.23. Let j ∈ {1, 2}. There exists a real analytic map Φ(j) ≡ (Φ1,(j) , Φ2,(j) , Φo,(j) ) : ]−δΦ , δΦ [2 → C 0,α (∂Ω1 )×C 0,α (∂Ω2 )×C 0,α (∂Ω o ) such that N (1 , 2 )[Φ(j) [1 , 2 ]] = (0, 0, 0, (δ1,j , δ2,j ))

∀(1 , 2 ) ∈] − δΦ , δΦ [2 . (10.61)

In particular, (φ1,(j) , φ2,(j) , φo,(j) ) = Φ(j) [1 , 2 ] 1 ,2 1 ,2 1 ,2

∀(1 , 2 ) ∈]0, δΦ [2 .

(10.62)

Proof. It suffices to set Φ(j) [1 , 2 ] ≡ N (1 , 2 )(−1) [0, 0, 0, (δ1,j , δ2,j )]

∀(1 , 2 ) ∈] − δΦ , δΦ [2 .

Then the statement is a consequence of Proposition 10.17 and of Lemma 10.22 (see also the argument in the proof of Proposition 10.6).   In particular, Proposition 10.23 implies that  Φi,(j) [1 , 2 ] dσ = δi,j ∀(1 , 2 ) ∈] − δΦ , δΦ [2 , i, j ∈ {1, 2}. (10.63) ∂Ωi

10.4.3 Analysis of System (10.40) We now turn to system (10.40) and prove a real analytic continuation result for the density μ1 ,2 . As for system (10.39), we begin by splitting the first equation of (10.40) over ∂Ω1 (1 , 2 ), ∂Ω2 (1 , 2 ), and ∂Ω o , and then we perform a change of variable in the integrals over the (1 , 2 )-dependent boundaries ∂Ω1 (1 , 2 ) and ∂Ω2 (1 , 2 ) (see also Section 8.4.2 for the case of one hole). Taking into account equalities (10.42) and (10.43) and defining the new densities

402

10 A Dirichlet Problem in a Domain with Two Small Holes

ψ11 ,2 (ξ) ≡ μ1 ,2 (1 p1 + 1 2 ξ)

∀ξ ∈ ∂Ω1 ,

ψ21 ,2 (ξ) ≡ μ1 ,2 (1 p2 ψo1 ,2 (x) ≡ μ1 ,2 (x)

∀ξ ∈ ∂Ω2 ,

+ 1 2 ξ)

(10.64)

∀x ∈ ∂Ω , o

we find that system (10.40) is equivalent to the system of the following three equations 1 1 ψ (ξ) − WΩ1 [ψ11 ,2 ](ξ) 2 1 ,2  + 2  −

∂Ω2

ψ21 ,2 (η) νΩ2 (η) · ∇S2 (p1 − p2 + 2 (ξ − η)) dση

ψ o (y) νΩ o (y) · ∇S2 (1 p1 + 1 2 ξ − y) dσy    i,(1) = f1 (ξ) − fi φ1 ,2 dσ − f o φo,(1) 1 ,2 dσ ∂Ω o

i=1,2

∂Ωi

∂Ω o

∀ξ ∈ ∂Ω1 , 1 2 ψ (ξ) − WΩ2 [ψ21 ,2 ](ξ) 2 1 ,2  + 2  −

∂Ω1

ψ11 ,2 (η) νΩ1 (η) · ∇S2 (p2 − p1 + 2 (ξ − η)) dση

ψ o (y) νΩ o (y) · ∇S2 (1 p2 + 1 2 ξ − y) dσy    = f2 (ξ) − fi φi,(2) dσ − f o φo,(2) 1 ,2 1 ,2 dσ ∂Ω o

i=1,2

∂Ωi

∂Ω o

∀ξ ∈ ∂Ω2 , 1 o ψ (x) + WΩ o [ψo1 ,2 ](x) 2 1 ,2 2   ψi1 ,2 (η) νΩi (η) · ∇S2 (x − 1 pi − 1 2 η) dση = f o (x) + 1 2 i=1

∂Ωi

∀x ∈ ∂Ω o , with the additional condition that  ψ11 ,2 dσ = 0 and ∂Ω1

 ∂Ω2

ψ21 ,2 dσ = 0.

(10.65)

Then for all (1 , 2 ) ∈] − δ1 , δ1 [×] − δ2 , δ2 [ we introduce the auxiliary operator M(1 , 2 ) ≡ (M1 (1 , 2 ), M2 (1 , 2 ), Mo (1 , 2 )) from C 1,α (∂Ω1 )0 × C 1,α (∂Ω2 )0 × C 1,α (∂Ω o ) to C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) × C 1,α (∂Ω o ) defined by

10.4 Moderately Close Holes in Dimension n = 2

403

1 M1 (1 , 2 )[ψ1 , ψ2 , ψ o ](ξ) ≡ ψ1 (ξ) − WΩ1 [ψ1 ](ξ) 2  + 2 ψ2 (η) νΩ2 (η) · ∇S2 (p1 − p2 + 2 (ξ − η)) dση ∂Ω2  − ψ o (y) νΩ o (y) · ∇S2 (1 p1 + 1 2 ξ − y) dσy

(10.66)

∀ξ ∈ ∂Ω1 ,

∂Ω o

1 M2 (1 , 2 )[ψ1 , ψ2 , ψ o ](ξ) ≡ ψ2 (ξ) − WΩ2 [ψ2 ](ξ) 2  + 2 ψ1 (η) νΩ1 (η) · ∇S2 (p2 − p1 + 2 (ξ − η)) dση  ∂Ω1 − ψ o (y) νΩ o (y) · ∇S2 (1 p2 + 1 2 ξ − y) dσy

(10.67)

∀ξ ∈ ∂Ω2 ,

∂Ω o

Mo (1 , 2 )[ψ1 , ψ2 , ψ o ](ξ) ≡ + 1 2

2   i=1

1 o ψ (x) + WΩ o [ψ o ](x) 2

(10.68)

ψ i (η) νΩi (η) · ∇S2 (x − 1 pi − 1 2 η) dση

∀x ∈ ∂Ω o ,

∂Ωi

for all (ψ1 , ψ2 , ψ o ) ∈ C 1,α (∂Ω1 )0 × C 1,α (∂Ω2 )0 × C 1,α (∂Ω o ). It follows that for (1 , 2 ) ∈]0, δ1 [×]0, δ2 [ the system of equations (10.40) is equivalent to M(1 , 2 )[ψ11 ,2 , ψ21 ,2 , ψo1 ,2 ] ⎛   ⎝ = f1 (ξ) − i=1,2

f2 (ξ) −

 ∂Ωi

  i=1,2

fi φi,(1) 1 ,2

dσ − ∂Ω o

f o φo,(1) 1 ,2 dσ, ⎞

 ∂Ωi

fi φi,(2) 1 ,2

dσ −

f ∂Ω o

o

φo,(2) 1 ,2

dσ, f

o⎠

(10.69) and the membership of ψ11 ,2 in C 1,α (∂Ω1 )0 and ψ21 ,2 in C 1,α (∂Ω2 )0 implies i,(j)

o,(j)

condition (10.65) (see (10.44) for the definition of φ1 ,2 and φ1 ,2 , i, j ∈ {1, 2}). Then, Equation (10.69) for (ψ11 ,2 , ψ21 ,2 , ψo1 ,2 ) is equivalent to Equation (10.40) for μ1 ,2 , with (ψ11 ,2 , ψ21 ,2 , ψo1 ,2 ) and μ1 ,2 related by (10.64). Accordingly, the existence and uniqueness result of Theorem 6.31 (i) implies that (10.69) has a unique solution. Namely, we have the following. Proposition 10.24. For all (1 , 2 ) ∈]0, δ1 [×]0, δ2 [ there exists one and only one solution (ψ11 ,2 , ψ21 ,2 , ψo1 ,2 ) ∈ C 1,α (∂Ω1 )0 ×C 1,α (∂Ω2 )0 ×C 1,α (∂Ω o ) of (10.69). By Proposition 10.24 we can define a map from ]0, δ1 [×]0, δ2 [ to C 1,α (∂Ω1 )0 × C (∂Ω2 )0 × C 1,α (∂Ω o ) that takes a pair (1 , 2 ) to the unique solution 1,α

(ψ11 ,2 , ψ21 ,2 , ψo1 ,2 )

(10.70)

404

10 A Dirichlet Problem in a Domain with Two Small Holes

of (10.69). Our aim in this subsection is to show that such a map has a real analytic continuation in a neighborhood of (0, 0). We begin by proving the following. Lemma 10.25. The map from ] − δ1 , δ1 [×] − δ2 , δ2 [ to L(C 1,α (∂Ω1 )0 × C 1,α (∂Ω2 )0 × C 1,α (∂Ω o ) , C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) × C 1,α (∂Ω o )) that takes (1 , 2 ) to M(1 , 2 ) is real analytic. Proof. The analyticity of M follows by definition (10.66)–(10.68), by Corollary A.29 of the Appendix on the analyticity of integral operators with real analytic kernel, and by the continuity of the linear boundary operators WΩ1 , WΩ2 , and WΩ o (see Theorem 4.37).   Then we consider the operator M(0, 0). By the definition of M in (10.66)– (10.68), we have the following. Proposition 10.26. For all (g1 , g2 , g o ) ∈ C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) × C 1,α (∂Ω o ) and all (ψ1 , ψ2 , ψ o ) ∈ C 1,α (∂Ω1 )0 × C 1,α (∂Ω2 )0 × C 1,α (∂Ω o ) we have M(0, 0)[ψ1 , ψ2 , ψ o ] = (g1 , g2 , g o ) if and only if

(10.71)

⎧ 1 o ⎪ ⎨− 2 ψ1 + WΩ1 [ψ1 ] = wΩ o [ψ ](0) − g1 , − 12 ψ2 + WΩ2 [ψ2 ] = wΩ o [ψ o ](0) − g2 , ⎪ ⎩1 o o o o 2 ψ + WΩ [ψ ] = g .

(10.72)

We now show that M(0, 0) is injective but not surjective. Lemma 10.27. The operator M(0, 0) is injective and its image Im M(0, 0) is the closed subspace of C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) × C 1,α (∂Ω o ) consisting of the triples (g1 , g2 , g o ) such that    1,(1) 2,(1) g1 Φ [0, 0] dσ + g2 Φ [0, 0] dσ + g o Φo,(1) [0, 0] dσ = 0 ∂Ω1

∂Ω o

∂Ω2

and 

(10.73)





g1 Φ1,(2) [0, 0] dσ + ∂Ω1

g2 Φ2,(2) [0, 0] dσ + ∂Ω2

g o Φo,(2) [0, 0] dσ = 0 ∂Ω o

(10.74) (cf. Proposition 10.23 for the definition of Φi,(j) [0, 0] and Φo,(j) [0, 0], i, j ∈ {1, 2}). Proof. To prove the lemma, we show that (10.71) has a solution (ψ1 , ψ2 , ψ o ) in C 1,α (∂Ω1 )0 × C 1,α (∂Ω2 )0 × C 1,α (∂Ω o ) if and only if the triple (g1 , g2 , g o ) in C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) × C 1,α (∂Ω o ) satisfies conditions (10.73) and (10.74). Moreover, if it exists, such a solution is unique.

10.4 Moderately Close Holes in Dimension n = 2

405

We note that (10.71) is a system of three equations: M1 (0, 0)[ψ1 , ψ2 , ψ o ] = g1 , M2 (0, 0)[ψ1 , ψ2 , ψ o ] = g2 , and Mo (0, 0)[ψ1 , ψ2 , ψ o ] = g o . We now consider the system consisting of the first two of such equations. The definitions of M1 and M2 in (10.66) and (10.67) imply that such system is equivalent to ! − 12 ψ1 + WΩ1 [ψ1 ] = wΩ o [ψ o ](0) − g1 , (10.75) − 12 ψ2 + WΩ2 [ψ2 ] = wΩ o [ψ o ](0) − g2 (see also Definition 4.27 of double layer potential). Then we denote by J the operator from C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) to C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) defined by   1 1 J ≡ − I + WΩ 1 , − I + WΩ 2 2 2 and we rewrite (10.75) as J[ψ1 , ψ2 ] = (wΩ o [ψ o ](0) − g1 , wΩ o [ψ o ](0) − g2 ) .

(10.76)

The compactness of WΩ1 and WΩ2 of Theorem 6.8 and the Fredholm Alternative of Theorem 5.8 imply that the last equation has a solution in C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) if and only if (wΩ o [ψ o ](0) − g1 , wΩ o [ψ o ](0) − g2 ) is orthogonal to the null space of   1 1 J t ≡ − I + WΩt 1 , − I + WΩt 2 . 2 2 By Lemmas 6.34 and 10.20 and by Proposition 10.23 the null space of J t is spanned by the pairs of functions (Φ1,(1) [0, 0] , Φ2,(1) [0, 0]) and (Φ1,(2) [0, 0] , Φ2,(2) [0, 0]). It follows that (10.76) has a solution if and only if the following two conditions hold,  (wΩ o [ψ o ](0) − g1 ) Φ1,(1) [0, 0] dσ ∂Ω1  (wΩ o [ψ o ](0) − g2 ) Φ2,(1) [0, 0] dσ = 0 + ∂Ω2

and

 ∂Ω1

(wΩ o [ψ o ](0) − g1 ) Φ1,(2) [0, 0] dσ  (wΩ o [ψ o ](0) − g2 ) Φ2,(2) [0, 0] dσ = 0. + ∂Ω2

Since

 ∂Ω h

Φi,(j) [0, 0] dσ = δi,j (cf. (10.63)) these conditions are equivalent to

406

10 A Dirichlet Problem in a Domain with Two Small Holes



 g1 Φ

1,(1)

[0, 0] dσ +

∂Ω1

and

g2 Φ2,(1) [0, 0] dσ = wΩ o [ψ o ](0)

(10.77)

g2 Φ2,(2) [0, 0] dσ = wΩ o [ψ o ](0) .

(10.78)

∂Ω2



 g1 Φ1,(2) [0, 0] dσ + ∂Ω1

∂Ω2

If conditions (10.77) and (10.78) are satisfied, then Theorem 6.24, which characterizes the null spaces of − 12 I + WΩ1 and − 12 I + WΩ2 , implies that (10.76) (and thus (10.75)) has one and only one solution (ψ1 , ψ2 ) in C 1,α (∂Ω1 )0 × C 1,α (∂Ω2 )0 . We now turn to equation Mo (0, 0)[ψ1 , ψ2 , ψ o ] = g o , which is equivalent to the third equation in (10.72). By Corollary 6.15, such equation has a unique solution ψ o ∈ C 1,α (∂Ω o ) for all given g o ∈ C 1,α (∂Ω o ). Moreover, by Theorem 6.5 the operator 12 I + WΩt o is transpose to 12 I + WΩ o . As a consequence, if ψ o is the solution of the third equation in (10.72), then we have     1 o o o,(1) o ψ + WΩ o [ψ ] Φo,(1) [0, 0] dσ g Φ [0, 0] dσ = 2 ∂Ω o ∂Ω o  1 (10.79)  2 1 o t o,(1) I + WΩ o Φ ψ [0, 0] dσ . = 2 ∂Ω o Since Φ(1) [0, 0] is a solution of (10.61) with (1 , 2 ) = (0, 0), Lemma 10.20 implies that   1 I + WΩt o [Φo,(1) [0, 0]] = −(δ1,1 + δ1,2 )νΩ o · (∇S2 )|∂Ω o = −νΩ o · (∇S2 )|∂Ω o 2 and thus the last integral in the right-hand side of (10.79) equals  − ψ o νΩ o · ∇S2 dσ . ∂Ω o

Then, by (10.79) and by the Definition 4.27 of double layer potential we obtain that  g o Φo,(1) [0, 0] dσ = −wΩ o [ψ o ](0) . ∂Ω o

Similarly, we can show that  g o Φo,(2) [0, 0] dσ = −wΩ o [ψ o ](0) . ∂Ω o

Hence, conditions (10.77) and (10.78) are equivalent to (10.73) and (10.74). The lemma is now proved.   Since M(0, 0) is not invertible we cannot exploit the real analyticity of the map that takes an invertible operator to its inverse to prove that the map that takes (1 , 2 ) to the triple in (10.70) is real analytic. Then we resort to Lemma 8.25 and we obtain

10.4 Moderately Close Holes in Dimension n = 2

407

the following Proposition 10.28 where we prove a real analytic continuation result for the map that takes (1 , 2 ) to the triple (ψ11 ,2 , ψ21 ,2 , ψo1 ,2 ). Proposition 10.28. Let δΦ be as in Proposition 10.23. There exist 0 < δΨ ≤ δΦ and a real analytic map Ψ ≡ (Ψ1 , Ψ2 , Ψ o ) : ] − δΨ , δΨ [2 → C 1,α (∂Ω1 )0 × C 1,α (∂Ω2 )0 × C 1,α (∂Ω o ) such that 6 7 M(1 , 2 ) Ψ 1 [1 , 2 ], Ψ 2 [1 , 2 ], Ψ o [1 , 2 ] ⎛    = ⎝f 1 − fi Φi,(1) [1 , 2 ] dσ − i=1,2



 

f2 −

i=1,2

f o Φo,(1) [1 , 2 ] dσ, ∂Ω o

∂Ωi

⎞ f o Φo,(2) [1 , 2 ] dσ, f o ⎠

fi Φi,(2) [1 , 2 ] dσ − ∂Ω o

∂Ωi

(10.80) for all (1 , 2 ) ∈] − δΨ , δΨ [2 (cf. Proposition 10.23 for the definition of Φi,(j) and Φo,(j) , i, j ∈ {1, 2}). In particular, (ψ11 ,2 , ψ21 ,2 , ψo1 ,2 ) = Ψ []

∀(1 , 2 ) ∈]0, δΨ [2 .

(10.81)

Proof. To show the existence of Ψ , we plan to apply Lemma 8.25 with E = R2 , X=C

U =] − δΦ , δΦ [2 ,

1,α

(∂Ω1 )0 × C

1,α

ε = (1 , 2 ) ,

(∂Ω2 )0 × C 1,α (∂Ω o ) ,

Y = C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) × C 1,α (∂Ω o ) , A = M,

Z = R2 ,

with the functions f (of Lemma 8.25) defined by ⎛    fi Φi,(1) [1 , 2 ] dσ − f (1 , 2 ) = ⎝f1 − i=1,2

f2 −



  i=1,2

f o Φo,(1) [1 , 2 ] dσ, ∂Ω o

∂Ωi

⎞

f o Φo,(2) [1 , 2 ] dσ, f o ⎠ ,

fi Φi,(2) [1 , 2 ] dσ − ∂Ω o

∂Ωi

for all (1 , 2 ) ∈] − δΦ , δΦ [2 , and with B = (B1 , B2 ) given by    Bj (1 , 2 )[g1 , g2 , g o ] = gi Φi,(j) [1 , 2 ] dσ + g o Φo,(j) [1 , 2 ] dσ , i=1,2

∂Ωi

∂Ω o

408

10 A Dirichlet Problem in a Domain with Two Small Holes

for all j ∈ {1, 2}, (1 , 2 ) ∈] − δΦ , δΦ [2 , and (g1 , g2 , g o ) ∈ C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) × C 1,α (∂Ω o ). Then we have to verify that the assumptions of Lemma 8.25 are satisfied with this specific choice. As a first step, we observe that our map A is real analytic by Lemma 10.25. Moreover, Proposition 10.23 implies that Φ is real analytic, and accordingly the maps f and B are real analytic. In particular, B is continuous. The first condition (8.59) of Lemma 8.25 concerns the equality of the null space of B(0, 0) and the image of A(0, 0). In our case such equality is a consequence of Lemma 10.27. The second condition (8.60) requires that Bj (1 , 2 ) [A(1 , 2 )[ψ1 , ψ2 , ψ o ] − f (1 , 2 )] = 0

(10.82)

for all j ∈ {1, 2}, (1 , 2 ) ∈] − δΦ , δΦ [2 and (ψ1 , ψ2 , ψ o ) ∈ C 1,α (∂Ω1 )0 × C 1,α (∂Ω2 )0 × C 1,α (∂Ω o ). To prove that (10.82) holds true, we remember equality (10.63) and we deduce by a straightforward computation that Bj (1 , 2 )[f (1 , 2 )] = 0

∀j ∈ {1, 2} , (1 , 2 ) ∈] − δΦ , δΦ [2 .

(10.83)

Moreover, Bj (1 , 2 )[A(1 , 2 )[ψ1 , ψ2 , ψ o ]]   = Mi (1 , 2 )[ψ1 , ψ2 , ψ o ] Φi,(j) [1 , 2 ] dσ i=1,2



∂Ωi

Mo (1 , 2 )[ψ1 , ψ2 , ψ o ] Φo,(j) [1 , 2 ] dσ

+ ∂Ω o

for all j ∈ {1, 2} and (1 , 2 ) ∈] − δΦ , δΦ [2 . Then, changing the variable of integration in the integrals over ∂Ω1 and ∂Ω2 to obtain integrals over the (1 , 2 )dependent boundaries ∂Ω1 (1 , 2 ) and ∂Ω2 (1 , 2 ) and taking into account definition (4.71) of double layer potential, we can verify that Mi (1 , 2 )[ψ1 , ψ2 , ψ o ](ξ) 1 = μ1 ,2 (1 pi + 1 2 ξ) + WΩ(1 ,2 ) [μ1 ,2 ](1 pi + 1 2 ξ) , 2 for all i ∈ {1, 2} and ξ ∈ ∂Ωi , and Mo (1 , 2 )[ψ1 , ψ2 , ψ o ](x) =

1 μ , (x) + WΩ(1 ,2 ) [μ1 ,2 ](x) 2 1 2

with μ1 ,2 ∈ C 1,α (∂Ω(1 , 2 )) which satisfies the conditions μ1 ,2 (1 pi + 1 2 ξ) = ψi (ξ)

∀i ∈ {1, 2} , ξ ∈ ∂Ωi

and μ1 ,2 (x) = ψ o (x)

∀x ∈ ∂Ω o

∀x ∈ ∂Ω o

10.4 Moderately Close Holes in Dimension n = 2

409

(here we have to follow backward the calculation carried out at the beginning of this section to introduce the operator M(1 , 2 )). By equalities (10.44) and (10.62), we have Φ1,(j) (ξ) = 1 2 τ(j) (1 p1 + 1 2 ξ) ∀ξ ∈ ∂Ω1 , 1 ,2 1 ,2 Φ2,(j) (ξ) = 1 2 τ(j) (1 p2 + 1 2 ξ) 1 ,2 1 ,2

∀ξ ∈ ∂Ω2 ,

Φo,(j) (x) 1 ,2

∀x ∈ ∂Ω o ,

=

τ(j) (x) 1 ,2 (j)

for all j ∈ {1, 2} and (1 , 2 ) ∈]0, δΦ [2 , where τ1 ,2 ∈ C 0,α (∂Ω(1 , 2 )) is the unique solution of system (10.39). It follows that Bj (1 , 2 )[A(1 , 2 )[ψ1 , ψ2 , ψ o ]]     1 μ1 ,2 (1 pi + 1 2 ξ) + WΩ(1 ,2 ) [μ1 ,2 ](1 pi + 1 2 ξ) 1 2 = 2 ∂Ωi i=1,2 

 + ∂Ω o



(1 pi + 1 2 ξ) dσ × τ(j) 1 ,2

1 μ , (x) + WΩ(1 ,2 ) [μ1 ,2 ](x) τ(j) (x) dσ 1 ,2 2 1 2

∀j ∈ {1, 2}.

Then, changing the variable of integration in the integrals over ∂Ω1 and ∂Ω2 , we verify that Bj (1 , 2 )[A(1 , 2 )[ψ1 , ψ2 , ψ o ]]    1 μ1 ,2 + WΩ(1 ,2 ) [μ1 ,2 ] τ(j) dσ = 1 ,2 2 ∂Ω(1 ,2 ) for all j ∈ {1, 2} and (1 , 2 ) ∈]0, δΦ [2 . Since the operator transpose to 12 I + WΩ(1 ,2 ) (cf. Theorem 6.5), we deduce that

1 2I

t + WΩ( is 1 ,2 )

Bj (1 , 2 )[A(1 , 2 )[ψ1 , ψ2 , ψ o ]]  1  2 1 t dσ I + WΩ(1 ,2 ) τ(j) μ1 ,2 = , 1 2 2 ∂Ω(1 ,2 ) for all j ∈ {1, 2} and (1 , 2 ) ∈]0, δΦ [2 . Hence, we have Bj (1 , 2 )[A(1 , 2 )[ψ1 , ψ2 , ψ o ]] = 0

∀j ∈ {1, 2} , (1 , 2 ) ∈]0, δΦ [2

by the first equation of (10.39). We now observe that the function from ] − δΦ , δΦ [2 to R that takes (1 , 2 ) to Bj (1 , 2 )[A(1 , 2 )[ψ i , ψ o ]] is real analytic. Then the Identity Principle for real analytic functions of Proposition 2.16 implies that Bj (1 , 2 )[A(1 , 2 )[ψ1 , ψ2 , ψ o ]] = 0

∀j ∈ {1, 2} , (1 , 2 ) ∈] − δΦ , δΦ [2 . (10.84) Now (10.82) follows by (10.83) and (10.84). Finally, to verify that all the conditions of Lemma 8.25 are verified we have to show that Im A(0, 0) has a closed direct

410

10 A Dirichlet Problem in a Domain with Two Small Holes

complement in Y . To do so, we observe that B(0, 0) is a bounded operator from Y to R2 and we recall that Im A(0, 0) = Ker B(0, 0). It follows that Im A(0, 0) has finite positive co-dimension in Y . Accordingly, it has a closed direct complement of finite dimension in Y (see Corollary 2.7). Now that the assumptions of Lemma 8.25 are verified, we can deduce that there exist δΨ ∈]0, δΦ ] and a real analytic map Ψ from ] − δΨ , δΨ [2 to C 1,α (∂Ω1 ) × C 1,α (∂Ω2 ) × C 1,α (∂Ω o ) that satisfies (10.80). Then (10.81) follows by Proposition 10.24.  

ξ ξ 10.4.4 The Auxiliary Functions HΩ , HΩ , and HxΩ 1 2

o

In this subsection we describe the relation between the limiting densities Φi,(j) [0, 0] ξ ξ and Φo,(j) [0, 0], with i, j ∈ {1, 2}, and certain auxiliary functions HΩ , HΩ , and 1 2 o Ω Hx , which will be be introduced below (see also Lemmas 8.48 and 8.49). As we shall see, such functions play a role in the description of the limit behavior of u1 ,2 . For the functions Φi,(j) [0, 0] we have the following. ξ be the only Lemma 10.29. Let i, j ∈ {1, 2} and ξ ∈ R2 \ ∂Ωi be fixed. Let HΩ i 1,α 2 solution in Cloc (R \ Ωi ) of the Dirichlet boundary value problem ⎧ ξ ⎪ in R2 \ Ωi , ⎨ΔHΩi = 0 ξ for all η ∈ ∂Ωi , HΩi (η) = S2 (ξ − η) ⎪ ⎩ ξ HΩi is harmonic at infinity.

Then we have ξ (η) . vΩi [Φi,(j) [0, 0]](ξ) = δi,j lim HΩ i η→∞

(10.85)

+ [Φi,(j) [0, 0]] = vΩi [Φi,(j) [0, 0]]|Ωi is constant on Ωi Moreover, the restriction vΩ i and we have + 0 [Φi,(j) [0, 0]] = δi,j lim HΩ (η) on Ωi . (10.86) vΩ i i η→∞

Proof. The reader may observe that the proof (and the statement) is very similar to 1,α that of Lemma 8.48. If u ∈ Cloc (R2 \ Ωi ) is harmonic in R2 \ Ωi and at infinity, then Proposition 6.39 implies that there exists ψ ∈ C 1,α (∂Ωi )0 such that − u = wΩ [ψ] + lim u(η) . i η→∞

− Then, by the jump formula for wΩ [ψ] of Theorem 6.6 we have i

  1 u|∂Ωi = − I + WΩi [ψ] + lim u(η) . η→∞ 2

10.4 Moderately Close Holes in Dimension n = 2

411

Since − 12 I + WΩi and − 12 I + WΩt i are transpose to one another (cf. Theorem 6.5) we deduce that  u|∂Ωi Φi,(j) [0, 0] dσ ∂Ωi    1  − ψ + WΩi [ψ] Φi,(j) [0, 0] dσ + lim u(η) Φi,(j) [0, 0] dσ = η→∞ 2 ∂Ωi ∂Ωi   1  ψ − Φi,(j) [0, 0] + WΩt i [Φi,(j) [0, 0]] dσ = 2 ∂Ωi  Φi,(j) [0, 0] dσ . (10.87) + lim u(η) η→∞

∂Ωi

By equality (10.61) the densities Φ1,(j) [0, 0] and Φ2,(j) [0, 0] solve the first, second, fourth, and fifth equation of (10.59) with g1 = 0, g2 = 0, ρ1 = δ1,j , and ρ2 = δ2,j . Accordingly we have  1 − Φi,(j) [0, 0] + WΩt i [Φi,(j) [0, 0]] = 0 and Φi,(j) [0, 0] dσ = δi,j . 2 ∂Ωi (10.88) By (10.87) it follows that  u|∂Ωi Φi,(j) [0, 0] dσ = δi,j lim u(η) . (10.89) η→∞

∂Ωi

ξ Thus, by Definition 4.18 of single layer potential and by taking u = HΩ in (10.89), i we have  i,(j) vΩi [Φ [0, 0]](ξ) = S2 (ξ − η)Φi,(j) [0, 0](η) dση ∂Ωi  ξ ξ = HΩ (η)Φi,(j) [0, 0](η) dση = δi,j lim HΩ (η) . i i η→∞

∂Ωi

Moreover, by the first equation of (10.88) and by the jump formula of the normal derivative of the single layer potential of Theorem 6.6 we have + [Φi,(j) [0, 0]]|∂Ωi = 0 . νΩi · ∇vΩ i

Then the uniqueness result for the Neumann boundary value problem of Theo+ rem 6.2 (i) imply that vΩ [Φi,(j) [0, 0]] is constant in Ωi . Then the validity of last i the statement of the lemma follows.   As observed for (8.146), the function ξ GΩi (ξ, η) ≡ S2 (ξ − η) − HΩ (η) i

∀ξ, η ∈ R2 \ Ωi , ξ = η

is the Dirichlet Green’s function for the exterior domain R2 \ Ωi .

(10.90)

412

10 A Dirichlet Problem in a Domain with Two Small Holes

In the following Lemma 10.30 we study the limiting value of Φo,(j) . Instead of confining ourselves to Φo,(j) [0, 0], we consider also Φo,(j) [0, 2 ] with 2 ∈]−δΦ , δΦ [. In this way we obtain an information that will be useful to analyze the asymptotic behavior of u1 ,2 in Section 10.4.6. o

Lemma 10.30. Let x ∈ Ω o be fixed. Let HxΩ be the unique solution in C 1,α (Ω o ) of the Dirichlet problem ! o ΔHxΩ = 0 in Ω o , o HxΩ (y) = S2 (x − y) for all y ∈ ∂Ω o . Then we have

o

vΩ o [Φo,(j) [0, 2 ]](x) = −HxΩ (0) for j ∈ {1, 2} and for all 2 ∈] − δΦ , δΦ [. Proof. The proof (and the statement) is very similar to that of Lemma 8.49. If u ∈ C 1,α (Ω o ) and Δu = 0 in Ω o , then Corollary 6.15 implies that there exists μ ∈ C 1,α (∂Ω o ) such that u|∂Ω o = ( 12 I + WΩ o )[μ]. Moreover, by the uniqueness of the solution of the Dirichlet problem of Theorem 3.6 and by the jump formulas for the + o double layer potential of Theorem 6.6 we have u = wΩ o [μ] in Ω . Let j ∈ {1, 2} 1 1 t be fixed. Since 2 I + WΩ o and 2 I + WΩ o are transpose operators (cf. Theorem 6.5), we have   1  μ + WΩ o [μ] Φo,(j) [0, 2 ] dσ u|∂Ω o Φo,(j) [0, 2 ] dσ = o 2 ∂Ω o ∂Ω 1  μ Φo,(j) [0, 2 ] + WΩt o [Φo,(j) [0, 2 ]] dσ = 2 ∂Ω o (10.91) for all 2 ∈] − δΦ , δΦ [. By equality (10.61) and by the definition of N o,(j) in (10.51) we have 1 o,(j) Φ [0, 2 ] + WΩt o [Φo,(j) [0, 2 ]] = − (δ1,j + δ2,j ) νΩ o · (∇S)|∂Ω o 2 = −νΩ o · (∇S)|∂Ω o for all 2 ∈] − δΦ , δΦ [. Then (10.91), Definition 4.27 of double layer potential, and the third Green Identity of Theorem 6.9 imply that  u|∂Ω o Φo,(j) [0, 2 ] dσ ∂Ω o  (10.92) + μ(y) νΩ o (y) · ∇S2 (y) dσy = −wΩ =− o [μ](0) = −u(0) . ∂Ω o

o

Thus, by Definition 4.18 of single layer potential and by taking u = HxΩ , we have

10.4 Moderately Close Holes in Dimension n = 2

413



S2 (x − y)Φo,(j) [0, 2 ](y) dσy

vΩ o [Φo,(j) [0, 2 ]](x) = 

∂Ω o o

= ∂Ω o

o

Ω o,(j) Hx|∂Ω [0, 2 ] dσ = −HxΩ (0) . o Φ

  As observed for (8.148), the function o

o

GΩ (x, y) ≡ S2 (x − y) − HxΩ (y)

∀x, y ∈ Ω o , x = y

(10.93)

is the Dirichlet Green’s function for the domain Ω o .

10.4.5 Representation of u1 ,2 in Terms of Analytic Maps In Theorem 10.35 below we write the maps that takes (1 , 2 ) to suitable restrictions of u1 ,2 and of the rescaled function u1 ,2 (1 pj + 1 2 ·) in terms of real analytic maps of 1 and 2 and of elementary functions of log 1 and log(1 2 ). To do so, we exploit the representation formula (10.38), the real analyticity results of Propositions 10.23 and 10.28, and the auxiliary functions of Section 10.4.4. In the following Propositions 10.31–10.34 we consider one by one the terms that appear in the representation formula (10.38) and we introduce certain auxiliary maps U [1 , 2 ] and V [1 , 2 ], a vector valued functions F(1 , 2 ), and matrix valued functions R(1 , 2 ) and Λ(1 , 2 ) that we then use in the representation Theorem 10.35. In what follows, we set (10.94) δ∗ ≡ δΨ Then, by Lemma 10.22 and Proposition 10.28 we have δ∗ = δΨ ≤ δΦ < min{δ1 , δ2 } .

(10.95)

+ In Proposition 10.31 we consider the term wΩ( [μ1 ,2 ] that appears in (10.38). 1 ,2 )

Proposition 10.31. For each (1 , 2 ) ∈] − δ∗ , δ∗ [2 there exists and is unique a function U [1 , 2 ] ∈ C 1,α (Ω(1 , 2 )) such that + o U [1 , 2 ](x) = wΩ o [Ψ [1 , 2 ]](x) 2   νΩk (η) · ∇S2 (x − 1 pk − 1 2 η)Ψk [1 , 2 ](η) dση + 1 2 k=1

(10.96)

∂Ωk

for all x ∈ Ω o \ Ω1 (1 , 2 ) ∪ Ω2 (1 , 2 ). Moreover, the following statements hold. (i) For all (1 , 2 ) ∈]0, δ∗ [2 we have + U [1 , 2 ] = wΩ( [μ1 ,2 ] , 1 ,2 )

414

10 A Dirichlet Problem in a Domain with Two Small Holes

where μ1 ,2 ∈ C 1,α (∂Ω(1 , 2 )) is the solution of (10.40). / ΩM . Let δM ∈]0, δ∗ ] be such (ii) Let ΩM be an open subset of Ω o such that 0 ∈ that ΩM ∩ Ωk (1 , 2 ) = ∅ for all (1 , 2 ) ∈] − δM , δM [×] − δ∗ , δ∗ [ and for all k ∈ {1, 2}. Then there exists a real analytic map UM : ] − δM , δM [×] − δ∗ , δ∗ [→ C 1,α (ΩM ) such that U [1 , 2 ]|ΩM = uo|ΩM + 1 2 UM [1 , 2 ]

(10.97)

for all (1 , 2 ) ∈] − δM , δM [×] − δ∗ , δ∗ [, where uo ∈ C 1,α (Ω o ) is the unique solution of  Δuo = 0 in Ω o , (10.98) uo = f o on ∂Ω o . (j)

(iii) Let j ∈ {1, 2}. Let Ωm be an open bounded subset of R2 \ Ωj . Let δm ∈]0, δ∗ ] be such that  1 pj +  1  2 Ω m ⊆ Ω o and (1 pj + 1 2 Ωm ) ∩ Ωk (1 , 2 ) = ∅ (j)

(j)

for all (1 , 2 ) ∈] − δm , δm [2 , where k = 1 if j = 2 and k = 2 if j = 1. Then there exists a real analytic map (j) (j) (j) 2 Um : ] − δm , δm [ → C 1,α (Ωm )

such that (j) U [1 , 2 ](1 pj + 1 2 ξ) = Um [1 , 2 ](ξ)

for all ξ ∈ Ωm and (1 , 2 ) ∈ (] −

(j) (j) δm , δm [\{0})2 .

(j) Um [0, 0](ξ) = uo (0) + uj (ξ) − lim uj (η) η→∞

(10.99)

Moreover, ∀ξ ∈ Ωm ,

1,α where uj ∈ Cloc (R2 \ Ωj ) is the solution of ⎧ in R2 \ Ωj , ⎨ Δuj = 0 on ∂Ωj , u = fj ⎩ j uj is harmonic at infinity.

(10.100)

(10.101)

Proof. We observe that we cannot use the expression on the right-hand side of (10.96) to define U [1 , 2 ] on the whole of Ω(1 , 2 ). Indeed, the integrals over ∂Ω1 and ∂Ω2 might have a singular kernel for x on ∂Ω1 (1 , 2 ) or ∂Ω2 (1 , 2 ). Then we observe that, by equality (10.81) of (ψ11 ,2 , ψ21 ,2 , ψo1 ,2 ) and Ψ [1 , 2 ], by equality (10.64) of (ψ11 ,2 , ψ21 ,2 , ψo1 ,2 ) and μ1 ,2 , and by a computation based on the rule of change of variable in integrals, the function on the right+ [μ1 ,2 ] on hand side of (10.96) coincides with the double layer potential wΩ( 1 ,2 )

10.4 Moderately Close Holes in Dimension n = 2

415

Ω o \ Ω1 (1 , 2 ) ∪ Ω2 (1 , 2 ) (see also Definition 4.27 of double layer potential). + [μ1 ,2 ] ∈ C 1,α (Ω(1 , 2 )) (cf. Theorem 4.31), it follows that the Since wΩ( 1 ,2 ) the function on the right-hand side of (10.96) has a unique continuous extension to Ω(1 , 2 ). Accordingly, there is a unique function U [1 , 2 ] ∈ C 1,α (Ω(1 , 2 )) + [μ1 ,2 ] on for which (10.96) holds true. Moreover, we have U [1 , 2 ] = wΩ( 1 ,2 ) Ω(1 , 2 ) and statement (i) is verified. We now prove statement (ii). By equality (10.80) and by the definition (10.68) of Mo , we have 1 ( I + WΩ o )[Ψ o [1 , 2 ]] = f o 2

∀(1 , 2 ) ∈] − δ∗ , δ∗ [2 such that 1 2 = 0.

Corollary 6.15 implies that equation 1 ( I + WΩ o )[μo ] = f o 2

(10.102)

has a unique solution μo ∈ C 1,α (∂Ω o ). Accordingly, we have Ψ o [1 , 2 ] = μo for all (1 , 2 ) ∈] − δ∗ , δ∗ [2 with 1 2 = 0.

(10.103)

Since the map that takes (1 , 2 ) to Ψ o [1 , 2 ] is real analytic from ] − δ∗ , δ∗ [ o ∞ }i,j=0 ⊆ to C 1,α (∂Ω o ), there exist δ∗∗ ∈]0, δ∗ ] and a family of functions {Ψi,j 1,α o C (∂Ω ) such that Ψ o [1 , 2 ] =

∞ 

o i1 j2 Ψi,j

∀(1 , 2 ) ∈] − δ∗∗ , δ∗∗ [2 ,

i,j=0

where the series converges normally in C 1,α (∂Ω o ). Then (10.103) implies that o o o = μo and that Ψi,0 = Ψ0,j = 0 for all i, j ∈ N \ {0}. Accordingly we have Ψ0,0 ∞ 

Ψ o [1 , 2 ] = μo + 1 2

o i1 j2 Ψi+1,j+1

∀(1 , 2 ) ∈] − δ∗∗ , δ∗∗ [2 .

i,j=0

Now let Ψ˜ o be the map from ] − δ∗ , δ∗ [2 to C 1,α (∂Ω o ) defined by Ψ˜ o [1 , 2 ] =

∞ 

o i1 j2 Ψi+1,j+1

∀(1 , 2 ) ∈] − δ∗∗ , δ∗∗ [2

i,j=0

and Ψ o [1 , 2 ] − μo Ψ˜ o [1 , 2 ] = 1 2

∀(1 , 2 ) ∈] − δ∗ , δ∗ [2 \] − δ∗∗ , δ∗∗ [2 .

We can verify that Ψ˜ o is real analytic and we have

416

10 A Dirichlet Problem in a Domain with Two Small Holes

Ψ o [1 , 2 ] = μo + 1 2 Ψ˜ o [1 , 2 ]

∀(1 , 2 ) ∈] − δ∗ , δ∗ [2 .

Moreover, by equality (10.102), by the jump formula for the double layer potential of Theorem 6.6 and by the uniqueness of the solution of the Dirichlet problem + o o (cf. Theorem 3.6) we have wΩ o [μ ] = u . It follows that + + ˜o o o wΩ o [Ψ [1 , 2 ]] = u + 1 2 wΩ o [Ψ [1 , 2 ]]

∀(1 , 2 ) ∈] − δ∗ , δ∗ [2 . (10.104)

Then we define + ˜o UM [1 , 2 ](x) ≡ wΩ o [Ψ [1 , 2 ]](x) 2   νΩk (η) · ∇S2 (x − 1 pk − 1 2 η)Ψk [1 , 2 ](η) dση + k=1

∂Ωk

(10.105) for all x ∈ ΩM and for all (1 , 2 ) ∈] − δM , δM [×] − δ∗ , δ∗ [. The validity of (10.97) follows by (10.96), (10.104), and (10.105). In addition, Theorem A.28 (i) on the properties of integral operators with a real analytic kernel and the mapping properties of the double layer potential of Theorem 4.31 (i) imply that the map from ]−δM , δM [×]−δ∗ , δ∗ [ to C 1,α (ΩM ) that takes (1 , 2 ) to UM [1 , 2 ] is real analytic. Finally, we prove statement (iii). We take k ≡ 1 if j = 2 and k ≡ 2 if j = 1. Then we set − (j) [1 , 2 ](ξ) ≡ wΩ o [Ψ o [1 , 2 ]](1 pj + 1 2 ξ) − wΩ [Ψj [1 , 2 ]](ξ) Um j  + 2 νΩk (η) · ∇S2 (pj − pk + 2 (ξ − η))Ψk [1 , 2 ](η) dση ∂Ωk (j)

(j)

for all ξ ∈ Ωm and (1 , 2 ) ∈] − δm , δm [2 . Then, Theorem A.28 (i) on the properties of integral operators with a real analytic kernel and the mapping properties of the double layer potential of Theorem 4.31 (i) and (ii) imply that the map that takes (j) (j) (j) (1 , 2 ) to Um [1 , 2 ] is real analytic from ] − δm , δm [2 to C 1,α (Ωm ). The validity of equality (10.99) can be deduced by a straightforward computation based on the rule of change of variables in integrals. We now verify (10.100). A straightforward computation shows that − (j) [0, 0](ξ) = wΩ o [Ψ o [0, 0]](0) − wΩ [Ψj [0, 0]](ξ) Um j

∀ξ ∈ Ωm .

(10.106)

Equality (10.104) with (1 , 2 ) = (0, 0) implies that + o o wΩ o [Ψ [0, 0]](0) = u (0) .

(10.107)

Moreover, by equality (10.80) with (1 , 2 ) = (0, 0) and by Proposition 10.26 we have

10.4 Moderately Close Holes in Dimension n = 2

417

1 − Ψj [0, 0] + WΩj [Ψj [0, 0]] = wΩ o [Ψ o [0, 0]](0) 2  2   fi Φi,(j) [0, 0] dσ + − fj + i=1

f o Φo,(j) [0, 0] dσ .

∂Ω o

∂Ωi

Now, equality (10.89) and the definition (10.101) of ui imply that   fi Φi,(j) [0, 0] dσ = ui|∂Ωi Φi,(j) [0, 0] dσ = δi,j lim ui (η) , ∂Ωi

η→∞

∂Ωi

for all i ∈ {1, 2}, and equality (10.92) and system (10.98) for uo imply that   f o Φo,(j) [0, 0] dσ = uo|∂Ω o Φo,(j) [0, 0] dσ = −uo (0) . ∂Ω o

∂Ω o

Hence,  2   1 o − Ψj [0, 0] + WΩj [Ψj [0, 0]] = u (0) − fj + δi,j lim ui (η) − uo (0) η→∞ 2 i=1 = −fj + lim uj (η) . η→∞

(see also (10.107)). Then, the jump formula for the double layer potential of Theorem 6.6 implies that − −wΩ [Ψj [0, 0]]|∂Ωj = fj − lim uj (η) j η→∞

and by the uniqueness of the solution of the exterior Dirichlet problem of Proposition 6.1 we deduce that − − wΩ [Ψj [0, 0]] = uj − lim uj (η) j η→∞

on R2 \ Ωj .

(10.108)  

Now, equality (10.100) follows by (10.106), (10.107), and (10.108). (j)

+ In the the following Proposition 10.32 we consider the factors vΩ( [τ1 ,2 ] with 1 ,2 ) j ∈ {1, 2} that appear in the representation formula (10.38).

Proposition 10.32. For all (1 , 2 ) ∈] − δ∗ , δ∗ [2 we denote by V [1 , 2 ] ≡ (V1 [1 , 2 ], V2 [1 , 2 ]) the element of C 1,α (Ω(1 , 2 ))2 defined by Vj [1 , 2 ](x) ≡ vΩ o [Φ

o,(j)

[1 , 2 ]](x) +

2   i=1

S2 (x − 1 pi − 1 2 η) Φi,(j) [1 , 2 ](η) dση ∂Ωi

418

10 A Dirichlet Problem in a Domain with Two Small Holes

for all x ∈ Ω(1 , 2 ) and j ∈ {1, 2}. Then the following statements hold. (i) For all (1 , 2 ) ∈]0, δ∗ [2 and j ∈ {1, 2} we have + Vj [1 , 2 ] = vΩ( [τ(j) ], 1 ,2 1 ,2 ) (j)

where τ1 ,2 ∈ C 0,α (∂Ω(1 , 2 )) is the solution of (10.39). (ii) Let ΩM be an open subset of Ω o such that 0 ∈ / ΩM . Let δM ∈]0, δ∗ ] be such that ΩM ∩ Ωi (1 , 2 ) = ∅ for all (1 , 2 ) ∈] − δM , δM [×] − δ∗ , δ∗ [ and for all i ∈ {1, 2}. Then there exists a real analytic map VM ≡ (VM,1 , VM,2 ) : ] − δM , δM [×] − δ∗ , δ∗ [→ C 1,α (ΩM )2 such that V [1 , 2 ]|ΩM = VM [1 , 2 ]

∀(1 , 2 ) ∈] − δM , δM [×] − δ∗ , δ∗ [ .

Moreover, for all j ∈ {1, 2} and all 2 ∈] − δ∗ , δ∗ [ we have o

VM,j [0, 2 ](x) = GΩ (x, 0)

∀x ∈ ΩM ,

(10.109)

o

where GΩ (x, y) is the Dirichlet Green function of Ω o (cf. (10.93)). (h) (iii) Let h ∈ {1, 2}. Let Ωm be an open bounded subset of R2 \ Ωi . Let δm ∈]0, δ∗ ] be such that  1 ph +  1  2 Ω m ⊆ Ω o and (1 ph + 1 2 Ωm ) ∩ Ωk (1 , 2 ) = ∅ (h)

(h)

for all (1 , 2 ) ∈] − δm , δm [2 , where k ≡ 1 if h = 2 and k ≡ 2 if h = 1. Then there exists a real analytic map (h)

(h)

(h) (h) (h) 2 ≡ (Vm,1 , Vm,2 ) : ] − δm , δm [ → C 1,α (Ωm )2 Vm

such that Vj [1 , 2 ](1 ph + 1 2 ξ) (h)

= Vm,j [1 , 2 ](ξ) + δh,j

1 1 log |1 2 | + (1 − δh,j ) log |1 | 2π 2π

∀ξ ∈ Ωm (10.110)

 2 (h) (h) for all (1 , 2 ) ∈ ] − δm , δm [\{0} and j ∈ {1, 2}. Moreover, (h)

o

ξ Vm,j [0, 0](ξ) = −H0Ω (0)+δh,j lim HΩ (η)+(1−δh,j )S2 (ph −pk ) (10.111) i η→∞

for all ξ ∈ Ωm and j ∈ {1, 2}, where k ≡ 1 if h = 2 and k ≡ 2 if h = 1.

10.4 Moderately Close Holes in Dimension n = 2

419 1,(j)

2,(j)

o,(j)

Proof. Statement (i) can be deduced by equality (10.62) of (φ1 ,2 , φ1 ,2 , φ1 ,2 ) and (j) 1,(j) 2,(j) o,(j) Φ(j) [1 , 2 ], by equality (10.44) of τ1 ,2 and (φ1 ,2 , φ1 ,2 , φ1 ,2 ), and by a calculation based on the rule of change of variable in integrals (see also Definition 4.18 of single layer potential). To prove statement (ii) we set VM [1 , 2 ] ≡ V [1 , 2 ]|ΩM

∀(1 , 2 ) ∈] − δM , δM [×] − δ2 , δ2 [ .

Then, the real analyticity of VM follows by Theorem A.28 (i) on the properties of the integral operators with a real analytic kernel and on the mapping properties of the single layer potential of Theorem 4.25. To prove (10.109) we observe that VM,j [0, 2 ](x) = vΩ o [Φo,(j) [0, 2 ]](x)+S2 (x)

2   i=1

Φi,(j) [0, 2 ] dσ

for all 2 ∈] − δ∗ , δ∗ [. Then Lemma 10.30 and equality (cf. (10.63)) imply that o

VM,j [0, 2 ](x) = −HxΩ (0) + S2 (x)

∀x ∈ ΩM

∂Ωi

 ∂Ωi

Φi,(j) [0, 2 ] dσ = δi,j

∀x ∈ ΩM , 2 ∈] − δ∗ , δ∗ [ . o

Hence, the validity of (10.109) follows by the definition of GΩ in (10.93). Finally, we consider statement (iii). We set k ≡ 1 if h = 2 and k ≡ 2 if h = 1. Then we observe that Vj [1 , 2 ](1 ph + 1 2 ξ) = vΩ o [Φo,(j) [1 , 2 ]](1 ph + 1 2 ξ)  + S2 (1 2 (ξ − η)) Φh,(j) [1 , 2 ](η) dση ∂Ωh  + S2 (1 (ph − pk ) + 1 2 (ξ − η)) Φh,(j) [1 , 2 ](η) dση ∂Ωk (h)

(h)

for all ξ ∈ Ωm and (1 , 2 ) ∈] − δm , δm [2 . Moreover, by Definition 3.2 of S2 we have 1 log |1 2 | + S2 (ξ − η) (10.112) S2 (1 2 (ξ − η)) = 2π for all ξ, η ∈ R2 , ξ = η, and (1 , 2 ) ∈ (R \ {0})2 , and S2 (1 (ph − pk ) + 1 2 (ξ − η)) =

1 log |1 | + S2 (ph − pk + 2 (ξ − η)) (10.113) 2π

for all ξ, η ∈ R2 and (1 , 2 ) ∈ (R \ {0})2 such that ph − pk + 2 (ξ − η) = 0. It follows that

420

10 A Dirichlet Problem in a Domain with Two Small Holes

Vj [1 , 2 ](1 ph + 1 2 ξ) = vΩ o [Φo,(j) [1 , 2 ]](1 ph + 1 2 ξ)  1 log |1 2 | + vΩh [Φh,(j) [1 , 2 ]](ξ) + Φh,(j) [1 , 2 ](η) dση 2π ∂Ωh  h k k,(j) + S2 (p − p + 2 (ξ − η)) Φ [1 , 2 ](η) dση ∂Ωk  1 log |1 | + Φk,(j) [1 , 2 ](η) dση 2π ∂Ωk (h)

(h)

for all ξ ∈ Ωm and (1 , 2) ∈] − δm , δm [2 (see also Definition 4.18 of single layer potential). Then equality ∂Ωh Φh,(j) [0, 0] dσ = δh,j (cf. (10.63)) implies that Vj [1 , 2 ](1 ph + 1 2 ξ) = vΩ o [Φo,(j) [1 , 2 ]](1 ph + 1 2 ξ) + vΩh [Φh,(j) [1 , 2 ]](ξ)  + S2 (ph − pk + 2 (ξ − η)) Φk,(j) [1 , 2 ](η) dση

(10.114)

∂Ωk

+ δh,j

1 1 log |1 2 | + δj,k log |1 | 2π 2π (h)

(h)

for all ξ ∈ Ωm and (1 , 2 ) ∈] − δm , δm [2 . We now set (h)

Vm,j [1 , 2 ](ξ) ≡ vΩ o [Φo,(j) [1 , 2 ]](1 ph + 1 2 ξ) + vΩh [Φh,(j) [1 , 2 ]](ξ)  + S2 (ph − pk − 2 (ξ − η))Φk,(j) [1 , 2 ](η) dση ∂Ωk (h)

(h)

for all ξ ∈ Ωm and (1 , 2 ) ∈] − δm , δm [2 . Then (10.110) follows by (10.114). Moreover, by Theorem A.28 (i) on the properties of the integral operators with a real analytic kernel and by the mapping properties of the single layer potential of (h) (h) Theorem 4.25 we can verify that the map from ] − δm , δm [2 to C 1,α (Ωm )2 that (h) (h) (h) takes (1 , 2 ) to Vm [1 , 2 ] ≡ (Vm,1 [1 , 2 ], Vm,2 [1 , 2 ]) is real analytic. To prove (10.111) we observe that (h)

Vm,j [0, 0](ξ) = vΩ o [Φo,(j) [0, 0]](0) + vΩh [Φh,(j) [0, 0]](ξ)  + S2 (ph − pk ) Φk,(j) [0, 0](η) dση ∂Ωk

for all ξ ∈ Ω  m . Then (10.111) follows by Lemma 10.30, by equality (10.85), and   by equality ∂Ωk Φk,(j) [0, 0] dσ = δk,j (cf. (10.63)).  (i) In the following Proposition 10.33 we consider the term ∂Ω(1 ,2 ) f1 ,2 τ1 ,2 dσ of (10.38). Proposition 10.33. Let F ≡ (F1 , F2 ) be the function from ] − δ∗ , δ∗ [2 to R2 defined by

10.4 Moderately Close Holes in Dimension n = 2

421

 Fj (1 , 2 ) ≡

f o Φo,(j) [1 , 2 ] dσ +

2  

∂Ω o

fi Φi,(j) [1 , 2 ] dσ ∂Ωi

i=1

for all j ∈ {1, 2} and (1 , 2 ) ∈] − δ∗ , δ∗ [2 . Then we have  Fj (1 , 2 ) = f1 ,2 τ(j) dσ 1 ,2

(10.115)

∂Ω(1 ,2 )

(j)

for all (1 , 2 ) ∈]0, δ∗ [2 and i ∈ {1, 2}, where τ1 ,2 ∈ C 0,α (∂Ω(1 , 2 )) is the solution of (10.39). Moreover, F is real analytic and Fj [0, 0] = −uo (0) + lim uj (η)

∀j ∈ {1, 2} ,

η→∞

(10.116)

where uo is the unique solution of (10.98) and uj with j ∈ {1, 2} is the unique solution of (10.101). Proof. Equality (10.115) can be deduced by a straightforward computation and using 1,(j) 2,(j) o,(j) (j) equality (10.62) of (φ1 ,2 , φ1 ,2 , φ1 ,2 ) and Φ(j) [1 , 2 ], equality (10.44) of τ1 ,2 1,(j) 2,(j) o,(j) and (φ1 ,2 , φ1 ,2 , φ1 ,2 ), and the rule of change of variable in integrals. The real analyticity of F is a consequence of the real analyticity of Φ(j) of Proposition 10.23 and of the continuity of the linear map that takes a function of C 0,α (∂Ω o ) to its integral over ∂Ω o and of the linear map that takes a function of C 0,α (∂Ωj ) to its integral over ∂Ωj . To prove (10.116) we observe that  Fj (0, 0) =

f o Φo,(j) [0, 0] dσ + ∂Ω o

i=1

 = ∂Ω o

2  

uo|∂Ω o

Φ

o,(j)

[0, 0] dσ +

fi Φi,(j) [0, 0] dσ ∂Ωi

2   i=1

ui|∂Ωi Φi,(j) [0, 0] dσ . ∂Ωi

(cf. (10.98) and (10.101)). Then (10.116) follows by (10.89) and (10.92).

 

Finally, we turn to the matrix ΛΩ(1 ,2 )− that appears in the representation formula (10.38) and prove the following. Proposition 10.34. Let R ≡ (Ri,j )(i,j)∈{1,2}2 be the function from ] − δ∗ , δ∗ [2 to the space of two by two real matrices M2 (R) defined by  Ri,j (1 , 2 ) ≡ − vΩ o [Φo,(j) [1 , 2 ]](1 pi + 1 2 ξ) + vΩi [Φi,(j) [1 , 2 ]](ξ) ∂Ωi   S2 (pi − pk + 2 (ξ − η))Φk,(j) [1 , 2 ](η) dση dσξ + ∂Ωk

for all (1 , 2 ) ∈] − δ∗ , δ∗ [2 and for alli, j ∈ {1, 2}. Here k ≡ 1 if i = 2 and k ≡ 2 if i = 1. Let Λ(1 , 2 ) ≡ Λi,j (1 , 2 ) (i,j)∈{1,2}2 be the matrix of M2 (R) defined by

422

10 A Dirichlet Problem in a Domain with Two Small Holes

Λ(1 , 2 ) ≡

1 2π



log(1 2 ) log 1 log 1 log(1 2 )

 + R(1 , 2 )

(10.117)

for all (1 , 2 ) ∈]0, δ∗ [2 . Then the following statements hold. (i) For all (1 , 2 ) ∈]0, δ∗ [2 we have Λ(1 , 2 ) = ΛΩ(1 ,2 )− ,

(10.118)

where ΛΩ(1 ,2 )− is the matrix in (10.41). (ii) The matrix Λ(1 , 2 ) is invertible for all (1 , 2 ) ∈]0, δ∗ [2 . (iii) The function from ] − δ∗ , δ∗ [2 to M2 (R) that takes (1 , 2 ) to R(1 , 2 ) is real analytic. (iv) We have o

0 Ri,j [0, 0] = −H0Ω (0) + δi,j lim HΩ (η) + (1 − δi,j ) S2 (pi − pj ) (10.119) j η→∞

for all i, j ∈ {1, 2}. Proof. To prove (10.118) we have to show that  Λi,j (1 , 2 ) = − VΩ(1 ,2 ) [τ(j) ] dσ 1 ,2 ∂Ωi (1 ,2 )

(j)

for all i, j ∈ {1, 2} and all (1 , 2 ) ∈]0, δ∗ [2 , where τ1 ,2 ∈ C 0,α (∂Ω(1 , 2 )) is the solution of (10.39). This last equality can be verified exploiting Definition 4.18 of single layer potential, equalities (10.112) and (10.113) for S2 , equal1,(j) 2,(j) o,(j) (j) ity (10.62) of (φ1 ,2 , φ1 ,2 , φ1 ,2 ) and Φ(j) [1 , 2 ], equality (10.44) of τ1 ,2 and 1,(j) 2,(j) o,(j) (φ1 ,2 , φ1 ,2 , φ1 ,2 ), the rule of change of variable in integrals, and equality (10.63). Moreover, Lemma 6.30 implies that ΛΩ(1 ,2 )− is invertible for all (1 , 2 ) in ]0, δ∗ [2 . Thus Λ(1 , 2 ) is invertible for all (1 , 2 ) ∈]0, δ∗ [2 and statement (ii) is proved. To prove statement (iii) we observe that    Ri,j (1 , 2 ) = − S2 (1 pi + 1 2 ξ − y)Φo,(j) [1 , 2 ](y) dσy ∂Ωi

∂Ω o i,(j)

+ VΩi [Φ [1 , 2 ]](ξ)   i k k,(j) S2 (p − p + 2 (ξ − η))Φ [1 , 2 ](η) dση dσξ + ∂Ωk

for all (1 , 2 ) ∈] − δ∗ , δ∗ [2 and for all i, j ∈ {1, 2}, with k ≡ 1 if i = 2 and k ≡ 2 if i = 1. Then, by the real analyticity of Φ(j) in Proposition 10.23, by Theorem A.28 (ii) on the properties of the integral operators with real analytic kernel, by Theorem 6.4 on the continuity of the linear map that takes φ ∈ C 0,α (∂Ωi ) to VΩi [φ] ∈ C 1,α (∂Ωi ), and by the continuity of the linear map that takes a func-

10.4 Moderately Close Holes in Dimension n = 2

423

tion of C 1,α (∂Ωi ) to its integral mean over ∂Ωi , we deduce that the function from ] − δ∗ , δ∗ [2 to M2 (R) that takes (1 , 2 ) to R(1 , 2 ) is real analytic. Finally, to verify statement (iv) we observe that  o,(j) o [0, 0]](0) + − vΩi [Φi,(j) [0, 0]](ξ) dσξ Ri,j (0, 0) = vΩ [Φ ∂Ωi  i k k,(j) + S2 (p − p ) Φ [0, 0] dσ ∂Ωk

for all i, j ∈ {1, 2}, where k ≡ 1 if i = 2 and k ≡ 2 if i =  1. Then (10.119) follows by Lemma 10.30, by equality (10.86), and by equality ∂Ωk Φk,(j) [0, 0] dσ = δj,k (cf. (10.63)).   We can now prove Theorem 10.35, where we write the maps that take (1 , 2 ) to suitable restrictions of u1 ,2 and of the rescaled functions u1 ,2 (1 ph + 1 2 ·), (j) with h ∈ {1, 2}, in terms of the real analytic maps UM , Um , VM , Vm , and F of Propositions 10.31, 10.32, and 10.33, of the matrix function Λ of Proposition 10.34, and of the singular functions log 1 and log(1 2 ). In what follows we retain the following notation: At denotes the transpose of a matrix A and A−1 denotes the inverse of an invertible matrix A. For the correct understanding of the matrix product below, we also remind that in this book all vectors and vector valued functions are understood to be columns. As a consequence, F(1 , 2 )t is a row vector for all (1 , 2 ) ∈] − δ∗ , δ∗ [2 . Theorem 10.35. The following statements hold. (i) Let ΩM be an open subset of Ω o such that 0 ∈ / ΩM . Let δM ∈]0, δ∗ ] be such that ΩM ∩ Ωk (1 , 2 ) = ∅ for all (1 , 2 ) ∈] − δM , δM [×] − δ∗ , δ∗ [ and for all k ∈ {1, 2}. Then we have u1 ,2 |ΩM = uo + 1 2 UM [1 , 2 ] + F(1 , 2 )t Λ(1 , 2 )−1 VM [1 , 2 ] for all (1 , 2 ) ∈]0, δM [×]0, δ∗ [, where uo is the unique solution of (10.98). (j) (ii) Let j ∈ {1, 2}. Let Ωm be an open bounded subset of R2 \ Ωj . Let δm ∈]0, δ∗ ] be such that  1 pj +  1  2 Ω m ⊆ Ω o and (1 pj + 1 2 Ωm ) ∩ Ωk (1 , 2 ) = ∅ (j)

(j)

for all (1 , 2 ) ∈] − δm , δm [2 , where k = 1 if j = 2 and k = 2 if j = 1. Then we have u1 ,2 (1 pj + 1 2 ·)|Ωm

  (j) (j) = Um [1 , 2 ] + F(1 , 2 )t Λ(1 , 2 )−1 Vm [1 , 2 ] + Z (j) (1 , 2 )

424

10 A Dirichlet Problem in a Domain with Two Small Holes (j)

for all (1 , 2 ) ∈]0, δm [2 , where Z (j) (1 , 2 ) ≡ (Z (j) (1 , 2 )1 , Z (j) (1 , 2 )2 ) is the vector of R2 defined by Z (j) (1 , 2 )i ≡ δi,j

1 1 log(1 2 ) + (1 − δi,j ) log 1 2π 2π

∀i, j ∈ {1, 2} .

Proof. Statement (i) follows by the representation formula (10.38), by Proposition 10.31 (i) and equality (10.97), by Proposition 10.32 (i) and (ii), by equality (10.115), and by Proposition 10.34 (i). Statement (ii) follows by the representation formula (10.38), by Proposition 10.31 (i) and equality (10.99), by Proposition 10.32 (i) and equality (10.110), by equality (10.115), and by Proposition 10.34 (i).   Finally, we derive by Theorem 10.35 the following Theorem 10.36 for the energy  |∇u1 ,2 |2 dx. Ω(1 ,2 ) Theorem 10.36. There exists δE ∈]0, δ∗ ] and two real analytic functions, E 1 : ] − δ E , δE [2 → R , E2 : ] − δE , δE [2 → Rn such that 

|∇u1 ,2 |2 dx = E1 (1 , 2 ) + F(1 , 2 )t Λ(1 , 2 )−1 E2 (1 , 2 )

Ω(1 ,2 )

for all (1 , 2 ) ∈]0, δE [2 . Moreover, we have    E1 (0, 0) = |∇uo |2 dx + Ωo

i=1,2

R2 \Ωi

|∇ui |2 dξ ,

where uo is the unique solution of (10.98) and u1 , u2 are the unique solutions of (10.101) with j = 1 and j = 2, respectively. Proof. Let ΩM and Ωm be as in statements (i) and (ii) of Theorem 10.35 and assume in addition that ∂Ω o ⊆ ΩM (1)

and

∂Ωi ⊆ Ωi,m

(2)

for i ∈ {1, 2} .

Let δE ≡ min{δM , δm , δm }. A calculation based on the first Green Identity of Theorem 4.2 and on the representation formulas of Theorem 10.35 shows that  |∇u1 ,2 |2 dx Ω(1 ,2 )    o o (i) o = f νΩ · ∇u dσ − fi νΩ i · ∇Um [1 , 2 ] dσ ∂Ω o

i=1,2



∂Ω i

f o νΩ o · ∇UM [1 , 2 ] dσ

+ 1 2 ∂Ω o

10.4 Moderately Close Holes in Dimension n = 2

425

+ F(1 , 2 )t Λ(1 , 2 )−1    × f o νΩ o · ∇VM [1 , 2 ] dσ − ∂Ω o

i=1,2

 ∂Ω i

(i) fi νΩ i · ∇Vm [1 , 2 ] dσ

for all (1 , 2 ) ∈] − δE , δE [2 (see also the proof of Theorem 10.9). Then we define E1 (1 , 2 ) as the sum of the first, second, and third term on the right-hand side of the equality above and E2 (1 , 2 ) as the sum of the terms in braces. The real analyticity of E1 and E2 follows by the real analyticity of the maps UM , Um , VM , and (j) Vm of Propositions 10.31 and 10.32, by the continuity of the linear maps from C 1,α (ΩM ) to C 0,α (ΩM )n and from C 1,α (Ωi,m ) to C 0,α (Ωi,m )n that take a function to its gradient, and by the continuity of the linear functions that take elements of C 0,α (∂ΩM )n and C 0,α (∂Ωi )n to their integral over ∂Ω o and ∂Ωi , i ∈ {1, 2}. The last statement of the theorem follows by equality (10.97), by the expression for (i) Um [0, 0] of Proposition 10.31 (iii), by the first Green Identity of Theorem 4.2, and by the first Green Identity in exterior domains of Corollary 4.7.  

10.4.6 Asymptotic Behavior of u1 ,2 as (1 , 2 ) → (0, 0) In this section we exploit Theorem 10.35 to obtain an asymptotic approximations of the solution u1 ,2 of problem (10.4) as the pair of parameters (1 , 2 ) approaches a singular value (0, 0). We will often find convenient to write the quotient 1/ log(1 2 ). To do so, we have to avoid the case when 1 2 = 1 and a zero appears at the denominator. Therefore, we shrink the set ] − δ1 , δ1 [×] − δ2 , δ2 [ of admissible pairs (1 , 2 ) and we assume, without loss of generality, that we have δ1 ∈]0, 1/δ2 [ . Then inequality δ∗ < min{δ1 , δ2 } (cf. (10.95)) ensures that 1 2 < 1 for all (1 , 2 ) (1) (2) in ]0, δ∗ [2 (we also recall that δM , δm , and δm of Theorem 10.35 are smaller than δ∗ ). In what follows, cof(A) denotes the cofactor matrix of a matrix A ∈ M2 (R), so that A−1 = (det A)−1 cof(A)t for all invertible matrices A ∈ M2 (R). Proposition 10.37 describes the inverse of the matrix Λ(1 , 2 ). The proof can be obtained by a direct computation based on equality (10.117) for Λ(1 , 2 ). Proposition 10.37. For all (1 , 2 ) ∈]0, δ∗ [2 we have det Λ(1 , 2 ) = and

1 ρ(1 , 2 ) log(1 2 ) 4π 2

426

10 A Dirichlet Problem in a Domain with Two Small Holes

Λ(1 , 2 )

−1

2π = ρ(1 , 2 )

!

1 log 1 − log( 1 2 )

log 1 − log( 1 2 ) 1



) cof(R(1 , 2 ))t , + 2π log(1 2 ) (10.120)

where the real number ρ(1 , 2 ) is given by ρ(1 , 2 ) ≡ log 2 +

log 1 log 2 log(1 2 )

− 2π(R1,2 (1 , 2 ) + R2,1 (1 , 2 ))

log 1 log(1 2 )

+ 2π(R1,1 (1 , 2 ) + R2,2 (1 , 2 )) + 4π 2 (R1,1 (1 , 2 )R2,2 (1 , 2 ) − R1,2 (1 , 2 )R2,1 (1 , 2 ))

1 . log(1 2 ) (10.121)

As observed in Proposition 10.34 (ii), Λ(1 , 2 ) is invertible for all (1 , 2 ) in ]0, δ∗ [2 . Then Proposition 10.37 implies that ρ(1 , 2 ) = 0

∀(1 , 2 ) ∈]0, δ∗ [2 .

At first, we focus on the ‘macroscopic’ behavior of the solution, that is, the behavior of the map that takes (1 , 2 ) to a restriction u1 ,2 |ΩM with ΩM ⊆ Ω o open and such that 0 ∈ / ΩM . In the following Proposition 10.38 we exploit Proposition 10.37 to refine the representation formula of Theorem 10.35 (i). / ΩM . Let δM ∈ Proposition 10.38. Let ΩM be an open subset of Ω o such that 0 ∈ ]0, δ∗ ] be such that ΩM ∩ Ωk (1 , 2 ) = ∅ for all (1 , 2 ) ∈] − δM , δM [×] − δ∗ , δ∗ [ and for all k ∈ {1, 2}. Then there exists a real analytic map XM ≡ (XM,1 , XM,2 ) : ] − δM , δM [×] − δ∗ , δ∗ [→ C 1,α (ΩM )2 such that u1 ,2 (x) = uo (x) + 1 2 UM [1 , 2 ](x) o log 2 1 (F1 (1 , 2 ) + F2 (1 , 2 )) GΩ (x, 0) + 2π log(1 2 ) ρ(1 , 2 )   log 1 1 − log( 1 t 1 2 ) F(1 , 2 ) + 2π XM [1 , 2 ](x) log 1 − log( 1 ρ(1 , 2 ) 1 2 ) + 4π 2

1 1 F(1 , 2 )t cof(R(1 , 2 ))t VM [1 , 2 ](x) log(1 2 ) ρ(1 , 2 ) (10.122)

for all x ∈ ΩM and all (1 , 2 ) ∈]0, δM [×]0, δ∗ [.

10.4 Moderately Close Holes in Dimension n = 2

427 o

Proof. Equality (10.109) implies that we have VM,j [0, 2 ] − GΩ (·, 0)Ω o = 0 for all 2 ∈] − δ∗ , δ∗ [ and j ∈ {1, 2}. Moreover, the real analyticity of VM of Proposition 10.32 (ii) implies that the map from ] − δM , δM [×] − δ∗ , δ∗ [ to C 1,α (ΩM ) that o takes (1 , 2 ) to VM,j [1 , 2 ]−GΩ (·, 0)|ΩM is real analytic. Then we can prove that the map from (] − δM , δM [\{0})×] − δ∗ , δ∗ [ to C 1,α (ΩM ) that takes (1 , 2 ) to o

VM,j [1 , 2 ] − GΩ (·, 0)|ΩM 1 has a real analytic extension to ] − δM , δM [×] − δ∗ , δ∗ [. If we denote such an extension by XM,j , then the map XM = (XM,1 , XM,2 ) is real analytic from ] − δM , δM [×] − δ∗ , δ∗ [ to C 1,α (ΩM )2 and we have o

VM,j [1 , 2 ](x) = GΩ (x, 0) + 1 XM,j [1 , 2 ](x)

∀x ∈ ΩM

for all (1 , 2 ) ∈] − δM , δM [×] − δ∗ , δ∗ [ and j ∈ {1, 2}. Then, by a straightforward computation based on Proposition 10.37 we have Λ(1 , 2 )−1 VM [1 , 2 ](x)

 Ωo  log 2 1 G (x, 0) o = 2π log(1 2 ) ρ(1 , 2 ) GΩ (x, 0)   log 1 1 − log( 1 1 2 ) + 2π XM [1 , 2 ](x) log 1 1 ρ(1 , 2 ) − log(  ) 1 2 + 4π 2

1 1 cof(R(1 , 2 ))t VM [1 , 2 ](x) log(1 2 ) ρ(1 , 2 )

∀x ∈ ΩM

for all (1 , 2 ) ∈]0, δ∗ [2 . To complete the proof we can now plug the expression  above into the representation formula for u1 ,2 |ΩM of Theorem 10.35 (i).  We plan to exploit the real analytic representation (10.122) to obtain an asymptotic expansion of u1 ,2 as (1 , 2 ) tends to (0, 0) in ]0, δ∗ [2 . To do so, we face the problem that log 1 lim (1 ,2 )∈]0,δ∗ [2 →(0,0) log(1 2 ) does not exist. As announced in the introduction of this section, we can overcome this difficulty by replacing 1 with a positive parameter t and by taking 2 = γ(t), where γ is a function from the interval ]0, 1[ to itself such that lim γ(t) = 0

t→0+

and such that the limit λ0 ≡ lim+ t→0

log t log(tγ(t))

(10.123)

428

10 A Dirichlet Problem in a Domain with Two Small Holes

exists finite in [0, +∞[ (we note that the membership of γ(t) in ]0, 1[ implies that λ0 ∈ [0, 1] when it exists). Then we can analyze the asymptotic behavior of the map that takes t > 0 small enough to ut,γ(t)|ΩM as t → 0+ . In the following Proposition 10.39 we compute the first and the second term of the corresponding asymptotic expansion. Proposition 10.39. Let ΩM be an open subset of Ω o such that 0 ∈ / ΩM . Let γ be a function from ]0, 1[ to itself such that limt→0+ γ(t) = 0 and such that the limit λ0 in (10.123) exists finite in [0, +∞[. Then we have ut,γ(t)|ΩM = uo|ΩM

 o 2π  1 lim u1 (η) + lim u2 (η) − 2uo (0) GΩ (·, 0)|ΩM η→∞ log(tγ(t)) 1 + λ0 η→∞   1 +o log(tγ(t)) +

as t → 0+ , where the approximation holds in the norm of C 1,α (ΩM ) and where o GΩ is the Dirichlet Green function of Ω o (cf. (10.93)), uo is the unique solution of (10.98), and u1 , u2 are the unique solutions of (10.101) with j = 1 and j = 2, respectively. Proof. We observe that the functions tγ(t) ,

t , log γ(t)

1 1 log(tγ(t)) log γ(t)

are all o(1/ log(tγ(t))) for t → 0+ . Moreover, by definition (10.121) of ρ(1 , 2 ) we can verify that 1 log γ(t) = lim . (10.124) 1 + λ0 t→0+ ρ(t, γ(t)) Then, taking (1 , 2 ) = (t, γ(t)) in the representation formula (10.122), we deduce that ut,γ(t)|ΩM (x) = uo|ΩM o log γ(t) 1 (F1 (t, γ(t)) + F2 (t, γ(t))) GΩ (·, 0)|ΩM + 2π log(tγ(t)) ρ(t, γ(t))   1 +o log(tγ(t))

for t → 0+ . By (10.124) we also have     1 1 log γ(t) 1 − =o log(tγ(t)) ρ(t, γ(t)) 1 + λ0 log(tγ(t)) Then, we can write

as t → 0+ .

10.4 Moderately Close Holes in Dimension n = 2

429

ut,γ(t)|ΩM = uo|ΩM o 1 1 + 2π (F1 (0, 0) + F2 (0, 0)) GΩ (·, 0)|ΩM log(tγ(t)) 1 + λ0   1 +o log(tγ(t))

as t → 0+ . Since (10.116) implies that F1 (0, 0) + F2 (0, 0) = lim u1 (η) + lim u2 (η) − 2uo (0) , η→∞

η→∞

 

we deduce the validity of statement.

We now turn to the rescaled function u1 ,2 (1 p +1 2 ·). Taking (1 , 2 ) = (t, γ(t)) with γ as above, we can compute the limit value of ut,γ(t) (tpj + tγ(t)·) as t → 0+ . By the real analytic representation of Theorem 10.35 (ii) and by the formula for Λ(1 , 2 )−1 in Proposition 10.37 we can prove the following. j

Proposition 10.40. Let j ∈ {1, 2}. Let Ωm be an open bounded subset of R2 \ Ωj . Let γ be a function from ]0, 1[ to itself such that limt→0+ γ(t) = 0 and such that the limit λ0 in (10.123) exists finite in [0, +∞[. Then we have as t → 0+ ,

ut,γ(t) (tpj + tγ(t)·)|Ωm = uj|Ωm + o(1)

where the approximation holds in C 1,α (Ωm ) and where uj is the unique solution of (10.101). Proof. By the representation formula of Theorem 10.35 (ii) and by the formula for Λ(1 , 2 )−1 of Proposition 10.37 we can write (j) ut,γ(t) (tpj + tγ(t)·)|Ωm = Um [t, γ(t)] ! )  log t 1 − log(tγ(t)) 2π cof(R(t, γ(t)))t t + F(t, γ(t)) + 2π log t − log(tγ(t)) 1 ρ(t, γ(t)) log(tγ(t))   (j) × Vm [t, γ(t)] + Z (j) (t, γ(t)) (j) [t, γ(t)] = Um

2π + F(t, γ(t)) ρ(t, γ(t))



t



1 log t − log(t,γ(t))

log t − log(t,γ(t)) 1

 Z (j) (t, γ(t))

 log t − log(t,γ(t)) (j) [t, γ(t)] Vm log t − log(t,γ(t)) 1  4π 2 cof(R(t, γ(t)))t  (j) + F(t, γ(t))t Vm [t, γ(t)] + Z (j) (t, γ(t)) ρ(t, γ(t)) log(tγ(t)) (10.125) 2π + F(t, γ(t)) ρ(t, γ(t)) t

1

430

10 A Dirichlet Problem in a Domain with Two Small Holes (j)

for all t > 0 small enough so that (t, γ(t)) ∈]0, δm [2 . By the definition of Z (j) in Theorem 10.35 (ii) we have   log t 1 − log(tγ(t)) 2π Z (j) (t, γ(t)) log t 1 ρ(t, γ(t)) − log(tγ(t))    1 log2 t δ1,j = . log(tγ(t)) − δ2,j ρ(t, γ(t)) log(tγ(t)) We now compute the limit of the scalar factor on the right-hand side as t → 0+ . In view of the limit relations (10.123) and (10.124), we have   1 log2 t lim log(tγ(t)) − log(tγ(t)) t→0+ ρ(t, γ(t))   log t log t log γ(t) log(tγ(t)) − = lim log γ(t) log γ(t) log(tγ(t)) t→0+ ρ(t, γ(t))     log(tγ(t)) 1 log t log t log t − = lim+ + 1− log γ(t) log γ(t) log γ(t) log(tγ(t)) t→0 1 + λ0   1 log t log γ(t) 1 = lim (1 + λ0 ) = 1 . 1+ = log γ(t) log(tγ(t)) 1 + λ0 t→0+ 1 + λ0 (10.126) We also observe that the definition of ρ in (10.121) implies that lim

t→0+

1 =0 ρ(t, γ(t))

(10.127)

and the definition of Z (j) in Theorem 10.35 implies that   1 1 δ1,j + (1 − δ1,j )λ0 (j) lim Z (t, γ(t)) = . 2π δ2,j + (1 − δ2,j )λ0 t→0+ log(tγ(t)) (j)

(10.128) (j)

Moreover, the real analyticity of Um of Proposition 10.31 (iii), of Vm of Proposition 10.32 (iii), of F of Proposition 10.33, and of R of Proposition 10.34 (iii) (j) (j) imply that the maps Um [t, γ(t)], Vm [t, γ(t)], F(t, γ(t)), and R(t, γ(t)) have a limiting value at t = 0. Then, by the limit relations (10.123), (10.126), (10.127), and (10.128) and by the representation formula (10.125) we have   δ1,j (j) lim ut,γ(t) (tpj + tγ(t)·)|Ωm = Um [0, 0] + F(0, 0)t δ2,j t→0+ in C 1,α (Ωm ). Now the statement follows by (10.100) and (10.116).

 

10.4 Moderately Close Holes in Dimension n = 2

431

Finally, for the energy of ut,γ(t) we have the following proposition. Proposition 10.41. If γ is a function from ]0, 1[ to itself such that limt→0+ γ(t) = 0 and such that the limit λ0 in (10.123) exists finite in [0, +∞[, then we have     |∇ut,γ(t) |2 dx = |∇uo |2 dx + |∇ui |2 dξ + o(1) Ω(t,γ(t))

Ωo

i=1,2

R2 \Ωi

as t → 0+ , where uo is the unique solution of (10.98) and u1 , u2 are the unique solutions of (10.101) with j = 1 and j = 2, respectively. Proof. It follows by the representation Theorem 10.36, by noting that the matrix Λ(t, γ(t))−1 tends to the zero matrix as t → 0+ (cf. equality (10.120)), and by the limit relations (10.123) and (10.127).  

Chapter 11

Nonlinear Boundary Value Problems in Domains with a Small Hole

Abstract This chapter is devoted to the application of the Functional Analytic Approach to nonlinear boundary value problems in domains with a small hole. We consider only the case where the nonlinearity appears in the boundary conditions. In our examples we deal with nonlinear Robin boundary conditions and nonlinear transmission conditions.

11.1 The Geometric Setting The geometric setting is the same as in Chapter 8. We fix once for all a natural number n ∈ N, n ≥ 2 and a real number α ∈]0, 1[ . Then we take two sets Ω i and Ω o in the n-dimensional Euclidean space Rn , the letter ‘i’ standing for ‘inner domain’ and the letter ‘o’ standing for ‘outer domain’. We assume that Ω i and Ω o satisfy the following condition Ω i and Ω o are bounded open connected subsets of Rn of class C 1,α , the exterior domains Ω i− and Ω o− are connected, and the origin 0 of Rn belongs both to Ω i and to Ω o . Then Ω i and Ω o have no holes and there exists a positive real number 0 such that Ω i ⊆ Ω o

∀ ∈] − 0 , 0 [ .

We denote by Ω() the perforated domain Ω() ≡ Ω o \ (Ω i )

∀ ∈] − 0 , 0 [

© Springer Nature Switzerland AG 2021 M. Dalla Riva et al., Singularly Perturbed Boundary Value Problems, https://doi.org/10.1007/978-3-030-76259-9 11

433

434

11 Nonlinear Boundary Value Problems in Domains with a Small Hole

and we observe that Ω() is an open bounded connected subset of Rn with boundary consisting of the union of the two sets ∂Ω o and ∂Ω i (cf., e.g., Theorem A.10). Moreover, for  in ]−0 , 0 [\{0} the domain Ω() is of class C 1,α (cf. Section 2.13). Instead, for  = 0 we have Ω(0) = Ω o \ {0} and Ω(0) is no longer a local strict hypograph of class C 0 (cf. Definition 2.37).

11.2 A Nonlinear Robin Problem In this section, we consider a nonlinear Robin problem in Ω(). First we introduce an ‘unperturbed’ problem with no hole in Ω o , and then we consider the case with a hole Ω i . To do so, we assign a function Go from ∂Ω o × R to R and we assume that the ‘unperturbed’ nonlinear Robin boundary value problem  Δu = 0 in Ω o , (11.1) ∂u o o ∂νΩ o (x) = G (x, u(x)) ∀x ∈ ∂Ω admits at least a solution u ˜o ∈ C 1,α (Ω o ). Here νΩ o denotes the outward unit normal o to ∂Ω . Moreover, we denote by Gou the partial derivative of Go with respect to the second variable in R and we assume that the function Gou (x, u ˜o (x))

∀x ∈ ∂Ω o

is less or equal to 0 and has a nonzero integral, an assumption which ensures that problem (11.1) does not have other solutions close to u ˜o . o Solutions as u ˜ can be shown to exist under reasonable assumptions on Go . For example, we might introduce a specific growth condition and exploit the topological degree, as in [164, Theorem 7.11]. Other existence results for problems such as (11.1) can be found in the works of Carleman [40], Nakamori and Suyama [231], Klingelh¨ofer [139–142], and Efendiev et al. [94]. We also mention the contribution of Begehr and Hsiao [18, 19] and Begehr and Hile [17] for problems in the plane. Next we make a hole in the domain Ω o and for each  ∈]0, 0 [ we consider the nonlinear Robin problem ⎧ ⎪ in Ω(), ⎨ Δu = 0 − ∂ν∂u i = 0 on ∂Ω i , (11.2) Ω ⎪ ⎩ ∂u (x) = Go (x, u(x)) ∀x ∈ ∂Ω o , ∂νΩ o in the perforated domain Ω(). Here νΩ i denotes the outward unit normal to ∂(Ω i ). We will prove that possibly shrinking 0 , problem (11.2) has a solution u(, ·) in C 1,α (Ω()) for all  ∈]0, 0 [. Moreover, the family of solutions ˜o in the C 1,α (ΩM ){u(, ·)}∈]0,0 [ satisfies the limiting relation lim→0+ u(, ·) = u

11.2 A Nonlinear Robin Problem

435

norm for all bounded open subsets ΩM of Ω o such that 0 ∈ / ΩM and it is unique in a local sense which we clarify below. Then we pose the following three questions: (i) Let x be fixed in Ω o \ {0}. What can be said on the map  → u(, x) when  > 0 is close to 0? (ii) Let ξ be fixed in Rn \ Ω i . What can be said on the map  → u(, ξ) when  > 0 is close to 0?  (iii) What can be said on the map  → Ω() |Du(, y)|2 dy when  > 0 is close to 0? In a sense, question (i) concerns the ‘macroscopic’ behavior of u(, ·), whereas question (ii) concerns the ‘microscopic’ behavior of u(, ·) and question (iii) deals with the energy of u(, ·). We plan to answer such questions in the spirit of the present book. We also remark that the content of the present section develops from [164] and [66]. We now briefly outline our strategy. We first convert problem (11.2) into a nonlinear integral equation by exploiting classical Potential Theory. Then we observe that the corresponding integral equations can be written, after an appropriate rescaling and with an appropriate choice of the functional variables, in a form that can be analyzed by means of the Implicit Function Theorem around the degenerate case  = 0 and we represent the unknown densities of the integral equation in terms of real analytic functions of . Next we exploit the integral representation of the solutions of problem (11.2) in terms of the unknown densities and we deduce both the existence of u(, ·) and its representation in terms of .

11.2.1 Formulation of a Nonlinear Robin Problem in Terms of Integral Equations Given a function G ∈ C 0 (∂Ω × R), we denote by NG the (nonlinear nonautonomous) composition operator from C 0 (∂Ω) to itself that maps v ∈ C 0 (∂Ω) to the function NG [v] defined by NG [v](t) ≡ G(t, v(t))

∀t ∈ ∂Ω .

We now transform our nonlinear Robin boundary value problem (11.1) into a problem for integral equations by means of the following. Proposition 11.1. Let Ω be an open bounded connected subset of Rn of class C 1,α . Let G ∈ C 0 (∂Ω × R) be such that NG maps C 0,α (∂Ω) to itself. Then the map + [μ] + c is a bijection from the set of pairs (μ, c) of that takes a pair (μ, c) to vΩ C 0,α (∂Ω)0 × R that satisfy the equation 1 LΩ,0,+ [μ, c] ≡ − μ + WΩt [μ] = NG [VΩ [μ] + c] , 2 (cf. definition (6.56)) to the set of the solutions u ∈ C 1,α (Ω) of the problem

(11.3)

436

11 Nonlinear Boundary Value Problems in Domains with a Small Hole



Δu = 0 in Ω , ∂u ∂νΩ = NG [u] on ∂Ω .

(11.4)

Proof. If (μ, c) ∈ C 0,α (∂Ω)0 × R satisfies Equation (11.3), then we know that + [μ] + c belongs to C 1,α (Ω) and is harmonic in Ω (cf. Theorem 4.25). Then the vΩ jump properties for the normal derivative of a single layer potential of Theorem 6.6 + [μ] + c imply the validity of the boundary condition in problem (11.4). Hence, vΩ satisfies problem (11.4). If u ∈ C 1,α (Ω) satisfies problem (11.4), then Theorem 6.48 (i) on the representation of a harmonic function as a sum of a single layer potential and of a constant + [μ] + c. ensures that there exists a unique (μ, c) ∈ C 0,α (∂Ω)0 × R such that u = vΩ The jump formulas for the normal derivative of a single layer of Theorem 6.6 and the boundary condition in (11.4) imply that the integral equation of (11.3) is satisfied. Hence, the map of the statement is a bijection.   We also note that the assumption that NG maps C 0,α (∂Ω) to itself is certainly satisfied, for example, when G is locally Lipschitz continuous (cf. Lemma A.24).

11.2.2 Formulation of Problems (11.1) and (11.2) in Terms of Integral Equations We now provide a formulation of the Robin problem (11.1) in Ω o and (11.2) in Ω() in terms of integral equations. To do so, we consider the following assumptions

NG o

Go ∈ C 0 (∂Ω o × R) , maps C 0,α (∂Ω o ) to itself .

In addition, we denote by G the function from ∂Ω() × R to R defined by  o G (x, a) if (x, a) ∈ ∂Ω o × R , G(x, a) ≡ 0 if (x, a) ∈ ∂Ω i × R .

(11.5)

(11.6)

We could apply directly Proposition 11.1. We note, however, that the corresponding representation formulas include an integration in ∂Ω() and thus on ∂Ω i , a set that depends on . In order to get rid of such a dependence, we properly rescale the restriction of the unknown function to ∂Ω i . Then we obtain the following Theorem 11.2. In what follows, we find convenient to use the abbreviation X0,α ≡ C 0,α (∂Ω i ) × C 0,α (∂Ω o ) .

Theorem 11.2. Let Go ∈ C 0 (∂Ω o × R) satisfy (11.5). Let M ≡ (M1 , M2 , M3 ) be the map from ] − 0 , 0 [×X0,α × R to X0,α × R defined by

11.2 A Nonlinear Robin Problem

437



1 M1 [, θi , θo , c](ξ) ≡ θi (ξ) + νΩ i (ξ) · ∇Sn (ξ − η)θi (η) dση (11.7) 2 ∂Ω i  νΩ i (ξ) · ∇Sn (ξ − y)θo (y) dσy ∀ξ ∈ ∂Ω i , + o ∂Ω  1 i o M2 [, θ , θ , c](x) ≡ − θo (x) + n−1 νΩ o (x) · ∇Sn (x − η)θi (η) dση 2 ∂Ω i  + νΩ o (x) · ∇Sn (x − y)θo (y) dσy ∂Ω o     o n−1 i o −G  Sn (x − η)θ (η) dση + Sn (x − y)θ (y) dσy + c ∂Ω i

∂Ω o

∀x ∈ ∂Ω o ,



 M3 [, θi , θo , c] ≡ n−1

θi dσ + ∂Ω i

θo dσ , ∂Ω o

for all (, θi , θo , c) ∈] − 0 , 0 [×X0,α × R. Let u[, θi , θo , c] be the function from Ω() to R defined by + [μ] + c , (11.8) u[, θi , θo , c] ≡ vΩ() where μ(x) ≡ θo (x)

∀x ∈ ∂Ω o ,

μ(x) ≡ θi (x/)

∀x ∈ ∂Ω i ,

(11.9)

for all (, θi , θo , c) ∈]0, 0 [×X0,α × R. If  ∈]0, 0 [, then the map u[, ·, ·, ·] is a bijection from the set of solutions (θi , θo , c) ∈ X0,α × R of equation M[, θi , θo , c] = 0 ,

(11.10)

to the set of solutions u ∈ C 1,α (Ω()) of problem (11.2). Proof. A simple computation based on the rule of change of variables in integrals over ∂Ω i shows that (θi , θo , c) solves equation (11.10) if and only the function μ of (11.9) and c solve the integral equation (11.3) with Ω = Ω(). Thus statement (i) follows by Proposition 11.1 (see also (11.6)).   We note that, contrary to problem (11.2), which has been defined only for  > 0, equation M[0, θi , θo , c] = 0 makes sense also for  = 0. We now show that in the degenerate case where  = 0, equation M[, θi , θo , c] = 0 is equivalent to a ‘limiting boundary value problem’. Theorem 11.3. Let Go ∈ C 0 (∂Ω o ×R) satisfy (11.5). A triple (θi , θo , c) in X0,α ×R satisfies equation (11.11) M[0, θi , θo , c] = 0 , if and only if both the following conditions are satisfied. (i) The pair (θo , c) ∈ C 0,α (∂Ω o ) × R is a solution of the system

438

11 Nonlinear Boundary Value Problems in Domains with a Small Hole

⎧ 1 o  ν o (x) · ∇Sn (x − y)θo(y) dσy ⎨ − 2 θ (x) + ∂Ω o Ω  o −G S (x − y)θo (y) dσy + c = 0 ∂Ω o n ⎩ o θ dσ = 0 . ∂Ω o

∀x ∈ ∂Ω o ,

(11.12)

(ii) θi is the only solution in C 0,α (∂Ω i ) of the integral equation  1 i θ (ξ) + νΩ i (ξ) · ∇Sn (ξ − η)θi (η) dση (11.13) 2 ∂Ω i  νΩ i (ξ) · ∇Sn (y)θo (y) dσy ∀ξ ∈ ∂Ω i . = ∂Ω o

For all (θo , c) in C 0,α (∂Ω o ) × R, let − + i o (ui0 [θo , c], uo0 [θo , c]) ≡ (vΩ i [θ ], vΩ o [θ ] + c) ,

(11.14)

where  θ ≡ i

1 I + WΩt 2

(−1) $

%

 νΩ i ·

∇Sn (y)θ (y) dσy o

(11.15)

∂Ω o

is the only solution in C 0,α (∂Ω i ) of Equation (11.13). Then the map (ui0 [·, ·], uo0 [·, ·]) is a bijection from the set of solutions (θo , c) in C 0,α (∂Ω o ) × R of system (11.12) 1,α (Ω i− ) × C 1,α (Ω o ) of the ‘limiting boundary to the set of solutions (ui , uo ) ∈ Cloc value problem’ ⎧ Δuo = 0 in Ω o , ⎪ ⎪ ⎪ ∂uo o o ⎪ ⎪ ∀x ∈ ∂Ω o , ⎨ ∂νΩo (x) = G (x, u (x)) i in Ω i , Δu = 0 (11.16) i ⎪ ∂u o i ⎪ i ⎪ (ξ) = −ν (ξ) · ∇u (0) ∀ξ ∈ ∂Ω , Ω ⎪ ∂ν ⎪ ⎩ Ωi limξ→∞ ui (ξ) = 0 . Moreover, uo0 [·, ·] is a bijection from the set of solutions (θo , c) in C 0,α (∂Ω o ) × R of problem (11.12) to the set of solutions uo ∈ C 1,α (Ω o ) of the nonlinear Robin problem (11.1) in Ω o . Proof. The equivalence of equation M[0, θi , θo , c] = 0 and of the system of the equations in (11.12) and (11.13) is obvious. Since Ω i− is connected, Corollary 6.15 implies that 12 I + WΩt i is a linear bijection from C 0,α (∂Ω i ) to itself. Since νΩ i belongs to C 0,α (∂Ω i , Rn ), the right-hand side of equation (11.13) is of class C 0,α (∂Ω i ) and accordingly Equation (11.13) has a unique solution in C 0,α (∂Ω i ) for each fixed θo ∈ C 0,α (∂Ω o ). By Proposition 11.1 with Ω = Ω o , it follows that uo [·, ·] is a bijection from the set of solutions of (11.12) to the set of solutions of (11.1) and thus from the set of solutions of (11.12) to the set of solutions of the first two equations of (11.16). If (θi , θo , c) solves equation (11.11), then uo [θo , c] solves the first two equations of (11.16). Moreover, θi must equal the right-hand side of (11.15) and thus the jump

11.2 A Nonlinear Robin Problem

439

formulas of Theorem 6.6 imply that  ∂ − i v i [θ ] = νΩ i · ∇Sn (y)θo (y) dσy = −νΩ i · ∇uo [θo , c](0) . ∂νΩ i Ω ∂Ω o Then the third and fourth equation of (11.16) are satisfied. Moreover,   1 i θ + WΩt i [θi ] dσ θi dσ = ∂Ω i ∂Ω i 2   =− νΩ i (ξ) · ∇Sn (y)θo (y) dσy dσξ = 0 , ∂Ω i

∂Ω o

− i and thus vΩ i [θ ] tends to zero at infinity (cf. Lemma 6.11 and Theorem 4.23). Hence, i o o o (u0 [θ , c], u0 [θ , c]) solves (11.16). 1,α (Ω i− ) × C 1,α (Ω o ) satisfies the limWe now assume that a pair (ui , uo ) ∈ Cloc iting boundary value problem (11.16) and show that there exists a solution (θo , c) in C 0,α (∂Ω o ) × R of system (11.12) such that (ui , uo ) = (ui0 [θo , c], uo0 [θo , c]). Since uo0 [·, ·] is a bijection, then there exists a unique (θo , c) ∈ C 0,α (∂Ω o ) × R such that system (11.12) is satisfied and uo = uo0 [θo , c]. We now define θi by means of (11.15) − i and we show that ui = vΩ i [θ ]. Then the jump formula for the normal derivative of − 1 i i t i a single layer potential of Theorem 6.6 implies that ∂ν∂ i vΩ i [θ ] = 2 θ + WΩ i [θ ]. Ω Since  νΩ i · ∇Sn (y)θo (y) dσy = −νΩ i · ∇uo0 [θo , c](0) = −νΩ i · ∇uo (0) , ∂Ω o

− i the definition of θi implies that the function vΩ i [θ ] satisfies the third and fourth equation of the limiting boundary value problem (11.16). As above we can prove  − i that equality (11.15) implies that ∂Ω i θi dσ = 0. Hence, vΩ i [θ ] tends to zero − i i at infinity (cf. Theorem 4.23). Then the difference u − vΩ i [θ ] is harmonic in Ω i− , tends to zero at infinity, and satisfies the Neumann boundary condition − ∂ i i i ∂ν i [u − vΩ i [θ ]] = 0 on ∂Ω . Then the uniqueness of the exterior Neumann Ω

problem for harmonic functions on the connected exterior domain Ω i− implies that − i i o i o o o ui − v Ω i [θ ] = 0 (cf. Theorem 6.2). Hence, (u , u ) = (u0 [θ , c], u0 [θ , c]). i o The injectivity of the map (u0 [·, ·], u0 [·, ·]) is a consequence of the injectivity of   uo0 [·, ·]. Theorems 11.2 and 11.3 reduce the analysis of the Robin problems (11.1) in Ω o and (11.2) in Ω() to that of equation M = 0. We shall show that under reasonable assumptions on the data, if problem (11.1) admits a solution u ˜o satisfying certain nondegeneracy conditions, then, for  sufficently small, problem (11.2) has a solution, which is unique in a local sense that we clarify in section 11.2.4. To show an existence result for problem (11.2), we apply the Implicit Function Theorem A.19 ˜o0 = uo0 [θ˜o , c˜]. to the equation M = 0 around a zero (0, θ˜i , θ˜o , c˜) of M such that u Thus we prove the following.

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11 Nonlinear Boundary Value Problems in Domains with a Small Hole

Theorem 11.4. Assume that Go is real analytic from ∂Ω o × R to R ,

(11.17)

i.e., Go has a real analytic extension on an open neighborhood of ∂Ω o × R. Assume that there exists a solution u ˜o ∈ C 1,α (Ω o ) of problem (11.1) such that Gou (x, u ˜o (x)) ≤ 0 ∀x ∈ ∂Ω o ,  Go (y, u ˜o (y)) dσy = 0 .

(11.18)

∂Ω o

Let (θ˜o , c˜) be the unique solution of (11.12) in C 0 (∂Ω o ) × R such that u ˜o = uo0 [θ˜o , c˜] . Let θ˜i be as in (11.15) with θo ≡ θ˜o . Then there exist  ∈]0, 0 [, an open neighborhood V˜ of (θ˜i , θ˜o , c˜) in X0,α × R, and a real analytic map (Θi , Θo , C) from ] −  ,  [ to V˜ such that the set of zeros of M in ] −  ,  [×V˜ coincides with the graph of (Θi , Θo , C). In particular, (Θi [0], Θo [0], C[0]) = (θ˜i , θ˜o , c˜) .

(11.19)

Proof. We plan to apply the Implicit Function Theorem A.19 to Equation (11.10) around the point (0, θ˜i , θ˜o , c˜). By assumption (11.17), by the analyticity of composition operators generated by analytic functions (cf. Theorem A.23 (ii)), by the linearity and continuity of the layer potential operators VΩ o , WΩt i , WΩt o (cf. Theorems 6.4 and 6.8), and by the analyticity of the nonlinear integral operators with analytic kernel of Theorem A.28 (ii), the operator M is real analytic. We now compute the differential ∂(θi ,θo ,c) M1 [0, θ˜i , θ˜o , c˜](θi , θo , c ) of M at (0, θ˜i , θ˜o , c˜) with respect to the variable (θi , θo , c). By definition of M, we have   1 i θ + WΩt i [θi ] − νΩ i · ∇Sn θo dσ , M[0, θi , θo , c] = 2 ∂Ω o 1 − θo + WΩt o [θo ] − Go ◦ (VΩ o [θo ] + c),  2 θo dσ

∀(θo , θi , c) ∈ X0,α × R .

∂Ω o

The first and third components of M[0, θi , θo , c] and the term − 12 I + WΩt o that appears in the second component are linear and continuous. Accordingly, such two components and the term − 12 I + WΩt o coincide at all points with their own differential. Instead, Go ◦ (VΩ o [θo ] + c) can be seen as the composition of the linear and continuous map (θo , c) → VΩ o [θo ] + c

11.2 A Nonlinear Robin Problem

441

with the composition operator NGo from C 1,α (∂Ω o ) to itself, which is defined by NGo [v] ≡ Go ◦ v

∀v ∈ C 1,α (∂Ω o ) .

Then we observe that NGo is differentiable by Theorem A.23 (ii) and, by the chain rule, the differential of Go ◦ (VΩ o [θo ] + c) at (θ˜o , c˜) coincides with the composition of the differential of NGo at VΩ o [θ˜o ] + c˜, which is delivered by the formula dNGo (VΩ o [θ˜o ] + c˜)[v] = Gou (VΩ o [θ˜o ] + c˜)v

∀v ∈ C 1,α (∂Ω o )

with the differential of the linear and continuous operator (θo , c) → VΩ o [θo ] + c , which coincides with the operator itself. Namely, we have   d(Go ◦ (VΩ o [θo ] + c))(θ˜o , c˜)[θo , c ] = Gou (VΩ o [θ˜o ] + c˜) VΩ o [θo ] + c for all (θo , c ) ∈ C 0,α (∂Ω o ) × R. We conclude that the differential of M at (0, θ˜i , θ˜o , c˜) with respect to the variable (θi , θo , c) is delivered by the formula 1 ∂(θi ,θo ,c) M1 [0, θ˜i , θ˜o , c˜](θi , θo , c )(ξ) = θi (ξ) 2  i + νΩ i (ξ) · ∇Sn (ξ − η)θ (η) dση ∂Ω i  − νΩ i (ξ) · ∇Sn (y)θo (y) dσy ∀ξ ∈ ∂Ω i ,

(11.20)

∂Ω o

∂(θi ,θo ,c) M2 [0, θ˜i , θ˜o , c˜](θi , θo , c )(x)  1 = − θo (x) + νΩ o (x) · ∇Sn (x − y)θo (y) dσy 2 ∂Ω o   o o o  −Gu (x, u ˜ (x)) Sn (x − y)θ (y) dσy + c ∂Ω o  i ˜o i o  ˜ ∂(θi ,θo ,c) M3 [0, θ , θ , c˜](θ , θ , c ) = θo dσ ,

∀x ∈ ∂Ω o ,

∂Ω o

for all (θi , θo , c ) ∈ X0,α × R. We now prove that ∂(θi ,θo ,c) M[0, θ˜i , θ˜o , c˜] is a linear homeomorphism. By the Open Mapping Theorem 2.2, it suffices to show that ∂(θi ,θo ,c) M[0, θ˜i , θ˜o , c˜] is a bijection from X0,α × R onto itself. Let (f i , f o , a) belong to X0,α × R. We must show that there exists a unique (θi , θo , c ) ∈ X0,α × R such that ∂(θi ,θo ,c) M[0, θ˜i , θ˜o , c˜](θi , θo , c ) = (f i , f o , a) .

(11.21)

We first observe that the second and third equation of (11.21) can be rewritten in the form

442

11 Nonlinear Boundary Value Problems in Domains with a Small Hole



  LΩ o ,bo ,+ [θo , c ],

θo dσ

= (f o , a) ,

(11.22)

∂Ω o

where 1 LΩ o ,bo ,+ [θo , c ] = − θo + WΩt o [θo ] + bo VΩ o [θo ] + bo c 2

on ∂Ω o

and bo (x) ≡ −Gou (x, u ˜o (x)) for all x ∈ ∂Ω o (we have already introduced the sym˜o ∈ C 1,α (Ω o ), bol LΩ o ,bo ,+ in (6.56)). Since NGou maps C 0,α (∂Ω o ) to itself and u o 0,α o we have b ∈ C (∂Ω ). Then Proposition 6.57 (i) ensures that Equation (11.22) has a unique solution (θo , c ) in C 0,α (∂Ω o ) × R. We now consider the first equation of (11.21). Since the function  νΩ i · ∇Sn (y)θo (y) dσy ∂Ω o

 belongs to C 0,α (∂Ω i ), f i + νΩ i · ∂Ω o ∇Sn (y)θo (y) dσy belongs to C 0,α (∂Ω i ). Since Ω i− is connected, then 12 I + WΩt i is a linear bijection from C 0,α (∂Ω i ) to itself, and accordingly there exists a unique θi ∈ C 0,α (∂Ω i ) that solves equation    1 I + WΩt i [θi ] = f i + νΩ i · ∇Sn (y)θo (y) dσy , 2 ∂Ω o i.e., the first equation of (11.21) (cf. Corollary 6.15). Hence, the linear operator ∂(θi ,θo ,c) M[0, θ˜i , θ˜o , c˜] is a continuous linear bijection and the statement of the theorem follows by the Implicit Function Theorem in Banach spaces (cf. Theorem A.19).   Remark 11.5. We can prove that solutions u ˜o ∈ C 1,α (Ω o ) of problem (11.1) as in (11.17) and (11.18) exist for a large class of functions Go . For example, Theorem 7.11 of [164] (see also the end of Appendix B and Proposition 7.13 of the same paper) shows that one can use a topological degree argument to prove that a solution exists when Go satisfies a suitable growth  condition. In particular, if there exist a function g ∈ C 0,α (∂Ω) with g ≥ 0 and ∂Ω g dσ = 0, and a real number δ ∈]0, 1[ such that |Go (x, a) + g(x)a| sup < +∞ . (1 + |a|)δ (x,a)∈∂Ω×R We can now introduce an -dependent family of solutions of (11.2). Definition 11.6. Let the assumptions of Theorem 11.4 hold. Then we set 6 7 u(, x) ≡ u , Θi [], Θo [], C[] (x) ∀x ∈ Ω() , for all  ∈]0,  [ (cf. (11.8)). We shall prove that possibly shrinking  , the family of solutions {u(, ·)}∈]0, [ is the only family of solutions of problem (11.2) that is close to u ˜o in a sense which we clarify in Section 11.2.4. First we analyze its behavior as  approaches 0.

11.2 A Nonlinear Robin Problem

443

11.2.3 Analytic Representation for the Family {u(, ·)}∈]0, [ The following statement shows that {u(, ·)}∈]0, [ can be continued real analytically for negative values of . Theorem 11.7. With the assumptions of Theorem 11.4, the following statements hold. (i) If ΩM is a bounded open subset of Ω o \ {0} such that 0 ∈ / ΩM , then there exist M ∈]0,  [ and a real analytic map UM from ] − M , M [ to C 1,α (ΩM ) such that ΩM ⊆ Ω()

∀ ∈] − M , M [ ,

(11.23)

∀ ∈]0, ΩM [ .

(11.24)

u(, ·)|ΩM = UM [](·)

Moreover, UM [0] = u ˜o|Ω . In particular, lim→0 u(, ·)|ΩM = u ˜o|Ω M

M

in the

1,α

C -norm. (ii) There exists a real analytic map U∂Ω o from ] −  ,  [ to C 1,α (∂Ω o ) such that u(, ·)|∂Ω o = U∂Ω o [](·)

∀ ∈]0,  [ .

(11.25)

Moreover, U∂Ω o [0] = u ˜o|∂Ω o . Proof. Since 0 has a positive distance from ΩM , there exists M ∈]0,  [ such that ΩM ⊆ Ω()

∀ ∈ [−M , M ] .

If  ∈]0, M [, then we have 6 7 u(, x) = u , Θi [], Θo [], C[] (x)  = Sn (x − y)Θ (y) dσy + C[]

∀x ∈ Ω() ,

∂Ω()

where Θ (y) ≡ Θo [](y)

∀y ∈ ∂Ω o ,

Θ (y) ≡ Θi [](y/)

∀y ∈ ∂Ω i .

Accordingly  u(, x) = n−1  +

Sn (x − η)Θi [](η) dση

(11.26)

∂Ω i

Sn (x − y)Θo [](y) dσy + C[]

∀x ∈ Ω() .

∂Ω o

Thus, we find natural to define a map UM from ] − M , M [ to C 1,α (ΩM ) by setting  n−1 UM [](x) ≡  Sn (x − η)Θi [](η) dση (11.27) ∂Ω i

444

11 Nonlinear Boundary Value Problems in Domains with a Small Hole

 Sn (x − y)Θo [](y) dσy + C[]

+

∀x ∈ ΩM ,

∂Ω o

for all  ∈] − M , M [. It clearly suffices to show that the right-hand side of (11.27) + defines a real analytic map from ] − M , M [ to C 1,α (ΩM ). Since vΩ o [·] is linear 0,α o 1,α o and continuous from C (∂Ω ) to C (Ω ) (cf. Theorem 4.25) and since ΩM ⊆ Ω o and Θo is real analytic from ] − M , M [ to C 0,α (∂Ω o ), we conclude that the second summand in the right-hand side of (11.27) defines a real analytic map from ]−M , M [ to C 1,α (ΩM ). We now consider the first summand. By the analyticity of the nonlinear integral operators with analytic kernel of Theorem A.28, the map from  ] − M , M [×L1 (∂Ω i ) to C 1,α (ΩM ) that takes (, f ) to ∂Ω i Sn (· − η)f (η) dση is real analytic. Since Θi is real analytic and C 0,α (∂Ω i ) ⊆ L1 (∂Ω i ) with continuous embedding, we conclude that the function from ] − M , M [ to C 1,α (ΩM ) that takes  to the first summand in the right-hand side of (11.27) is real analytic. Since C[·] is analytic, the proof of the analyticity of UM is complete. By Theorems 11.3 and 11.4, we have + ˜o + c˜ = u ˜o|ΩM . UM [0] = vΩ o [θ ]Ω M Then the limiting relation in  follows by the continuity of UM . We now consider statement (ii). We define U∂Ω o to be equal to the right-hand side of (11.27) for all x ∈ ∂Ω o and  ∈] −  ,  [. Then equality (11.27) for all x ∈ ∂Ω o and  in ] −  ,  [ implies the validity of (11.25). The analyticity of U∂Ω o follows by the same argument used above to prove the analyticity of UM . Then we have + ˜o U∂Ω o [0] = vΩ ˜= u ˜o|∂Ω o , o [θ ]|∂Ω o + c

 

and thus the proof is complete.

In a sense, the previous theorem describes the ‘macroscopic’ behavior of the family {u(, ·)}∈]0, [ as  is close to 0 and it implies that if we fix x ∈ Ω o \ {0}, then we can expand u(, x) into a convergent power expansion of  for  small enough. We now wish to understand the behavior of u(, ·) close to the boundary ∂Ω i of the cavity. To do so, we apply a dilation of factor . We observe that if ξ ∈ Rn \ Ω i , then ξ ⊆ Rn \ Ω i , and accordingly ξ ∈ Ω o \ Ω i = Ω() , for  small enough so that ξ ∈ Ω o . Then for such , it makes sense to consider the evaluation of u(, ·) at ξ. Namely, u(, ξ) . We note that the function u(, ξ) of ξ ∈ 1 Ω() is a ‘rescaled’ version of u(, x), with x ∈ Ω(), and we can think of it as the ‘microscopic’ version of u(, x). Then

11.2 A Nonlinear Robin Problem

445

we ask what can be said on the map  → u(, ξ) when  is close to 0. To answer in the spirit of the present book, we prove the following theorem. Theorem 11.8. Let the assumptions of Theorem 11.4 hold. Let Ωm be a bounded 1 from open subset of Rn \ Ω i . Then there exist m ∈]0,  [, a real analytic map Um 1,α 2 ] − m , m [ to C (Ωm ), and a real analytic map U from ] − m , m [ to R such that Ωm ⊆ Ω o ∀ ∈] − m , m [ , (11.28) 1 2 u(, ξ) = Um [](ξ) + δ2,n ( log )U [] ∀ξ ∈ Ωm ∀ ∈]0, m [ . 1 Moreover, Um [0] = u ˜o (0) and U 2 [0] = 0.

Proof. Let m ∈]0, 0 [ be such that Ωm ⊆ Ω o

∀ ∈ [−m , m ] \ {0} .

Then we have

1 Ω() ∀ ∈ [−m , m ] \ {0} .  By the definition of u(, ·), we have  u(, ξ) = Sn (ξ − η)Θi [](η) dση n−1 ∂Ω i  + Sn (ξ − y)Θo [](y) dσy + C[] o  ∂Ω  δ2,n = ( log ) Sn (ξ − η)Θi [](η) dση  + Θi [] dσ 2π ∂Ω i ∂Ω i  Sn (ξ − y)Θo [](y) dσy + C[] + Ωm ⊆

∂Ω o

for all ξ ∈ Ωm and for all  ∈]0, m [. Then we find natural to set 1 Um [](ξ)





≡

Sn (ξ − η)Θi [](η) dση  + ∂Ω i

Sn (ξ − y)Θo [](y) dσy + C[] ∂Ω o

for all ξ ∈ Ωm and for all  ∈] − m , m [, and  1 U 2 [] ≡ Θi [] dσ 2π ∂Ω i

∀ ∈] −  ,  [ .

− By the analyticity of Θi , Θo , C and by the linearity and continuity of vΩ from i [·]|Ω m 0,α i 0,α C (∂Ω ) to C (Ωm ) (cf. Theorem 4.25) and by the analyticity of nonlinear 1 and U 2 integral operators with analytic kernel of Theorem A.28 (i), the maps Um are analytic.

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11 Nonlinear Boundary Value Problems in Domains with a Small Hole

We now turn to prove the last part of the statement. By equality (11.19), we have (Θi [0], Θo [0], C[0]) = (θ˜i , θ˜o , c˜). Then the definition of (θ˜i , θ˜o , c˜), which must satisfy equality (11.15), and Lemma 6.11 imply that + ˜o 1 Um [0] = vΩ ˜= u ˜o (0) , o [θ ](0) + c   1 ˜i 1 1 2 i ˜ θ dσ = U [0] = θ + WΩt i [θ˜i ] dσ 2π ∂Ω i 2π ∂Ω i 2    1 = νΩ i (η) · ∇Sn (y)θ˜o (y) dσy dση = 0 . 2π ∂Ω i ∂Ω o

 

Thus the proof is complete.

The previous theorem implies that if ξ ∈ Ω i− and n ≥ 3, then we can expand u(, ξ) into a convergent power expansion of , for  > 0 small enough. For n = 2 instead, it shows that we can expand u(, ξ) into a convergent power expansion of  and ( log ), for  > 0 small enough.

11.2.4 Local Uniqueness of the Family {u(, ·)}∈]0,0 [ We now turn to the question of local uniqueness of the family {u(, ·)}∈]0,0 [ , and we prove the following. Theorem 11.9. Let the assumptions of Theorem 11.4 hold. If {εj }j∈N is a sequence in ]0, +∞[ that converges to 0 and if {uj }j∈N is a sequence of functions such that the function uj belongs to C 1,α (Ω(εj )) and solves (11.2) for  = εj for all j ∈ N , limj→∞ uj = u ˜o in C 1,α (ΩM ) for all bounded open subsets ΩM of Ω o \ {0} such that 0 ∈ / ΩM , then there exists j0 ∈ N such that uj (·) = u(εj , ·) for all j ≥ j0 . Proof. Possibly neglecting a finite number of indexes, we can assume that εj ∈]0,  [ for all j ∈ N. Since Go is real analytic, Lemma A.24 ensures that condition (11.5) is satisfied. Then Theorems 11.2 and 11.3 imply that there exist (θji , θjo , cj ) and (θ˜i , θ˜o , c˜) ∈ X0,α × R such that uj = u[εj , θji , θjo , cj ] , u ˜o = uo [θ˜o , c˜] , 0

M[εj , θji , θjo , cj ] = 0 , M[0, θ˜i , θ˜o , c˜] = 0 .

We rewrite equation M[, θi , θo , c] = 0 for an arbitrary (, θi , θo , c) in the domain of M in the following form M1 [, θi , θo , c] = 0 ,

(11.29)

11.2 A Nonlinear Robin Problem

447



1 − θo (x) + n−1 νΩ o (x) · ∇Sn (x − η)θi (η) dση 2 ∂Ω i  + νΩ o (x) · ∇Sn (x − y)θo (y) dσy o ∂Ω   −Gou (x, u ˜o (x)) n−1 Sn (x − η)θi (η) dση i ∂Ω   + Sn (x − y)θo (y) dσy + c ∂Ω o     = Go x, n−1 Sn (x − η)θi (η) dση + Sn (x − y)θo (y) dσy + c i ∂Ω o ∂Ω  −Gou (x, u ˜o (x)) n−1 Sn (x − η)θi (η) dση i ∂Ω   + Sn (x − y)θo (y) dσy + c ∀x ∈ ∂Ω o , ∂Ω o

M3 [, θ , θ , c] = 0 . i

o

  Next we denote by N [, θi , θo , c] ≡ Nl [, θi , θo , c] l=1,2,3 , the function from the set ] − 0 , 0 [×X0,α × R to X0,α × R such that N1 ≡ M 1 ,

N3 ≡ M 3 ,

and such that N2 equals the left-hand side of the second equation in (11.29). Thus Equation (11.29) can be rewritten as N1 [, θi , θo , c] = 0 ,

(11.30)

N2 [, θ , θ , c](x)     = Go x, n−1 Sn (x − η)θi (η) dση + Sn (x − y)θo (y) dσy + c i ∂Ω o ∂Ω  −Gou (x, u ˜o (x)) n−1 Sn (x − η)θi (η) dση ∂Ω i   + Sn (x − y)θo (y) dσy + c ∀x ∈ ∂Ω o , i

o

∂Ω o

N3 [, θ , θ , c] = 0 . i

o

By arguing precisely as we did to prove the analyticity of M, we can show that N is analytic. We now note that N [, ·, ·, ·] is linear and continuous in X0,α × R for all fixed  ∈] − 0 , 0 [. Since N is analytic, the map from ] − 0 , 0 [ to L (X0,α × R, X0,α × R) that takes  to the linear operator N [, ·, ·, ·] is analytic. We also note that N [0, ·, ·, ·] = ∂(θi ,θo ,c) M[0, θ˜i , θ˜o , c˜](·, ·, ·) ,

448

11 Nonlinear Boundary Value Problems in Domains with a Small Hole

and that the operator in the right-hand side is a linear homeomorphism (cf. the proof of Theorem 11.4 and (11.20)). Since the set of linear homeomorphisms is open in the set of linear and continuous operators, and since the map that takes a linear invertible operator to its inverse is real analytic (cf. Proposition A.15), there exists  ∈]0,  [ such that the map  → N [, ·, ·, ·](−1) is real analytic from ] −  ,  [ to L (X0,α × R, X0,α × R). Since M[εj , θji , θjo , cj ] = 0, the invertibility of N [εj , ·, ·, ·] and equality (11.30) guarantee that 1 2 uo|∂Ω o ]uj|∂Ω o , 0 (θji , θjo , cj ) = N [εj , ·, ·, ·](−1) 0, NGo [uj|∂Ω o ] − NGou [˜ whenever εj ∈]0,  [. Since Go is real analytic, then NGo [·] is also real analytic (cf. Theorem A.23 (ii)), and thus NGo [·] is also continuous. Accordingly, lim NGo [uj|∂Ω o ] − NGou [˜ uo|∂Ω o ]uj|∂Ω o = NGo [˜ uo|∂Ω o ] − NGou [˜ uo|∂Ω o ]˜ uo|∂Ω o ,

j→∞

(11.31) in C 0,α (∂Ω o ). Moreover, the analyticity of  → N [, ·, ·, ·](−1) guarantees that lim N [εj , ·, ·, ·](−1) = N [0, ·, ·, ·](−1) ,

j→∞

(11.32)

in L (X0,α × R, X0,α × R). Since the evaluation map from L (X0,α × R, X0,α × R) × X0,α × R to X0,α × R that takes a pair (R, v) to R[v] is bilinear and continuous, the limiting relations (11.31) and (11.32) imply that lim (θji , θjo , cj )

(11.33) 1 2 = lim N [j , ·, ·, ·](−1) 0, NGo [uj|∂Ω o ] − NGou [˜ uo|∂Ω o ]uj|∂Ω o , 0 j→∞ 1 2 uo|∂Ω o ] − NGou [˜ uo|∂Ω o ]˜ u|∂Ω o , 0 in X0,α × R . = N [0, ·, ·, ·](−1) 0, NGo [˜

j→∞

Since M[0, θ˜i , θ˜o , c˜] = 0, the right-hand side of (11.33) equals (θ˜i , θ˜o , c˜). Hence, lim (εj , θji , θjo , cj ) = (0, θ˜i , θ˜o , c˜) ,

j→∞

in ] −  ,  [×X0,α × R. Hence, there exists j0 ∈ N such that every (εj , θji , θjo , cj ) with j ≥ j0 belongs to the neighborhood ] −  , , [×V˜ of (0, θ˜i , θ˜o , c˜) introduced in Theorem 11.4. Since (εj , θji , θjo , cj ) are zeros of M, Theorem 11.4 implies that θji = Θi [εj ] ,

θjo = Θo [εj ]

cj = C[εj ] ,

11.2 A Nonlinear Robin Problem

449

for all j ≥ j0 . Accordingly, uj = u[εj , θji , θjo , cj ] = u[Θi [εj ], Θo [εj ], C[εj ]] = u(εj , ·) for all j ≥ j0 .

 

11.2.5 Analytic Representation for the Energy Integral of the Family {u(, ·)}∈]0, [ We now turn to the energy integral, and we prove the following. Theorem 11.10. Let the assumptions of Theorem 11.4 hold. Then there exists a real analytic map E from ] −  ,  [ to R such that  |∇y u(, y)|2 dy = E[] ∀ ∈]0,  [ . Ω()

Moreover, E[0] =

 Ωo

|∇˜ uo |2 dy.

Proof. By the first Green Identity of Theorem 4.2, we have  |∇y u(, y)|2 dy Ω()   ∂ ∂ u(, y) u(, y) dσy + u(, y) u(, y) dσy =− i ∂ν ∂ν i o Ωyo Ω ∂Ω ∂Ω y  = u(, y)Go (y, u(, y)) dσy , ∂Ω o

for all  ∈]0,  [. Then it suffices to take  E[] ≡ U∂Ω o [](y)Go (y, U∂Ω o [](y)) dσy ∂Ω o

for all  ∈] −  ,  [ (cf. Theorem 11.7). The analyticity of U∂Ω o [·] from ] −  ,  [ to C 1,α (∂Ω o ), the analyticity of NGo from C 0,α (∂Ω o ) to itself, which follows by the analyticity of Go and by Theorem A.23 (ii), the continuity of the pointwise product in C 0 (∂Ω o ), and the continuity of the linear map that takes a function of C 0 (∂Ω o ) to its integral ensure that E is real analytic on ] −  ,  [. Finally, equality ˜o|∂Ω o and the definition of E, ensure that E[0] = Ω o |∇˜ uo |2 dy.   U∂Ω o [0] = u

450

11 Nonlinear Boundary Value Problems in Domains with a Small Hole

11.3 A Nonlinear Transmission Problem In this section, we consider a nonlinear transmission problem in a domain with a small inclusion. First we introduce an ‘unperturbed’ problem with no hole (and no transmission) in Ω o , and then we consider the case with a small hole Ω i (cf. (8.1) and (8.2)). To do so, we assign a Dirichlet datum f o ∈ C 1,α (∂Ω o ) . As is well known, the ‘unperturbed’ Dirichlet problem  Δu = 0 in Ω o , u = f o on ∂Ω o ,

(11.34)

(11.35)

has one and only one solution u ˜o ∈ C 1,α (Ω o ) (cf. Theorem 6.27). i Next we make a hole Ω in the domain Ω o and we consider a nonlinear transmission problem in the perforated domain Ω() as in (8.3) and in the corresponding hole Ω i for each  ∈]0, 0 [. In order to introduce the nonlinear transmission conditions on ∂Ω i , we introduce an increasing real analytic diffeomorphism F from R onto R ,

(11.36)

and a real analytic map G from R to R .

(11.37)

Then we consider the nonlinear transmission problem ⎧ Δui = 0 ⎪ ⎪ o ⎪ ⎪ ⎨ Δu = 0 o u = F (ui ) ⎪ ∂uo ⎪ (x) = ⎪ ∂ν ⎪ ⎩ o Ωi o u =f

in Ω i , in Ω(), on ∂Ω i , ∂ui i i ∂ν Ω i (x) + G(u (x)) ∀x ∈ ∂Ω , on ∂Ω o ,

(11.38)

where νΩ i denotes the outward unit normal to ∂Ω i . Nonlinear transmission problems of this type arise in the study of the heat conduction in composite materials with different nonconstant thermal conductivities. More specifically, when the conductivities of the two materials are (nonlinear) functions of the temperature itself, we can perform a suitable change of the unknown functions (by means of the so called ‘Kirchhoff transformation’) and transform the boundary value problem into a boundary value problem for the Laplace equation with a nonlinear transmission condition on ∂Ω i as the one in (11.38). We present an application of this strategy in the case of a periodic problem in Section 13.8.2 (see also Mityushev and Rogosin [224, Chap. 5.1]). We note that in the simpler case in which F is affine and G equals 0, (11.38) becomes a standard linear transmission problem with a temperature jump on the interface.

11.3 A Nonlinear Transmission Problem

451

A priori, it is not clear why problem (11.38) should admit a classical solution. We shall prove that possibly shrinking 0 , problem (11.38) has a family of solutions (ui (, ·), uo (, ·)) in C 1,α (Ω i ) × C 1,α (Ω()) for all  ∈]0, 0 [, which has the property that lim ui (, ξ) = F (−1) (˜ uo (0))

→0+

in the C 1,α -norm in ξ ∈ Ω i ,

˜o (0) lim uo (, ξ) = u

→0+

in the C 1,α -norm for ξ in the closure of all bounded open subsets Ωm of Rn \ Ω i , lim uo (, x) = u ˜o (x)

→0+

in the C 1,α -norm for x in the closure of all bounded open subsets ΩM of Ω o such that 0 ∈ / ΩM . This family of solutions is also unique in a local sense which we clarify in section 11.3.3. Moreover, we answer the following questions (i) Let ξ be fixed in Ω i . What can be said on the map  → ui (, ξ) when  > 0 is close to 0? (ii) Let ξ be fixed in Rn \ Ω i . What can be said on the map  → uo (, ξ) when  > 0 is close to 0? (iii) Let x be fixed in Ω o \ {0}. What can be said on the map  → uo (, x) when  > 0 is close to 0?  (iv) What can be said on the map  → Ω i |∇y ui (, y)|2 dy when  > 0 is close to 0?  (v) What can be said on the map  → Ω() |∇y uo (, y)|2 dy when  > 0 is close to 0? In a sense, questions (i), (ii) concern the ‘microscopic’ behavior of ui (, ·) and uo (, ·), whereas question (iii) concerns the ‘macroscopic’ behavior of uo (, ·) and questions (iv), (v) deal with the energy of ui (, ·) and uo (, ·), respectively. To analyze the problem and answer the above questions we resort to the Functional Analytic Approach of this book. Accordingly, we first convert the nonlinear transmission problem (11.38) into a system of nonlinear integral equations by exploiting classical Potential Theory. Then we observe that, by changing the variables appropriately, we can obtain a functional equation that can be analyzed by means of the Implicit Function Theorem around the degenerate case where  = 0. We deduce that we can represent the unknown densities of the integral equations in terms of real analytic functions of  for n ≥ 3, and of , log−1 , and  log2  for n = 2. Next we go back to the integral representation of the solutions of problem (11.38) and we deduce both the existence of ui (, ·) and of uo (, ·) and the representation formulas that describe their dependence upon  and the dependence upon  of their energies. Doing so, we answer to questions (i)–(v).

452

11 Nonlinear Boundary Value Problems in Domains with a Small Hole

We observe that the content of the present chapter develops from [168]. Moreover, for the analysis of a similar problem, but with nonautonomous transmission conditions, we refer to the paper [225] of Molinarolo. For an existence result in the case of a big inclusion (that is, for  > 0 fixed) we refer to the paper [71] with Mishuris and for a local uniqueness result for the solutions of (11.38) themselves, rather than for the family of solutions, we mention the paper [72] with Molinarolo.

11.3.1 Formulation of the Nonlinear Transmission Problem in Terms of Integral Equations As a first step, we wish to convert the transmission problem (11.38) into a system of integral equations. By exploiting the representation Theorem 6.48 for harmonic functions, the solutions of (11.38) can be written as sums of a single layer potential and of a constant. Then by exploiting the classical jump formulas for single layer potentials (cf. Theorem 6.6), we obtain a system of integral equations on the -dependent domain ∂Ω i ∪ ∂Ω o . In order to get rid of  in the domain, we rescale our functional variables on ∂Ω i and finally obtain an -dependent system of integral equations on the boundaries ∂Ω i and ∂Ω o , which do not depend on . We present such a change of variable in the following Theorem 11.11. To do so, we find convenient to introduce the product space Y0,α defined by Y0,α ≡ C 0,α (∂Ω i )0 × R × C 0,α (∂Ω i ) × C 0,α (∂Ω o ) × R .

(11.39)

Theorem 11.11. Let F , G, f o be as in (11.36), (11.37), (11.34). Let  ∈]0, 0 [. Then the map that takes (ψ, ci , θi , θo , co ) to the pair of functions   + + − o i o vΩ i [ψ(·/)] + ci , vΩ o [θ ] + v Ω i [θ (·/)] + c is a bijection from the set of (ψ, ci , θi , θo , co ) in Y0,α , that solve the following system of integral equations  δ2,n ( log ) θi dσ (11.40) VΩ i [θi ](ξ) + 2π i ∂Ω    ∀ξ ∈ ∂Ω i , + Sn (ξ − y)θo (y) dσy + co = F VΩ i [ψ](ξ) + ci ∂Ω o  1 i t i θ (ξ) + WΩ i [θ ](ξ) + νΩ i (ξ) · ∇Sn (ξ − y)θo (y) dσy 2 ∂Ω o   1 ∀ξ ∈ ∂Ω i , = − ψ(ξ) + WΩt i [ψ](ξ) + G VΩ i [ψ](ξ) + ci 2  Sn (x − η)θi (η) dση + VΩ o [θo ](x) + co = f o (x)

n−1 ∂Ω i

∀x ∈ ∂Ω o ,

11.3 A Nonlinear Transmission Problem

453



 n−1

θi dσ + ∂Ω i

θo dσ = 0 , ∂Ω o

to the set of pairs (ui , uo ) of C 1,α (Ω i ) × C 1,α (Ω()), that solve the boundary value problem (11.38). Proof. By the representation Theorem 6.48 for harmonic functions as a sum of a single layer potential and a constant, if a pair of functions (ui , uo ) of C 1,α (Ω i ) × C 1,α (Ω()) solves the transmission problem (11.38), then there exists a unique quadruple (ω, ci , μ, co ) ∈ C 0,α (∂Ω i )0 × R × C 0,α (∂Ω())0 × R such that + i ui = vΩ i [ω] + c ,

+ uo = vΩ() [μ] + co .

(11.41)

Moreover, the jump formulas for the normal derivative of a single layer of Theorem 6.6 implies that (ω, ci , μ, co ) satisfies the following system of integral equations   Sn (x − y)μ(y) dσy + Sn (x − y)μ(y) dσy + co (11.42) i o ∂Ω ∂Ω   =F Sn (x − y)ω(y) dσy + ci ∀x ∈ ∂Ω i , ∂Ω i  1 μ(x) − νΩ() (x) · ∇Sn (x − y)μ(y) dσy 2 ∂Ω i  − νΩ() (x) · ∇Sn (x − y)μ(y) dσy ∂Ω o  1 = − ω(x) + νΩ i (x) · ∇Sn (x − y)ω(y) dσy 2 ∂Ω i   +G Sn (x − y)ω(y) dσy + ci ∀x ∈ ∂Ω i , ∂Ω i  Sn (x − y)μ(y) dσy ∂Ω i  + Sn (x − y)μ(y) dσy + co = f o (x) ∀x ∈ ∂Ω o . ∂Ω o

Conversely, if (ω, ci , μ, co ) satisfies system (11.42), then the pair of functions in (11.41) solves the transmission problem (11.38) in Ω(). Now we observe that (ω, ci , μ, co ) solves (11.42) if and only if the 5-tuple (ψ, ci , θi , θo , co ) of Y0,α with ψ(ξ) ≡ ω(ξ)

∀ξ ∈ ∂Ω i ,

θo (x) ≡ μ(x)

∀x ∈ ∂Ω o

satisfies system (11.40).

θi (ξ) ≡ μ(ξ)

∀ξ ∈ ∂Ω i ,  

454

11 Nonlinear Boundary Value Problems in Domains with a Small Hole

If the system of integral equations (11.40) has a family of solutions i o {(ψ[] , ci[] , θ[] , θ[] , co[] )}∈[0,0 [

in Y0,α that depends continuously on the parameter , then by taking the limit as  tends to 0 in (11.40), we obtain ⎧ + o vΩ o [θ[0] ](0) + co[0] = F (ci[0] ) , ⎪ ⎪ ⎪ o ⎪ 1 θi + W t i [θi ] + ν i (ξ) ·  ⎪ ∇Sn (−y)θ[0] (y) dσy Ω ⎨ 2 [0] [0] Ω ∂Ω o 1 t i = − 2 ψ[0] + WΩ i [ψ[0] ] + G(c[0] ) , (11.43) ⎪ o o o ⎪ ⎪ V [θ ] + c = f , Ωo [0] ⎪ [0] ⎪ ⎩ θo dσ = 0 . ∂Ω o [0] i o We now prove that (ψ[0] , ci[0] , θ[0] , θ[0] , co[0] ) is not the only solution in Y0,α of system (11.43). By the representation Theorem 6.48 for harmonic functions as a sum of a single layer potential and a constant, the last two equations uniquely determine + o o o and co[0] . Moreover, vΩ ˜o of the θ[0] o [θ[0] ] + c[0] coincides with the unique solution u Dirichlet problem (11.35). Then the first equation of the limit system (11.43) implies uo (0)). However, if we choose an arbitrary ψ[0] in C 0,α (∂Ω i ), that ci[0] = F (−1) (˜ i then the second equation of the limit system (11.43) has a unique solution θ[0] in 0,α i C (∂Ω ) (cf. Corollary 6.15). Hence, the limit system (11.43) does not determine uniquely the limiting value of a family of solutions of (11.38), even if we assume that the limiting value of the corresponding family of solutions of (11.40) does exist. In this sense, we may regard the limit system (11.43) as singular. To overcome this difficulty, we introduce a change of ‘functional’ variables in system (11.40), so that we can obtain a nonsingular limiting system for a family of solutions that does have a limit as  tends to zero. To do so, we exploit the following remark.

Remark 11.12. Let F , G, f o be as in (11.36), (11.37), (11.34). Let (θ˜o , c˜o ) be the unique element of C 0,α (∂Ω o )0 × R such that + ˜o u ˜ o = vΩ ˜o , o [θ ] + c

(11.44)

where u ˜o is the unique solution of the ‘unperturbed’ Dirichlet problem (11.35) in o Ω . If  ∈]0, 0 [, then the map from Y0,α to itself that maps (ψ, ci1 , θi , θ1o , co1 ) to the 5-tuple   δ2,n ( log ) uo (0)) + ci1 + θi dσ, (11.45) ψ, F (−1) (˜ 2πF  (F (−1) (˜ uo (0))) ∂Ω i  θi , θ˜o + θ1o , c˜o + co1 is a bijection.

11.3 A Nonlinear Transmission Problem

455

We exploit the ‘functional’ variables of Remark 11.12 to transform the system of integral equations (11.40) into a new system, which we later show to yield a nonsingular limiting system when  tends to zero. Theorem 11.13. Let F , G, f o be as in (11.36), (11.37), and (11.34). Let  i  u [, ψ, ci1 , θi , θ1o , co1 ], uo [, ψ, ci1 , θi , θ1o , co1 ] be the pair of functions of C 1,α (Ω i ) × C 1,α (Ω()) defined by + (−1) o ui [, ψ, ci1 , θi , θ1o , co1 ](x) ≡ vΩ (˜ u (0)) i [ψ(·/)](x) + F  ( log ) δ 2,n +ci1 + θi dσ 2πF  (F (−1) (˜ uo (0))) ∂Ω i  n−1 = Sn (x − η)ψ(η) dση + F (−1) (˜ uo (0)) ∂Ω i  δ2,n ( log ) i +c1 + θi dσ ∀x ∈ Ω i , 2πF  (F (−1) (˜ uo (0))) ∂Ω i

(11.46)

+ − o o i uo [, ψ, ci1 , θi , θ1o , co1 ](x) ≡ vΩ ˜o (x) + vΩ o [θ1 ](x) + c1 + u i [θ (·/)](x)  + o o ˜o (x) + n−1 Sn (x − η)θi (η) dση ∀x ∈ Ω() = vΩ o [θ1 ](x) + c1 + u ∂Ω i

for all (, ψ, ci1 , θi , θ1o , co1 ) in ]0, 0 [×Y0,α . If  ∈]0, 0 [, then the map (ui [, ·, ·, ·, ·, ·], uo [, ·, ·, ·, ·, ·]) is a bijection from the subset of Y0,α consisting of the 5-tuples (ψ, ci1 , θi , θ1o , co1 ) that solve the following system of integral equations, 

 Sn (ξ −

1

∇˜ uo (βξ)ξ dβ (11.47)  

= F  (F (−1) (˜ uo (0))) VΩ i [ψ](ξ) + ci1 +  (1 − β)F  F (−1) (˜ uo (0)) 0    δ2,n log  i i θ dσ dβ +β VΩ i [ψ](ξ) + c1 + 2πF  (F (−1) (˜ uo (0))) ∂Ω i  2  δ2,n log  i × VΩ i [ψ](ξ) + ci1 + θ dσ ∀ξ ∈ ∂Ω i , 2πF  (F (−1) (˜ uo (0))) ∂Ω i  1 i θ (ξ) + WΩt i [θi ](ξ) +  νΩ i (ξ) · ∇Sn (ξ − y)θ1o (y) dσy 2 ∂Ω o  + νΩ i (ξ) · ∇Sn (ξ − y)θ˜o (y) dσy i

VΩ i [θ ](ξ) +

∂Ω o

y)θ1o (y) dσy

+

co1

+

0 1

∂Ω o

1 = − ψ(ξ) + WΩt i [ψ](ξ) 2 uo (0)) + ci1  + +G VΩ i [ψ](ξ) + F (−1) (˜

δ2,n ( log ) 2πF  (F (−1) (˜ uo (0)))



 θi dσ ∂Ω i

456

11 Nonlinear Boundary Value Problems in Domains with a Small Hole

∀ξ ∈ ∂Ω i ,

 n−2 

∂Ω i

n−2 ∂Ω i

Sn (x − η)θi (η) dση + VΩ o [θ1o ](x) + co1 = 0  θi dσ + θ1o dσ = 0 ,

∀x ∈ ∂Ω o ,

∂Ω o

to the set of solutions (ui , uo ) ∈ C 1,α (Ω i ) × C 1,α (Ω()) of the transmission problem (11.38). Proof. By Theorem 11.11 and by Remark 11.12, it suffices to show that the 5-tuple (ψ, ci1 , θi , θ1o , co1 ) in Y0,α solves the system of integral equations (11.47) if and only if the 5-tuple of (11.45) satisfies the system of integral equations (11.40). By plugging the 5-tuple of (11.45) into system (11.40), we obtain   δ2,n ( log ) VΩ i [θi ](ξ) + θi dσ + Sn (ξ − y)θ˜o (y) dσy + c˜o (11.48) 2π ∂Ω i ∂Ω o  Sn (ξ − y)θ1o (y) dσy + co1 + ∂Ω o    δ2,n ( log ) (−1) o i i = F VΩ i [ψ](ξ) + F (˜ u (0)) + c1 + θ dσ 2πF  (F (−1) (˜ uo (0))) ∂Ω i ∀ξ ∈ ∂Ω i ,



1 i θ (ξ) + WΩt i [θi ](ξ) + νΩ i (ξ) · ∇Sn (ξ − y)θ˜o (y) dσy 2 ∂Ω o  + νΩ i (ξ) · ∇Sn (ξ − y)θ1o (y) dσy ∂Ω o

1 = − ψ(ξ) + WΩt i [ψ](ξ) 2 +G VΩ i [ψ](ξ) + F

(−1)

o

(˜ u (0)) +

ci1

δ2,n ( log ) + 2πF  (F (−1) (˜ uo (0)))



 i

θ dσ ∂Ω i

∀ξ ∈ ∂Ω i ,

 n−1 ∂Ω i

Sn (x − η)θi (η) dση + VΩ o [θ˜o ](x) + c˜o + VΩ o [θ1o ](x) + co1 = f o (x) 

 n−1

θ˜o dσ + 

θi dσ + ∂Ω i

∀x ∈ ∂Ω o ,

 ∂Ω o

∂Ω o

θ1o dσ = 0 .

By the Taylor Formula, we have   (−1) o F F (˜ u (0)) + t = u ˜o (0) + F  (F (−1) (˜ uo (0)))t 

1

+t2 0

(1 − β)F  (F (−1) (˜ uo (0)) + βt) dβ

∀t ∈ R ,

11.3 A Nonlinear Transmission Problem



457

1

u ˜o (ξ) − u ˜o (0) = 

∇˜ uo (βξ) · ξ dβ

∀ξ ∈ ∂Ω i .

0

Then by taking t = VΩ i [ψ](ξ) + ci1 +

δ2,n ( log ) 2πF  (F (−1) (˜ uo (0)))

 θi dσ , ∂Ω i

by plugging the corresponding expression in the first equation of (11.48), by replacing  Sn (ξ − y)θ˜o (y) dσy + c˜o = u ˜o (ξ) ∂Ω o

(cf. (11.44)) with



1

∇˜ uo (βξ) · ξ dβ ,

u ˜o (0) +  0

and by dividing the corresponding expression by , we obtain the first equation of (11.47). Next, we observe that the second equation of (11.48) coincides with the sec˜o = f o on ∂Ω o, the third equaond equation of (11.47). Since VΩ o [θ˜o ](x) + c˜o = u tion of (11.48) is equivalent to the third equation of (11.47). Since ∂Ω o θ˜o dσ = 0, the fourth equation of (11.48) is equivalent to the fourth equation of (11.47).   Hence, we are now reduced to analyze system (11.47). At first glance, we can see that system (11.47) contains the term log , which is singular at 0, only in case n = 2. Then we understand that case n = 2 needs a separate treatment. Our first step to analyze system (11.47) is to note that by letting  tend to zero, we obtain a system, which we address to as the ‘limiting system’, and which has the following form.  Sn (−y)θ1o (y) dσy + co1 + ∇˜ uo (0)ξ (11.49) VΩ i [θi ](ξ) + ∂Ω o

∀ξ ∈ ∂Ω i , = F  (F (−1) (˜ uo (0))) VΩ i [ψ](ξ) + ci1 1 i θ (ξ) + WΩt i [θi ](ξ) + νΩ i (ξ) · ∇˜ uo (0) 2   1 = − ψ(ξ) + WΩt i [ψ](ξ) + G F (−1) (˜ uo (0)) ∀ξ ∈ ∂Ω i ,  2 δ2,n 

∂Ω i

δ2,n ∂Ω i

Sn (x)θi (η) dση + VΩ o [θ1o ](x) + co1 = 0  θi dσ + θ1o dσ = 0 .

∀x ∈ ∂Ω o ,

∂Ω o

To analyze this limiting system, we need the following technical statement. n 1,α Lemma 11.14. Let Ω be a bounded open  connected subset of R of class C . 1 t ˜ ˜ Then there exists φ ∈ Ker − 2 I + WΩ such that the function VΩ [φ] is constant  and ∂Ω φ˜ dσ = 1

458

11 Nonlinear Boundary Value Problems in Domains with a Small Hole

Proof. Since Ω is connected, the number κ + of connected components of Ω equals  1 and we have 1 = κ + = dim Ker − 12 I + WΩt (see Theorem 6.24 and fol  lowing comment). Thus there exists φ˜ ∈ Ker − 12 I + WΩt \ {0}. By the jump formula for the normal derivative of a single layer potential of Theorem 6.6, we + ˜ + ˜ [φ] = 0 on ∂Ω and thus vΩ [φ] is constant in Ω. Then Lemma 6.22 and have ∂ν∂Ω vΩ  ˜ ˜ condition φ = 0 imply that ∂Ω φ dσ = 0. Possibly multiplying φ˜ by a suitable    constant, we can assume that ∂Ω φ˜ dσ = 1. We can now prove an existence and uniqueness result for the solution of a system of boundary integral equations, that we need to solve system (11.49). Theorem 11.15. Let Ω be a bounded open connected subset of Rn of class C 1,α such that the exterior of Ω is connected. Let a ∈]0, +∞[. Let (f, g) belong to C 1,α (∂Ω) × C 0,α (∂Ω). Then the system of integral equations ⎧ on ∂Ω , ⎨ VΩ [θ] − a(VΩ [ψ] + c) = f 1 1 t t θ + W [θ] + ψ − W [ψ] = g on ∂Ω , (11.50) Ω Ω 2 ⎩ 2 ψ dσ = 0 , ∂Ω has one and only one solution (θ, ψ, c) ∈ C 0,α (∂Ω) × C 0,α (∂Ω) × R. Proof. We first assume that problem (11.50) does have a solution (θ, ψ, c) in C 0,α (∂Ω) × C 0,α (∂Ω) × R, and we prove that such a solution must necessarily be delivered by a certain formula. By the first equation, we must have VΩ [θa−1 − ψ] − c = f a−1 .

(11.51)

We claim that θa−1 − ψ and c are uniquely determined by f a−1 . Ultimately, this will be a consequence of Theorem 6.47, which implies that f a−1 can be written in a unique way as the sum of a constant function and a single layer potential with density that has zero integral on ∂Ω. We cannot, however, use directly Theorem 6.47,  because the integral ∂Ω θa−1 − ψ dσ might be different from zero. Then we ob0,α ˜ ˜ serve that Lemma 11.14  implies that there exists φ ∈ C (∂Ω) such that VΩ [φ] is constant on ∂Ω and ∂Ω φ˜ dσ = 1. Thus there exists a unique a ˜ ∈ R such that  θa−1 − ψ − a ˜φ˜ dσ = 0 (11.52) ∂Ω

and we have

˜ + VΩ [φ]˜ ˜ a − c = f a−1 . ˜φ] VΩ [θa−1 − ψ − a

(11.53)

By (11.52), we have 

θa−1 − ψ dσ =

a ˜= ∂Ω



θ dσa−1 .

(11.54)

∂Ω

By Theorem 6.47, there exists a unique pair (ω, d) ∈ C 0,α (∂Ω)0 × R such that VΩ [ω] + d = f a−1 .

(11.55)

11.3 A Nonlinear Transmission Problem

459

By (11.52) and (11.53), we must necessarily have ˜ , θ = a(ψ + ω + a ˜φ)

˜ a − d, c = VΩ [φ]˜

(11.56)

with a ˜ as in (11.54). Then the second equation of (11.50) implies that $   % 1 1˜ 1 t t t ˜ (a + 1)ψ + (a − 1)WΩ [ψ] = g − a ω + WΩ [ω] + ˜ . φ + WΩ [φ] a 2 2 2 (11.57) By integrating the last equation on ∂Ω, by invoking Lemma 6.11, and by equalities    ˜ ω dσ = 0 , ψ dσ = 0 , φ dσ = 1 , ∂Ω

∂Ω

∂Ω



we obtain

g dσ − a˜ a,

0= ∂Ω

and thus a ˜=a

−1

 g dσ .

(11.58)

∂Ω

By Theorem 6.51, and by inequality −1
0 such that 0 0 0 ∂ϕf 0 0 0 ≤ Aj+1 . ∀l ∈ {1, . . . , n − 1} 0 ∂xl 0 j,α C (Bn−1 (xo ,r)) for all f ∈ Fm,α . Hence, we have ϕf C j+1,α (Bn−1 (xo ,r)) ≤

sup

|ϕf | +

Bn−1 (xo ,r) o

0 n−1  l=1

0 0 ∂ϕf 0 0 0 0 ∂xl 0

C j,α (Bn−1 (xo ,r))

≤ |y | + δ + (n − 1)Aj+1 ≡ M(r,δ),j+1 ,

for all f ∈ Fm,α (cf. (2.32)). Hence, the constant M(r,δ),j does exist for all j in {1, . . . , m} and thus in particular for j = m. Accordingly, we can take M(r,δ) ≡   M(r,δ),m and the proof is complete. Then we can readily deduce the following simplified form of Corollary A.21. Corollary A.22. Let n ∈ N \ {0, 1}. Let m ∈ N \ {0}, α ∈ [0, 1]. Let ι, σ ∈]0, +∞[, ι < σ. Let Ω be a Whitney regular bounded open subset of Rn−1 × R. Let (xo , y o ) belong to Ω. In case (m, α) = (1, 0) we also introduce an increasing function ω1,0 from [0, +∞[ to itself such that ω1,0 (t) > 0 for t ∈]0, +∞[ and lim ω1,0 (t) = 0 .

t→0+

Let Fm,α be defined as in Definition A.20. Let δ ∗ ∈]0, +∞[. Then there exist r1 ∈ ]0, +∞[, δ ∈]0, δ ∗ [, M ∈]0, +∞[ such that Bn−1 (xo , r) × [y o − δ, y o + δ] ⊆ Ω

∀r ∈]0, r1 ]

and such that for each (f, r) ∈ Fm,α ×]0, r1 ] there exists one and only one function ϕf ∈ C m,α (Bn−1 (xo , r)) such that (A.16)–(A.18) hold true and ϕf C m,α (Bn−1 (xo ,r)) ≤ M

∀(f, r) ∈ Fm,α ×]0, r1 ] .

(A.30)

Appendix A

633

Proof. By Corollary A.21 (i), the set S is not empty. Possibly shrinking first δ and then r, we can see that there exist r1 ∈]0, +∞[ and δ < δ ∗ such that (r, δ) ∈ S for all r ∈]0, r1 ]. Then Corollary A.21 (ii) implies the existence and uniqueness of ϕf and by applying Corollary A.21 (ii) with the pair (r1 , δ), we deduce the existence of M such that ϕf C m,α (Bn−1 (xo ,r1 )) ≤ M for all f ∈ Fm,α . Since   ϕf C m,α (Bn−1 (xo ,r)) ≤ ϕf C m,α (Bn−1 (xo ,r1 )) , inequality (A.30) holds true.

A.7 Composition Operators Let Ω be a bounded open subset of Rn . Let Ω1 be an open subset of R. Let G ∈ C 0 (Ω × Ω1 ). We denote by NG the (nonautonomous) composition operator from C 0 (Ω, Ω1 ) to C 0 (Ω) which maps v ∈ C 0 (Ω, Ω1 ) to the function NG [v] defined by NG [v](x) ≡ G(x, v(x))

∀x ∈ Ω .

Here the letter ‘N ’ stands for Nemytskii operator. Similarly, if G belongs to C 0 (∂Ω × Ω1 ), we denote by NG the (nonautonomous) composition operator from C 0 (∂Ω, Ω1 ) to C 0 (∂Ω) which maps v ∈ C 0 (∂Ω, Ω1 ) to the function NG [v] defined by ∀x ∈ ∂Ω . NG [v](x) ≡ G(x, v(x)) If G is sufficiently regular, then NG is known to map a Schauder space C m,α to itself. Here we mention the work of Berkolajko, Bondarenko, Rutitskij, Zabrejko, Dr´abek, and of Sobolevskij mainly for case m = 0, as reported in the extensive monograph of Appell and Zabrejko [14]. We need the following result on (nonautonomous) composition operators. Theorem A.23. Let n ∈ N \ {0, 1}. Let α ∈]0, 1]. Let Ω be a bounded open connected subset of Rn of class C 1,α . Let Ω1 be an open subset of R. Let m ∈ {0, 1}. (i) If G is an analytic function from Ω × Ω1 to R, then the composition operator NG is analytic from C m,α (Ω, Ω1 ) to C m,α (Ω) and the Fr´echet differential of NG at a point u ˜ ∈ C m,α (Ω, Ω1 ) is delivered by the formula dNG (˜ u)[v](x) = Gu (x, u ˜(x))v(x)

∀x ∈ Ω ,

for all v ∈ C m,α (Ω), where Gu denotes the partial derivative of G with respect to its last variable. (ii) If G is an analytic function from ∂Ω × Ω1 to R, then the composition operator NG is analytic from C m,α (∂Ω, Ω1 ) to C m,α (∂Ω), and the Fr´echet differential ˜ ∈ C m,α (∂Ω, Ω1 ) is delivered by the formula of NG at a point u dNG (˜ u)[v](x) = Gu (x, u ˜(x))v(x)

∀x ∈ ∂Ω ,

for all v ∈ C m,α (∂Ω), where Gu denotes the partial derivative of G with respect to its last variable.

634

Appendix A

In the above statement, the assumption that G is an analytic function from Ω × Ω1 to R means that there exist an open neighborhood W of Ω × Ω1 in Rn × R and an ˜ ˜ from W to R such that the restriction G ˜ analytic function G |Ω×Ω1 of G to Ω × Ω1 coincides with G. Similarly, we understand that G being analytic on ∂Ω × Ω1 means that G has a real analytic extension on an open neighborhood of ∂Ω × Ω1 in Rn × R. In the case where G does not depend on its first variable x—the so-called autonomous case—the proof of statement (i) can be found in B¨ohme and Tomi [28, p. 10] and Henry [122, pp. 29, 54]. For the proof of statement (i) with G that depends also on x—the so-called nonautonomous case—we refer to Valent [269, Theorem 5.2]. Statement (ii) can be proved by the same argument of the proof of statement (i). For a treatment of the continuity and differentiability of autonomous composition operators in Banach algebras with applications to function spaces, we refer to [158, 159]. We also need the following elementary lemma. Lemma A.24. Let n ∈ N \ {0}. Let α ∈]0, 1]. Let D be a compact subset of Rn . Let G be a locally Lipschitz continuous function from D × R to R. Then the composition operator NG from C 0,α (D) to itself defined by NG [v](x) ≡ G(x, v(x))

∀x ∈ D

for all v ∈ C 0,α (D) maps bounded subsets of C 0,α (D) to bounded subsets of C 0,α (D). Proof. Let A be a bounded subset of C 0,α (D). Let a ∈]0, +∞[ be such that |v| + 1 ≤ a

∀v ∈ A .

Since G is Lipschitz continuous on the compact set D × [−a, a], the statement follows by the elementary inequality |G(x, v(x)) − G(y, v(y))| ≤ Lip (G|D×[−a,a] )(|x − y| + |v(x) − v(y)|) ≤ Lip (G|D×[−a,a] )[(diam (D))1−α + |v : D|α ]|x − y|α for all v ∈ A.

∀x, y ∈ D ,  

In order to consider the dependence upon both the functions of a composition, we need an extension of the composition operator of Theorem A.23. Then we introduce the following (slightly restated) result of Preciso [247, Prop. 1.1, p. 101] (see also Preciso [246, Proposition 4.2.16]), which also involves the notion of Roumieu class (cf. (2.7)). Proposition A.25. Let n1 , n2 ∈ N \ {0}, ρ ∈]0, +∞[. Let Ω1 be a bounded open subset of Rn1 . Let Ω2 be a bounded open connected subset of Rn2 of class C 1 . Then 0 (Ω1 ) × C 1,α (Ω2 , Ω1 ) to C 1,α (Ω2 ) defined the composition operator T from Cω,ρ by 0 (Ω1 ) × C 1,α (Ω2 , Ω1 ) , T [u, v] ≡ u ◦ v ∀(u, v) ∈ Cω,ρ

Appendix A

635

is real analytic. For other results on the differentiability of the composition operator T in Schauder spaces, we refer to Valent [269], and to [157–159].

A.8 Integral Operators with Real Analytic Kernel In the book we need several times to prove real analyticity results for integral operators with real analytic kernels. The first time it happens is in the proof of Proposition 8.2, where we introduce the family of operators V˜ i () which take a density θo ∈ C 0,α (∂Ω o ) to the function  V˜ i ()[θo ](ξ) ≡ Sn (ξ − y)θo (y) dσy ∀ξ ∈ ∂Ω i (A.31) ∂Ω o

and we want to show that the map that takes  ∈] − 0 , 0 [ to V˜ i () ∈ L(C 0,α (∂Ω o ) , C 1,α (∂Ω i )) is real analytic. As we shall see, a result of this kind can be deduced from a proposition proven in [175] in a more general setting (cf. Theorem A.27 below, see also Corollary A.29). For the sake of completeness and to illustrate the kind of arguments that are involved, we provide a direct proof of the following Lemma A.26. Lemma A.26. Let n ∈ N \ {0, 1}. Let α ∈]0, 1[. Let Ω i and Ω o be bounded open connected subsets of Rn of class C 1,α , with connected exteriors Ω i− and Ω o− , and such that the origin 0 of Rn belongs both to Ω i and to Ω o . Assume that 0 ∈]0, +∞[ and that Ω i ⊆ Ω o for all  ∈] − 0 , 0 [. Then the map from ] − 0 , 0 [ to L(C 0,α (∂Ω o ) , C 1,α (∂Ω i )) that takes  to the linear operator V˜ i () defined in (A.31) is real analytic. Proof. We fix 1 ∈]0, 0 [. By the continuity of the multiplication by a scalar in Rn and by the compactness of the set [−1 , 1 ] × Ω i , the set ' ( x : (, x) ∈ [−1 , 1 ] × Ω i = ∪∈[−1 ,1 ] Ω i is a compact subset of the open set Ω o . Accordingly, we have   ∪∈[−1 ,1 ] Ω i ∩ ∂Ω o = ∅ . Then

  dist ∪∈[−1 ,1 ] Ω i , ∂Ω o > δ.

for some positive δ. Hence,

636

Appendix A

  dist Ω i , ∂Ω o > δ for all  ∈ [−1 , 1 ]. In particular,   dist ∂Ω i , ∂Ω o > δ for all  ∈ [−1 , 1 ]. Then, if we set (∂Ω i )δ/(41 ) ≡ {x ∈ Rn : dist(x, ∂Ω i ) < δ/(41 )} and (∂Ω o )δ/4 ≡ {x ∈ Rn : dist(x, ∂Ω o ) < δ/4}, the triangular inequality implies that   dist (∂Ω i )δ/(41 ) , (∂Ω o )δ/4 > δ/2 for all  ∈] − 1 , 1 [. In particular, ξ − y = 0

∀(, ξ, y) ∈] − 1 , 1 [×(∂Ω i )δ/(41 ) × (∂Ω o )δ/4 .

Since the fundamental solution Sn is real analytic on Rn \ {0}, and since the map from ] − 1 , 1 [×(∂Ω i )δ/(41 ) × (∂Ω o )δ/4 to Rn \ {0} that takes (, ξ, y) to ξ − y is real analytic, it follows that the map from ] − 1 , 1 [×(∂Ω i )δ/(41 ) × (∂Ω o )δ/4 to R that takes (, ξ, y) to Sn (ξ − y) is real analytic. Hence, for each (# , ξ# , y# ) in ] − 1 , 1 [ × (∂Ω i )δ/(41 ) × (∂Ω o )δ/4 there exist real numbers C# > 0, M# > 0 n and a# j,β,γ for all j ∈ N and β, γ ∈ N , such that    #  j+|β|+|γ| aj,β,γ  ≤ C# M#

(A.32)

and such that Sn (ξ − y) =

∞ 



j β γ a# j,β,γ ( − # ) (ξ − ξ# ) (y − y# )

l=0 (j,β,γ)∈N×Nn ×Nn ,|(j,β,γ)|=l

for all (, ξ, y) in the open neighborhood ]# − 1/M# , # + 1/M# [×Bn (ξ# , 1/M# ) × Bn (y# , 1/M# ) of (# , ξ# , y# ) (cf. Theorem 2.17). Then, by setting  β γ a# a# j (ξ, y) ≡ j,β,γ (ξ − ξ# ) (y − y# ) (β,γ)∈Nn ×Nn

for all (ξ, y) ∈ Bn (ξ# , 1/M# ) × Bn (y# , 1/M# ) and all j ∈ N, we see that

(A.33)

Appendix A

637

Sn (ξ − y) =

∞ 

j a# j (ξ, y)( − # )

j=0

for | − # | < 1/M# . A computation based on (A.32) shows that     #  |β| |γ| aj,β,γ  |ξ − ξ# | |y − y# | (β,γ)∈Nn ×Nn



j ≤ C # M#

(M# |ξ − ξ# |)

|β|

(M# |y − y# |)

|γ|

.

(β,γ)∈Nn ×Nn

Since M# |ξ − ξ# | and M# |y − y# | are smaller than one when (ξ, y) belongs to Bn (ξ# , 1/M# ) × Bn (y# , 1/M# ), we deduce that the series on the right-hand side converges on Bn (ξ# , 1/M# ) × Bn (y# , 1/M# ). Accordingly, the functions a# j are real analytic on Bn (ξ# , 1/M# ) × Bn (y# , 1/M# ) (cf. (A.33)). Moreover, we have 



∂ξβ ∂yγ a# j (ξ, y) =



a# j,β,γ

(β,γ)∈Nn ×Nn , β≥β  ,γ≥γ 

  β! γ! (ξ − ξ# )β−β (y − y# )γ−γ (β − β  )! (γ − γ  )!

for all β  , γ  ∈ Nn and all (ξ, y) ∈ Bn (ξ# , 1/M# ) × Bn (y# , 1/M# ). By inequality (A.32) we deduce that       β γ # ∂ξ ∂y aj (ξ, y)  β! γ! j+|β  |+|γ  | β−β  γ−γ  ≤ C # M# η# z#   (β − β )! (γ − γ )! n n   (β,γ)∈N ×N , β≥β ,γ≥γ





j+|β |+|γ |

= C # M#



β∈Nn , β≥β 

β! β−β  η#  (β − β )!

 γ∈Nn , γ≥γ 

γ! γ−γ  z#  (γ − γ )! (A.34)

for all (ξ, y) ∈ Bn (ξ# , 1/M# ) × Bn (y# , 1/M# ), where η# and z# are the vectors of Rn defined by η# ≡ (M# |ξ − ξ# |, . . . , M# |ξ − ξ# |) and z# ≡ (M# |y − y# |, . . . , M# |y − y# |) . Since |η#,l | < 1 for all l ∈ {1, . . . , n}, a computation based on the equality  β∈Nn

shows that

β η# =

n 8 l=1

1 (1 − η#,l )

638

 β∈Nn , β≥β 

β! β−β  η# =  (β − β )! =

 ∂ηβ



l=1

n 8 l=1

n 8

1 (1 − ηl )

βl ! (1 − η#,l )

βl +1

Appendix A

 |η=η#

=

β! (1 − M# |ξ − ξ# |)|β  |+n

(cf. John [130, Chap. 3]). Similarly, we verify that  γ∈Nn , γ≥γ 

γ! γ! γ−γ  z = . # (γ − γ  )! (1 − M# |y − y# |)|γ  |+n

Then, by inequality (A.34) we deduce that      β γ #  ∂ξ ∂y aj (ξ, y) j+|β  |+|γ  |

≤ C # M#

β! γ!  (1 − M# |ξ − ξ# |)|β |+n (1 − M# |y − y# |)|γ  |+n

for all (ξ, y) ∈ Bn (ξ# , 1/M# ) × Bn (y# , 1/M# ). In particular, if m belongs to {0, . . . , m} and |β  |+|γ  | = m , (ξ, y) ∈ Bn (ξ# , 1/(2M# )) × Bn (y# , 1/(2M# )), we have        β γ # j m M# . ∂ξ ∂y aj (ξ, y) ≤ C# m !22n+m M# We conclude that, for each m ∈ N, there exists C#,m > 0 such that j a# j C m (Bn (ξ# ,1/(2M# ))×Bn (y# ,1/(2M# ))) < C#,m M#

∀j ∈ N .

Then we consider a finite cover of the compact subset (∂Ω i )δ/(51 ) × (∂Ω o )δ/5 of (∂Ω i )δ/(41 ) × (∂Ω o )δ/4 consisting of open sets (k)

(k)

(k)

(k)

Bn (ξ# , 1/(2M# )) × Bn (y# , 1/(2M# )), with k ∈ {1, . . . , κ}, constructed as Bn (ξ# , 1/(2M# )) × Bn (y# , 1/(2M# )) above. For each index k in #,(k) {1, . . . , κ} we have functions aj with j ∈ N which are analytic from the set (k)

(k)

(k)

(k)

Bn (ξ# , 1/M# ) × Bn (y# , 1/M# ) to R and such that #,(k)

aj

  (k) (k) (k) (k) C m Bn (ξ# ,1/(2M# ))×Bn (y# ,1/(2M# ))



for some C#,m,k > 0. Moreover, Sn (ξ − y)

(k)

< C#,m,k (M# )j

∀j ∈ N

Appendix A

=

∞ 

639 #,(k)

aj

(k)

(ξ, y)( − # )j

(k)

(k)

(k)

∀(ξ, y) ∈ Bn (ξ# , 1/M# ) × Bn (y# , 1/M# )

j=0 #,(h)

and thus a standard inductive argument shows that each two functions aj

and

#,(k) aj

with h, k ∈ {1, . . . , κ} must coincide in the intersection of their domains. Namely, we have #,(k) #,(h) (ξ, y) = aj (ξ, y) aj for all (ξ, y) in   (k) (k) (k) (k) Bn (ξ# , 1/M# ) × Bn (y# , 1/M# )   (h) (h) (h) (h) ∩ Bn (ξ# , 1/M# ) × Bn (y# , 1/M# )

∀j ∈ N

(the equality being vacuously true when the intersection is empty). Accordingly, for #,(k) each j ∈ N we can glue together the functions aj with k ∈ {1, . . . , κ} and # i o define a function Aj from (∂Ω )δ/(51 ) × (∂Ω )δ/5 to R such that #,(k)

A# j (ξ, y) = aj

(ξ, y)

for all (ξ, y) in     (k) (k) (k) (k) Bn (ξ# , 1/(2M# )) × Bn (y# , 1/(2M# )) ∩ (∂Ω i )δ/(51 ) × (∂Ω o )δ/5 and for all k ∈ {1, . . . , κ}. Then we can verify that there exist Cm > 0 and M > 0 such that A# j C m ((∂Ω i )δ/(5

1)

×(∂Ω o )δ/5 )

< Cm M j

∀j ∈ N .

Moreover, we have Sn (ξ − y) =

∞ 

j A# j (ξ, y)( − # )

j=0

for all  ∈]# − 1/M, # + 1/M [ and all(ξ, y) ∈ (∂Ω i )δ/(51 ) × (∂Ω o )δ/5 . It i o defined by follows that the map from ] − 1 , 1 [ to C m ∂Ωδ/(5 × ∂Ωδ/5 1)   →

 (ξ, y) → Sn (ξ − y)

(A.35)

is real analytic. Then, by exploiting the classical theorem  on differentiation under  integral sign we verify that the map from the space C m (∂Ω i )δ/(51 ) × (∂Ω o )δ/5    to the space L C 0,α (∂Ω o ) , C m (∂Ω i )δ/(51 ) defined by

640

Appendix A

 h →



 θ →

h(·, y)θ(y) dσy

(A.36)

∂Ω o

is linear and bounded (it would be possible to consider a larger space for θ, e.g., L∞ (∂Ω o ), but we do not need it for our proof). Hence, by taking m ≥ 2, by considering the composition of the maps in (A.35) and (A.36), and by the boundedness of the restriction operator from C m ((∂Ω i )δ/(51 ) ) to C 1,α (∂Ω i ), we deduce that the operator V˜ i defined by (A.31) is real analytic from ] − 1 , 1 [ to L(C 0,α (∂Ω o ) , C 1,α (∂Ω i )) for all 1 ∈]0, 0 [. Thus, it is real analytic from the interval ] − 0 , 0 [ to L(C 0,α (∂Ω o ) , C 1,α (∂Ω i )) and the validity of the lemma is proved.   We now introduce a more general result of [175, Prop. 4.1], which we do not prove here. Theorem A.27. Let h1 , h2 ∈ N \ {0}. Let m ∈ N, α ∈]0, 1]. Let Y be a topological space. Let M be a σ-algebra of parts of Y containing the Borel sets of Y . Let μ be measure on M. Let Z be a Banach space. Let W be a nonempty open subset of Rh1 × Rh2 × Z. Let G be a real analytic map from W to R. Then the following statements hold. (i) Let n ∈ N \ {0}. Let Ω1 be a bounded open connected subset of Rn . Let Ω1 be regular in the sense of Whitney. Let  ˜ F ≡ (Ψ, φ, z) ∈ C m,α (Ω1 , Rh1 ) × Cb0 (Y, Rh2 ) × Z :  Ψ (Ω1 ) × φ(Y ) × {z} ⊆ W . ˜ G from F˜ × L1 (Y ) to C m,α (Ω1 ) defined by Then the map H  ˜ HG [Ψ, φ, z, f ](x) ≡ G(Ψ (x), φ(y), z)f (y) dμy

∀x ∈ Ω1

Y

for all (Ψ, φ, z, f ) ∈ F˜ × L1 (Y ), is real analytic. (ii) Let n ∈ N \ {0, 1}. Let Ω1 be a bounded open connected subset of Rn of class C max{1,m},α . Let   F ≡ (ψ, φ, z) ∈ C m,α (∂Ω1 , Rh1 ) × Cb0 (Y, Rh2 ) × Z :  ψ(∂Ω1 ) × φ(Y ) × {z} ⊆ W .  from F  × L1 (Y ) to C m,α (∂Ω1 ) defined by Then the map HG   HG [ψ, φ, z, f ](x) ≡ G(ψ(x), φ(y), z)f (y) dμy ∀x ∈ ∂Ω1 , Y

Appendix A

641

for all (ψ, φ, z, f ) ∈ F  × L1 (Y ), is real analytic. We now show that Theorem A.27 implies the validity of the following statement, which we actually exploit a number of times in this book. Theorem A.28. Let n ∈ N \ {0, 1}. Let α ∈]0, 1]. Let Ω be a bounded open subset of Rn of class C 1,α . (i) Let Ω1 be a bounded open subset of Rn . Let W be an open subset of Rn . Let J1 , J2 be open intervals of R such that

1 x − 2 y : 1 ∈ J1 , 2 ∈ J2 , x ∈ Ω1 , y ∈ ∂Ω ⊆ W . If G is an analytic function from W to R and m ∈ N, then the map from J1 × J2 × L1 (∂Ω) to C m,α (Ω1 ) that takes (1 , 2 , θ) to the function  G(1 x − 2 y)θ(y) dσy ∀x ∈ Ω1 ∂Ω

is real analytic. (ii) Let Ω1 be a bounded open subset of Rn of class C 1,α . Let W be an open subset of Rn . Let J1 , J2 be open intervals of R such that {1 x − 2 y : 1 ∈ J1 , 2 ∈ J2 , x ∈ ∂Ω1 , y ∈ ∂Ω} ⊆ W . If G is an analytic function from W to R, then the map from J1 × J2 × L1 (∂Ω) to C 1,α (∂Ω1 ) that takes (1 , 2 , θ) to the function  G(1 x − 2 y)θ(y) dσy ∀x ∈ ∂Ω1 ∂Ω

is real analytic. (iii) Let Ω1 be a bounded open subset of Rn of class C 1,α . Let h1 ∈ N \ {0}. Let m ∈ {0, 1}. Let W be an open subset of Rh1 × R × R. Let G be an analytic function from W to R. Then the map from the set

A ≡ (ψ, ) ∈ C m,α (∂Ω1 , Rh1 ) × R : ψ(∂Ω1 ) × [0, 1] × {} ⊆ W to C m,α (∂Ω1 ) that takes (ψ, ) to the function 

1

G(ψ(x), β, ) dβ

∀x ∈ ∂Ω1

0

is analytic. (iv) Let Ω1 be a bounded open subset of Rn . Let m ∈ N. Let W be an open subset of Rn × R × R. Let G be an analytic function from W to R. Then the map from the set

J ≡  ∈ R : Ω1 × [0, 1] × {} ⊆ W to C m,α (Ω1 ) that takes  to the function

642

Appendix A



1

∀x ∈ Ω1

G(x, β, ) dβ 0

is analytic. Proof. In order to prove statement (i) it suffices to show that if (˜ 1 , ˜2 ) ∈ J1 × J2 , then there exists η ∈]0, +∞[ such that B1 (˜ 1 , η) × B1 (˜ 2 , η) ⊆ J1 × J2 , and such that the map of statement (i) is analytic on B1 (˜ 1 , η)×B1 (˜ 2 , η)×L1 (∂Ω). To shorten our notation, we set J˜1 ≡ B1 (˜ 1 , η)

J˜2 ≡ B1 (˜ 2 , η) .

Since {˜ 1 } × {˜ 2 } × Ω1 × ∂Ω is compact and contained in the open subset {(1 , 2 , x, y) ∈ R2+2n : 1 x − 2 y ⊆ W } of R2+2n , then there exists η ∈]0, +∞[ such that {1 x − 2 y : 1 ∈ J˜1 , 2 ∈ J˜2 , x ∈ Ω1 + Bn (0, 2η), y ∈ (∂Ω) + Bn (0, 2η)} ⊆ W . ˜1 of Rn of class C ∞ such By Lemma 2.70, there exists a bounded open subset Ω that ˜1 ⊆ Ω1 + Bn (0, η) , ˜1 ⊆ Ω Ω1 ⊆ Ω and thus we have ' ( ˜1 , y ∈ ∂Ω ⊆ W . 1 x − 2 y : 1 ∈ J˜1 , 2 ∈ J˜2 , x ∈ Ω ˜ be the map from W ˜ ≡ {(ξ, η) ∈ Rn × Rn : ξ − η ∈ W } defined by Let G ˜ η) ≡ G(ξ − η) G(ξ,

˜ . ∀(ξ, η) ∈ W

˜1 have pairwise disjoint closures Since the finitely many connected components of Ω (cf. Lemma 2.38) and are regular in the sense of Whitney and Ω is of class C 1,α , Theorem A.27 (i) implies that the map from ( '   ˜1 ) × C 0 (∂Ω) : Ψ Ω ˜1 − φ(∂Ω) ⊆ W , F˜ ≡ (Ψ, φ) ∈ C m,α (Ω ˜1 ) defined by to C m,α (Ω  ˜ ˜ [Ψ, φ, θ](x) ≡ H G

G(Ψ (x) − φ(y))θ(y) dσy

˜1 ∀x ∈ Ω

∂Ω

for all (Ψ, φ, θ) ∈ F˜ × L1 (∂Ω) is analytic. Since the map from J˜1 × J˜2 × L1 (∂Ω) to F˜ × L1 (∂Ω) that takes (1 , 2 , θ) to (1 IΩ˜ , 2 I∂Ω , θ) is analytic and the restric1

Appendix A

643

tion is linear and continuous in Schauder spaces, then the composite function that ˜ ˜ [1 I , 2 I∂Ω , θ] takes (1 , 2 , θ) to the restriction H ˜1 G |Ω1 is analytic, i.e., the map Ω 1 ˜ ˜ of statement (i) is analytic on J1 × J2 × L (∂Ω). The proof of statement (ii) follows the lines of the proof of the analyticity of the map of statement (i) on J˜1 × J˜2 × L1 (∂Ω) by exploiting Theorem A.27 (ii) instead ˜1 because we of Theorem A.27 (i). In this case we do not need to introduce the set Ω know that the finitely many connected components of Ω1 satisfy the assumptions of Theorem A.27 (ii) (cf. Lemma 2.38). (iii) Since Ω1 has finitely many connected components of class C 1,α (cf. Lemma 2.38), Theorem A.27 (ii) implies that the map from the set   F ≡ (ψ, φ, ) ∈ C m,α (∂Ω1 , Rh1 ) × C 0 ([0, 1]) × R :  ψ(∂Ω1 ) × φ([0, 1]) × {} ⊆ W , to C m,α (∂Ω1 ) defined by   HG [ψ, φ, , θ](x) ≡

G(ψ(x), φ(β), )θ(β) dβ

∀x ∈ ∂Ω1

[0,1]

for all (ψ, φ, , θ) ∈ F  × L1 ([0, 1]) is analytic. Since the map from the set A to F  × L1 ([0, 1]) that takes (ψ, ) to the quadruple (ψ, I[0,1] , , 1) is analytic, then the 1  [ψ, I[0,1] , , 1] = 0 G(ψ(·), β, ) dβ is composite function that takes (ψ, ) to HG analytic, i.e., the map of statement (iii) is analytic. (iv) It suffices to show that if ˜1 ∈ J, then there exists η ∈]0, +∞[ such that B1 (˜ 1 , η) ⊆ J and such that the map of statement (iv) is analytic on B1 (˜ 1 , η). To shorten our notation, we set J˜1 ≡ B1 (˜ 1 , η) . Since the compact set Ω1 × [0, 1] × {˜ 1 } is contained in the open set W , then there exists η ∈]0, +∞[ such that (Ω1 + Bn (0, 2η)) × ([0, 1] + B1 (0, 2η)) × B1 (˜ 1 , 2η) ⊆ W . ˜1 of Rn of class C ∞ such that By Lemma 2.70, there exists an open subset Ω ˜1 ⊆ Ω1 + Bn (0, η) , ˜1 ⊆ Ω Ω1 ⊆ Ω and thus we have ˜1 + Bn (0, η)) × [0, 1] × B1 (˜ (Ω 1 , η) ⊆ W .

644

Appendix A

˜1 have pairwise disjoint closures Since the finitely many connected components of Ω (cf. Lemma 2.38) and are regular in the sense of Whitney and Ω is of class C 1,α , Theorem A.27 (i) implies that the map from F˜1

( '   ˜1 ) × C 0 ([0, 1]) × R : Ψ Ω ˜1 × φ([0, 1]) × {} ⊆ W ≡ (Ψ, φ, ) ∈ C m,α (Ω

˜1 ) defined by to C m,α (Ω  ˜ ˜ [Ψ, φ, , θ](x) ≡ H G,1

G(Ψ (x), φ(β), )θ(β) dβ

˜1 ∀x ∈ Ω

[0,1]

for all (Ψ, φ, , θ) ∈ F˜1 ×L1 (∂Ω) is analytic. Since the map from J˜1 to F˜1 ×L1 (∂Ω) that takes  to (IΩ˜ , I[0,1] , , 1) is analytic and the restriction is linear and continu1 ous in Schauder spaces, then the composite function that takes  to the restriction ˜ ˜ [I , I[0,1] , , 1] ˜ H ˜1 G,1 Ω |Ω1 is analytic, i.e., the map of statement (iv) is analytic on J1 .   In addition, we have the following corollary of Theorem A.28 (ii). Corollary A.29. Let n ∈ N \ {0, 1}. Let α ∈]0, 1]. Let Ω, Ω1 be bounded open subsets of Rn of class C 1,α . Let W be an open subset of Rn . Let J1 , J2 be open intervals of R such that {1 x − 2 y : 1 ∈ J1 , 2 ∈ J2 , x ∈ ∂Ω1 , y ∈ ∂Ω} ⊆ W . Let G be an analytic function from W to R. For all (1 , 2 ) ∈ J1 × J2 , let H(1 , 2 ) be the linear operator that takes θ ∈ L1 (∂Ω) to the function  G(1 x − 2 y)θ(y) dσy ∀x ∈ ∂Ω1 . ∂Ω

Then the map from J1 × J2 to L(L1 (∂Ω) , C 1,α (∂Ω1 )) that takes (1 , 2 ) to H(1 , 2 ) is real analytic. Proof. By Theorem A.28 (ii), the map H from J1 × J2 × L1 (∂Ω) to C 1,α (∂Ω1 ) that takes (1 , 2 , θ) to H(1 , 2 , θ) ≡ H(1 , 2 )[θ] is real analytic. Since H is linear and continuous with respect to the variable θ, we have H(1 , 2 ) = dθ H(1 , 2 , θ)

∀(1 , 2 , θ) ∈ J1 × J2 × L1 (∂Ω) .

Since the right-hand side equals a partial Fr`echet differential of an analytic map, the right-hand side is analytic. Hence H is analytic on J1 × J2 × L1 (∂Ω). Since H(·, ·) is independent of its last variable, we conclude that H is analytic on J1 × J2 .   For the case of a linear integral operator in Roumieu spaces, we prove the following.

Appendix A

645

Proposition A.30. Let n ∈ N \ {0}. Let Ω, Ω1 be bounded open subsets of Rn . Let W be an open subset of Rn which contains the set

Ω1 − Ω = x − y : x ∈ Ω1 , y ∈ Ω . Let G be an analytic map from W to R. Then there exists ρ0 ∈]0, +∞[ such that the 0 (Ω1 ) that takes f to the function linear operator from L1 (Ω) to Cω,ρ 0  G(x − y)f (y) dy ∀x ∈ Ω1 Ω

is continuous. Proof. Since Ω1 −Ω is a compact subset of W and G is analytic in W , the analiticity criterion of Theorem 2.17 implies that there exist ρ0 , M ∈]0, +∞[ such that sup ξ∈Ω1 −Ω

|Dβ G(ξ)| ≤ M

|β|! |β| ρ0

∀β ∈ Nn .

Then we have    |β|  β  ρ0  sup sup Dx G(x − y)f (y) dy  n β∈N |β|! x∈Ω1 Ω   |β|   ρ ≤ sup 0 sup  Dβ G(x − y)f (y) dy  β∈Nn |β|! x∈Ω1 Ω |β|

= sup β∈Nn

ρ0 |β|!

sup ξ∈Ω1 −Ω

|Dβ G(ξ)|f L1 (Ω) ≤ M f L1 (Ω)

∀f ∈ L1 (Ω) ,

and thus the statement follows by the definition of norm in Roumieu spaces.

 

A.9 Sard’s Theorem Theorem A.31 (of Sard). Let n ∈ N. Let Ω be an open subset of Rn . If f is a continuously differentiable function from Ω to R, then the Lebesgue measure of the set f ({x ∈ Ω : ∇f (x) = 0}) is equal to zero. For a proof, we refer for example to Br¨ocker and J¨anich [36, (6.1), p. 56].

646

Appendix A

A.10 Theorem of Invariance of Domain Theorem A.32 (of Invariance of Domain). Let n ∈ N. Let Ω be an open subset of Rn . If a function f from Ω to Rn is continuous and injective, then f maps open subsets of Ω to open subsets of Rn . For a proof, we refer for example to Deimling [87, Theorem 4.3].

A.11 Mollifiers Let n ∈ N \ {0}. Let η ∈ D(Rn ) be such that  supp η ⊆ Bn (0, 1) ,

η ≥ 0,

η dx = 1 . Rn

In order to prove the existence of a function η as above, we consider the function h from Rn to [0, +∞[ defined by ! 1 − 1−|x| 2 if |x| < 1 , h(x) ≡ e 0 if |x| ≥ 1 . It is well-known that h ∈ Cc∞ (Rn ). Then we set η(x) ≡  and we also have

 Rn

h(x) h(y) dy Rn

∀x ∈ Rn ,

η dx = 1. Next we set η (x) ≡ −n η(x/)

∀x ∈ Rn

for all  ∈]0, +∞[. Then the family {η }∈]0,+∞[ is said to be a family of standard mollifiers. Clearly,  η ≥ 0 , η dx = 1 ∀ ∈]0, +∞[ supp η ⊆ Bn (0, ) , Rn

and we have the following known result. Proposition A.33. Let n ∈ N \ {0}. Let {η }∈]0,+∞[ be a family of standard mollifiers. Let f be an essentially bounded and measurable function from Rn to R. Let U be an open subset of Rn . If f is continuous on U , then lim η ∗ f = f

→0

uniformly on the compact subsets of U . For a proof, we refer for example to Folland [103, Theorem 8.14].

Appendix A

647

A.12 The Partition of Unity Lemma A.34 (of the Partition of Unity). Let n ∈ N \ {0}. Let {Vλ }λ∈Λ be a family of open subsets of Rn such that V ≡ λ∈Λ Vλ is not empty. Then there exists a sequence {θj }j∈N of functions of Cc∞ (V ) such that (i) θj ≥ 0 for all j ∈ N. (ii) If j ∈ N, then there exists λ(j) ∈ Λ such that supp θj ⊆ Vλ(j) . (iii) The family of sets {supp θj }j∈N is locally finite. Namely, if x ∈ V , then there exists a neighborhood Wx of x in V such that Wx ∩ supp θj = ∅ holds true for at most a finite number of j ∈ N. (Equivalently, if K is a compact subset of V , we have K ∩ supp θj = ∅ for at most a finite number of j ∈ N.) +∞ (iv) j=0 θj (x) = 1 for all x ∈ V . For a proof, we refer for example to Mitrea [214, Theorem 13.34]. The sequence {θj }j∈N is said to be a partition of unity subordinate to the open cover {Vλ }λ∈Λ of V . We also note that V ⊆ ∪j∈N supp θj ⊆ ∪j∈N Vλ(j) ⊆ V . Then we have the following immediate consequence of the lemma of the partition of unity in the case of compact subsets of Rn . Lemma A.35 (of the Partition of Unity on a Compact Set). Let n ∈ N \ {0}. Let K be a compact subset of Rn . Let {Vl }kl=1 be a finite family of open subsets of Rn k such that K ⊆ l=1 Vl . Then there exist ϕ1 ∈ D(V1 ), . . . , ϕk ∈ D(Vk ) with ϕl ≥ 0 for all l ∈ {1, . . . , k} such that k 

ϕl (x) = 1

∀x ∈ K .

l=1

Proof. Let {θj }j∈N be a partition of unity subordinate to the cover {Vl }kl=1 of k Vl . Let m ∈ N such that K ∩ supp θj = ∅ for all j > m. Then we l=1 be m ∞ have j=0 θj (x) = j=0 θj (x) = 1 for all x ∈ K. Next we define inductively the sets {Λl }kl=1 as follows Λ1 ≡ {j ∈ {0, . . . , m} : supp θj ⊆ V1 } , Λl+1 ≡ {j ∈ {0, . . . , m} : supp θj ⊆ Vl+1 } \

l 

Λs

if l < k ,

s=1

and we set ϕl ≡

 j∈Λl

θj for all l ∈ {1, . . . , k}.

 

Then we can deduce the validity of the following classical lemma on the existence of a cut off function.

648

Appendix A

Lemma A.36 (of the Cut Off Function). Let Ω be an open subset of Rn . Let K be a compact subset of Ω. Then there exists ϕ ∈ D(Ω) such that ϕ = 1 on K. Proof. It suffices to set V1 ≡ Ω and to apply the previous lemma.

 

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Index

Symbols A0λ (R), 225 A1λ (R), 227 BX (p, r), 16 Cc∞ (K), 20 Cc∞ (Rn ), 20 C 0,α (D), 27 Cb0,α (D), 27 C 0,ω(·) (D), 25 0,ω(·) Cb (D), 25 0,ω(·) Cloc (D), 25 C 0 (U, V ), 12 0 Cω,ρ (Ω), 24 Cb0 (U, Y ), 12 0 Cq,ω,ρ (Rn ), 516 ω C (Ω, Y ), 623 Cqk,β (S[ΩQ ]− ), 491 Cqk,β (S[ΩQ ]), 491 Cqk (S[ΩQ ]− ), 491 Cqk (S[ΩQ ]), 491 C m,0 , 59 m,α Cloc (Ω), 59 m,α (Ω), 59 Cloc C m (Ω), 19 C m (Ω), 19 C m (Ω, R), 19 Cbm (Ω), 19 DX , 11 D∂Ω , 167 Df , 19 E2πiq−1 z , 485 Ga , 485 L(X, Y ), 15

L1loc (Ω), 21 Lp μ (Z), 13 Pq,n [ΩQ , f ], 517 Q, 484 Rq,n , 488 Sq,n , 488 VΩ [φ], 183 t Wq,Ω [φ], 498 Q t WΩ [φ], 184 WΩ [ψ], 165, 183 X0 , 13 Y X , 11 Γ (1/2), 15 Γ (α), 14  · C 0,ω(·) (D) , 25 b

 · L(X,Y ) , 16  · p , 13  · L(2) (X×Y,Z) , 621  · L(k) (X;Y ) , 621  · Km,α , 135 − , 13 A δx0 , 21 ≡, 11 λΩ (x, y), 29 ˚, 12 U μ-a.a., 13 μ-a.e., 13 μm (α, β), 63 ∇, 19 ωn , 15 κ + , 36 κ − , 37 |f : D|ω(·) , 25 c Ω,λ , 53

© Springer Nature Switzerland AG 2021 M. Dalla Riva et al., Singularly Perturbed Boundary Value Problems, https://doi.org/10.1007/978-3-030-76259-9

667

668 c Ω,λ , 53 cΩ,λ , 53 c[Ω], 30 c∗ Ω , 234 cΩ,α , 82 d2 f , 621 dk f , 621 f (−1) , 12 mn , 14 ms , 14 mM , 14 pK,m , 20 q, 484 sn , 15 + vq,Ω [φ], 494 Q

− vq,Ω [φ], 494 Q vΩ [φ], 151 + vΩ [φ], 156 − [φ], 156 vΩ + [ψ], 497 wq,Ω Q

− [ψ], 497 wq,Ω Q U + , 12 U − , 12 Wq,ΩQ [ψ], 498 B± n (0, ρ), 75 Bn , 13 Bn (x, ρ), 13 C, 11 K, 11 Mn (K), 12 N, 11 On (K), 12 R, 11 S[ΩQ ], 490 S[ΩQ ]− , 490 Z, 11 Z(a), 485 D  (Ω), 22 D(K), 20 D(Ω), 20  K0,1;e,0 , 234  , 234 K0,1 Km,α;o , 135 Km,α , 135 L(X, Y ), 16 L(2) (X; Y ), 621 L(2) (X × Y, Z), 621 L(k) (X; Y ), 621 (k) Ls (X; Y ), 622 LM , 14 Ln , 14 P[k, ϕ], 225

Index S  (Rn ), 22 S(Rn ), 22 Im T , 16 Ker T , 16 Lip(f ), 27 Lip (D), 27 diam (U ), 11 sgn, 12 cof(A), 425 vq,ΩQ [φ], 494 + wΩ [ψ], 163 − [ψ], 163 wΩ wΩ [ψ], 160 wq,ΩQ [ψ], 496 A affine tangent space to a set at a point, 39 algebraic direct complement, 17 direct sum, 17 supplement, 17 analytic function between normed spaces, 623 between normed spaces at a point, 622 in an open subset of Rn , 23 area element, 14, 50 B ball, 13 bilinear and continuous maps between normed spaces, 621 form, 176 boundary, 11 bounded linear operator, 16 C chain rule for functions between normed spaces, 620 closure, 11 compact perturbation of a Fredholm operator, 176 map, 18 composition of functions in a Schauder spaces, 63 operator, 633 cone of semi-tangent vectors, 39 connected components of a set of class C 0 , 36 continuity of the pointwise product in H¨older spaces, 28 of the pointwise product in Schauder spaces, 60

Index coordinate cylinder, 35 cover of a set, 617 D decreasing function, 12 derivative in the sense of distributions, 22 diagonal, 11 diameter, 11 diffeomorphism of class C k or C ω , 623 differentiable function between normed spaces, 620 differential manifold of class C m,α , 99 manifold of class C 1 , 14 of a bilinear form between normed spaces, 621 of a function between normed spaces, 620 of order k of a function between normed spaces, 621 Dini regularization, 625 Dirac measure, 21 Dirichlet problem definition of the exterior problem, 180 definition of the interior problem, 180 existence of the solution for the interior problem, 195 existence of the solution for the exterior problem, 201 existence of the solution of the periodic problem, 507 in a domain with a small hole, 262 interior problem for the Poisson equation in Schauder spaces, 257 uniqueness for the interior problem, 114 uniqueness of the solution for the exterior problem, 181 uniquess of the solution of the periodic problem, 503 distribution, definition of, 21 Divergence Theorem, 124 double layer potential behavior at infinity, 160 compactness, 186 definition, 160 jump formula, 162 Schauder regularity, 163 dual system, 177 E effective conductivity, 546 embedding of H¨older spaces, 27 of Schauder spaces, 59 Euler Gamma function, 14

669 extension of Schauder functions in the boundary of an open set, 102 of Schauder functions in the closure of an open set, 76 exterior, 12 F first Green Identity, 126 in exterior domains, 129 Fredholm Alternative, 176 Alternative in a dual system, 177 index, 176 operator, 175 function (strictly) decreasing, 12 (strictly) increasing, 12 of class C k between normed spaces, 622 positively homogeneous, 134 which represents a boundary as a graph in a coordinate cylinder, 35 G Gamma function, 14 geodesic distance, 29 gradient, 19 H H¨older constant, 25, 27 continuous, 25 exponent, 27 space, 25 spaces are complete, 26 homeomorphism, 14 I identity map, 12 Identity Principle, 24 image of a linear operator, 16 Implicit Function Theorem, of Dini, 625 increasing function, 12 inductive topology, 616 interior, 12 J Jacobian matrix, 19 jump formula for the double layer potential, 162 for the normal derivative of a single layer potential, 158 for the partial derivatives of a single layer potential, 157

670 K kernel of a linear operator, 16 of an integral operator, 134, 225 Kronecker symbol, 12 L Lebesgue number of an open cover, 617 space, 13 lemma of the uniform cylinders, 67 linear topology, 15 Lipschitz continuous function, 27 local constraint, 67 Lipschitz homeomorphism, 32 strict hypograph of class C 0 , 36 strict hypograph of class C m,α , 65 locally constant function, 12 convex topology, 20 integrable, 21 longitudinal flow, 526 M measure of the unit ball, 15 of the unit sphere, 15 Mixed problem existence of the solution, 211 for the Poisson equation in a perforated domain, 347 in a domain with a small hole, 338 uniqueness of the solution, 182 N Nemytskii operator, 633 Neumann problem definition of the exterior problem, 180 definition of the interior problem, 180 existence of the solution, 206 existence of the solution the periodic problem, 507 interior problem for the Poisson equation in Schauder spaces, 258 uniqueness up to constants of the solution, 181 uniquess up to constants of the solution the periodic problem, 503 Newtonian potential of a distribution of C −1,α (Ω), 253 of a function of C 0,α (Ω), 224

Index partial derivatives, 229 Schauder regularity, 250 Schauder regularity in case of a negative exponent, 251 Schauder regularity of the first order partial derivatives, 249 non-ideal transmission problem, 544 norm on a quotient, 16 normally convergent series, 15 null space of a linear operator, 16 O Open Mapping Theorem, 16 orthogonal matrix, 12 P parametrization, 14 for a manifold, 100 in a coordinate cylinder, 36 of Ω around p, 72 partial derivative of a Newtonian potential, 229 of a volume potential, 228 of a volume potential with a differentiable density, 229 partition of unity on a compact set, 647 on an open set, 647 periodic analog of the fundamental solution, 485 boundary value problems, 503 domains, 491 double layer potential, 496 layer potentials, 490 Newtonian potential, 517 single layer potential, 494 volume potential, 517 periodic analog of the fundamental solution of the Laplace operator, 488 points to the exterior, 44 the interior, 43 positively homogeneous function, 134 projection definition of, 17 of X onto X1 along X2 , 17 punctured neighborhood, 118 Q quasilinear heat transmission problem, 589 quotient topology, 16, 616

Index R Rademacher Theorem, 620 Theorem for Lipschitz sets, 50 range of a linear operator, 16 rapidly decreasing function, 22 real analytic function between normed spaces, 622 regular in the sense of Whitney, 30 Robin interior nonlinear problem for the Laplace equation, 434 interior problem for the Poisson equation in Schauder spaces, 260 Robin problem with linear conditions, 217 with nonlinear conditions in a domain with a small hole, 434 Roumieu class, 24 regularity of the volume potential, 254 S Schauder regularity of the double layer potential, 163 of the first order partial derivatives of the Newtonian potential, 249 of the Newtonian potential, 250 of the Newtonian potential in case of a negative exponent, 251 of the single layer potential, 158 Schauder space on the boundary of an open subset of Rn , 99 with negative exponent, 108 Schwartz space, 22 second differential of a function between normed spaces, 621 second Green Identity, 126 in exterior domains, 130 semi-tangent vector, 38 set of class C m,α , 73 in the form with the diffeomorphisms, 72 in the form with the local constraints, 67 set of class C m , 73 single layer potential behavior at infinity, 156 continuity, 154 definition, 151 jump formula for the normal derivative, 158 jump formula for the partial derivatives, 157 Schauder regularity, 158 singular kernel, 225

671 singularly perturbed periodic domains, 514 solution in the sense of distributions, 22 Steklov eigenvalue problem, 350 problem on a domain with a small hole, 351 strict hypograph, 35 strictly decreasing function, 12 increasing function, 12 strong singularity of a kernel, 225 sup-norm, 12 support of a complex valued function, 12 T tangent space to a set at a point, 39 tangential derivative, 105 gradient, 104 Taylor series of a function between normed spaces, 623 tempered distribution, 22 test function, 20 theorem of the homomorphism, 615 thermal resistance, 544 third Green Identity, 126 in exterior domains, 130 topological direct complement, 18 direct sum, 18 supplement, 18 trace theorem for Schauder spaces, 102 transmission problem with linear boundary conditions, 214 with nonlinear conditions in a domain with a small hole, 450 transpose operator with respect to a dual system, 177 two-phase dilute composite, 544 U uniform cylinders, 67 V Vitali Convergence Theorem, 619 volume potential, 227 partial derivatives, 228 regularity in Roumieu classes, 254 Schauder regularity in case the first order partial derivatives of the kernel have a strong singularity, 248 with a differentiable density: partial derivatives, 229

672 W weak derivative, 22 solution, 22

Index weakly singular kernel, 225 Whitney constant, 30 regular in the sense of, 30