Dual Variational Approach to Nonlinear Diffusion Equations 3031245822, 9783031245824

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Progress in Nonlinear Differential Equations and Their Applications PNLDE Subseries in Control 102

Gabriela Marinoschi

Dual Variational Approach to Nonlinear Diffusion Equations

Progress in Nonlinear Differential Equations and Their Applications

PNLDE Subseries in Control Volume 102

Series Editor Jean-Michel Coron, University of Paris-Sorbonne, Paris, France Editorial Board Members Viorel Barbu, Alexandru Ioan Cuza University, Ia¸si, Romania Piermarco Cannarsa, University of Rome “Tor Vergata”, Rome, Italy Karl Kunisch, University of Graz, Graz, Austria Gilles Lebeau, University of Nice Sophia Antipolis, Nice, Paris, France Tatsien Li, Fudan University, Shanghai, China Shige Peng, Shandong University, Jinan, China Eduardo Sontag, Northeastern University, Boston, Massachusetts, USA Enrique Zuazua, Autonomous University of Madrid, Madrid, Spain

The PNLDE Subseries in Controls is dedicated to the publication of cutting edge developments from the intersection of control theory with the analysis of differential equations. The subseries aims to include research monographs, graduate level textbooks, polished notes arising from lectures and seminars, as well as refereed proceedings of focused conferences. Topics from across the spectrum, ranging from the theoretical to the applied, will be included in the subseries.

Gabriela Marinoschi

Dual Variational Approach to Nonlinear Diffusion Equations

Gabriela Marinoschi Bucuresti, Romania

ISSN 1421-1750 ISSN 2374-0280 (electronic) Progress in Nonlinear Differential Equations and Their Applications ISSN 2731-7366 ISSN 2731-7374 (electronic) PNLDE Subseries in Control ISBN 978-3-031-24582-4 ISBN 978-3-031-24583-1 (eBook) https://doi.org/10.1007/978-3-031-24583-1 Mathematics Subject Classification: 35B80, 35K55, 35Q93, 47H06, 47J30, 49J40 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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 book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The aim of this book is to present a dual variational formulation of solutions to nonlinear diffusion equations with general nonlinearities as null minimizers of appropriate energy functionals. This formulation issued from physics and initially established as a principle by Brezis and Ekeland in 1976 was continued in some other authors’ works after the 1980s and was systematically treated and developed in a unifying framework by Ghoussoub in a monograph in 2009. Our purpose is to show how this method turns out to be a convenient tool for proving the existence of the solutions to nonlinear diffusion equations when other methods may fail. This is the case of evolution equations with nonautonomous operators, with low regular data, or with singular diffusion coefficients. In all these situations which are difficultly treated or may be not treated by other methods, this formulation, essentially employing the dual Legendre–Fenchel relations, proves to be an elegant and versatile technique involving knowledge from the theory of maximal monotone operators, semigroup theory, convex functions, and their subdifferentials. Besides its effectiveness in proving the existence of solutions to nonlinear partial differential equations, this technique is useful in the treatment of inverse problems and optimal control problems which are approached in this book as well. The dual variational formulation is illustrated in the book by diffusion equations, with general nonlinearities provided by potentials having various stronger or weaker properties, which actually can represent mathematical models associated to various real physical processes. Each chapter develops recipes of how to treat particular important cases by developing different arguments in the proofs. This book can serve as a text for doctoral and post-doctoral students, as well as for researchers. It assumes knowledge of functional analysis, operator theory, and convex analysis but it is self-contained. I am grateful to Mrs. Elena Mocanu for the Latex final processing of this book. Bucuresti, Romania

Gabriela Marinoschi

vii

Introduction

This monograph is aimed at showing how the dual variational formulation of nonlinear diffusion equations as minimization problems for appropriate energy functionals can be used to prove the existence of the solutions to these equations and to solve associated optimal control problems. The fact that the solutions of many equations with nonlinear, or even linear but non-self-adjoint operators are viewed as critical points of appropriate associated functionals is a tradition in physics. A deeper investigation into the origins and evolution of these ideas arising from physics is found in the extensive Ghoussoub’s monograph [48]. The minima of these functionals turn out actually to be the solutions searched to the partial differential equations, not because they are critical points, but also because they have the property of being the null minimizers of these functionals. This principle revealed by physicists in an original intuitive form was rigorously stated about 50 years ago by some mathematicians and we refer first to two papers written by Brezis and Ekeland (see [29, 30]). They formulated a minimum principle for certain evolution equations in Hilbert spaces, proved in particular for the classical heat equation. Since this principle was extended to the Banach spaces too, it is generically presented below as follows. Let us consider the infinite dimensional Cauchy problem in a Banach space X, dy dt (t) + Ay(t)

= f (t), a.e. t ∈ (0, T ) y(0) = y0

(CP )

which may be the operatorial form associated to a system of equations with initial and boundary conditions. In problem (CP ), A : X → X ' is a nonlinear operator ' of a potential type, X ' is the dual of X, y0 ∈ X, and f ∈ Lp (0, T ; X' ), with 1 ≤ ' p < ∞. This means that there exists a proper, convex, lower semicontinuous (l.s.c.) function ϕ : X → R ∪ {+∞} such that A = ∂ϕ, where ∂ϕ is the subdifferential of ϕ. Generally, A can be a multivalued operator and then (CP ) becomes a differential inclusion, the sign “=” being replaced by “e”. To this problem one associates the functional

ix

x

Introduction

) ( ( ) dy 1 ∗ J0 (y) = − X,X' dt + ||y(T )||2X ϕ(y) + ϕ f − dt 2 0 (J0 ) involving the function ϕ and its conjugate ϕ ∗ , and formulates the minimization problem (

T

Min {J0 (y); y ∈ K},

(P0 )

where { dy ' K = y ∈ Lp (0, T ; X); ∈ Lp (0, T ; X' ), ϕ(y) ∈ L1 (0, T ), )dt ( } dy ∗ ϕ f− ∈ L1 (0, T ), y(0) = y0 , dt and p, p ' are conjugate numbers, 1/p + 1/p ' = 1. Here, dy dt is taken in the sense of X' -valued distributions on (0, T ), and X,X' is the pairing between the dual spaces X and X' . The conjugate function ϕ ∗ : X' → R ∪ {+∞} is defined by ϕ ∗ (w) = sup (X,X' − ϕ(u)), w ∈ X' . u∈X

It was conjectured that a null minimizer of J0 , that is a minimizer which is also a zero of J0 is a solution to (CP ). The ingredient which connects these two formulations is the system of the two duality relations (see [44]), called later the Legendre–Fenchel relations, which establishes the following two facts: ϕ(u) + ϕ ∗ (w) − X,X' ≥ 0, for all u ∈ X, w ∈ X' ,

(LF 1)

ϕ(u) + ϕ ∗ (w) − X,X' = 0, if and only if w ∈ ∂ϕ(u).

(LF 2)

By the second relation, the vanishing of the left-hand side happens only if the dual variable w belongs to the subdifferential of ϕ, which generally is multivalued. Now, it is obvious that if y is a solution to (CP ), then it is a solution to (P0 ), and more exactly a solution which minimizes J0 at zero. The converse part of this assertion, that is, a solution to (P0 ) which minimizes J0 to zero is a solution to (CP ) is that challenging one. A general recipe for proving this implication does not exist, it rather depends on the particularities of the potential of the nonlinearity arising in the diffusion term and on the good choice of the functional. Nayroles presented in [63] an analogous result, by considering a Cauchy problem and a periodic problem for an elastoplastic material with constitutive equation of Maxwell type. The solutions to these problems were identified as the minima of certain functionals defined on a Hilbert space, and a theorem asserting that the minima exist was stated. Later, in [6] and [7], the min-max method was

Introduction

xi

used to identify the value of the infimum and to establish existence and uniqueness under suitable growth conditions on the convex potential. These previous results were continued in a series of articles by Ghoussoub and co-authors, among which we cite the papers [47, 49]. The topic was exhaustively developed in the monograph [48], in which the author offered a unifying framework for this theory and gave a variant of the Brezis–Ekeland principle able to prove global existence and uniqueness of solutions for several basic first-order and nonlinear evolution equations, by introducing the concept of self-duality of variational problems. The self-dual vector fields introduced in this monograph are viewed as natural extensions of gradients of convex energies, and hence of self-adjoint positive operators, which usually drive dissipative systems, but also provide representations for the superposition of such gradients with skew-symmetric operators, which normally generate conservative flows. Examples from heat equation, porous media, transport equations, Navier–Stokes equations, and equations with the operator A not derived from a potential were studied. We refer the reader to the paper [76] in which minimum principles for quasi-linear linear transport equations in the steady and nonstationary case are studied, as well as to the further applications based on the Brezis–Ekeland principle presented in [15, 40, 61, 62, 74, 75]. We also recall the paper [78], where evolution equations with (possibly non-cyclicallymonotone) maximal monotone operators are reformulated as a null-minimization principle and the extensions of the theory based on Fitzpatrick’s results developed in [79] and [80]. More recently, papers [22] and [27] focused on how selfdual variational calculus leads to variational solutions of various stochastic partial differential equations driven by monotone vector fields. In the first part of this work, we address especially the existence of solutions to nonlinear diffusion equations on a bounded domain Ω ⊂ RN , ∂y (t, x) − ∆β(t, x, y(t, x)) = f (t, x), (t, x) ∈ Q := (0, T ) × Ω, ∂t

(NDE)

with initial and boundary conditions and with a general time and space depending nonlinearity β(t, x, ·) provided by a potential j (t, x, ·). Here, β can be a continuous function of y, or a discontinuous one, situation in which the jumps are filled in, completing thus β(t, x, ·) to a multivalued maximal monotone operator. The purpose is to show, by a variational formulation involving the duality Legendre– Fenchel relations, that the solution of the nonlinear problem is given by the null minimizer of an appropriate minimization problem for a convex functional involving the potential j (t, x, ·) and its conjugate j ∗ (t, x, ·). Depending on the properties of the potential, this purpose could be achieved more easier or not, in some cases being necessary certain additional conditional results. As example, the minimization problem may be

xii

Introduction

{

(

Minimize J (y, w) = (y,w)∈U

Q

(j (t, x, y(t, x)) + j ∗ (t, x, w(t, x)))dxdt } ( y(t, x)w(t, x)dxdt , −

(P )

Q

subject to ∂y − ∆w = f, on Q, ∂t

(EQ)

with boundary and initial conditions, where U, the admissible set, is appropriately defined, in correspondence with the properties of the potential j : Q × R → R ∪ {+∞}. For instance, '

U = {(y, w) ∈ Lp (Q) × Lp (Q); j (·, ·, y(·, ·)), j ∗ (·, ·, w(·, ·)) ∈ L1 (Q), (y, w) satisfies (EQ)}, if p ∈ (1, ∞). Cases when p = 1, p = ∞ or a special case (y, w) ∈ L1 (Q) × L1 (Q) arising from mathematical models of real processes are treated as well. This monograph relies on partial results obtained in the author’s papers regarding the duality approach of nonlinear diffusion equations, completed with some new results. It is structured in six chapters based on the papers [20, 26, 38, 58–60], respectively, with studies corresponding to various particular properties of β. Each chapter begins with the setup of the appropriate functional framework and the introduction of the minimization problem for a functional J (y, w), generically denoted (P ). This can be viewed as an optimal control problem in which the controller w acts on the state y such that to minimize the cost J (y, w). That is why the proofs proceed with the steps for solving optimal control problems. These include the proof of the existence in the state system, the proof of the existence of at least a solution to the minimization problem, possibly completed by the uniqueness in some “good” cases or under supplementary assumptions and continues with the characterization of the necessary conditions that the optimal control together with the optimal state should satisfy. In some cases, this part requires the study of intermediate approximating problems. In some situations it is not possible to directly prove that the minimizer of the chosen J is a null minimizer. However, this may seem to be a good candidate for the solution to the nonlinear equation, reason for which it can be viewed as a generalized or variational solution. Further, if (P ) is modified by including restrictions in the admissible set for the control w or the state y, then the generalized solution which minimizes the functional to zero turns out to be quite a weak solution in some sense to the nonlinear equation. Also, a helpful technique is a regularization of β and j by the Yosida and Moreau approximations, respectively, followed by the proof of a priori estimates and a passing to the limit technique. This kind of minimization problems is relevant in several problems occurring in applied sciences and physics, since equation (N DE) with a particular β can model

Introduction

xiii

various physical phenomena. Steady-state or dynamic nonlinear diffusion processes in fluids, heat transfer or population dynamics are some main examples. A domain where this problem has a particular relevance is the image denoising variational technique. The first three chapters are devoted to the duality approach of nonlinear diffusion equations (NDE) with the potential j corresponding to three types of nonlinearities. Thus, in Chap. 1 it is investigated the situation in which j has a polynomial growth with the exponent m ≥ 2. This is the defining property for the so-called slow diffusion in fluids, whose representative is the porous media equation. This can be considered the most regular situation in which the fact that y, a null minimizer in (P ), is the unique solution to (CP ) can be proved without other additional ' assumptions, in the duality Lm -Lm . In Section 1.2 there is treated a singular (N DE) modeling a free boundary value problem with fast diffusion (m ∈ (1, 2)), arising from a process infiltration in porous media with the formation of two phases separated by an free interface. This is solved by technical proofs in the duality L∞ (L∞ )' and β turns out to be a measure. In Chap. 2, the functions j and j ∗ have less regular properties, e.g., both of them behave such that j (t, x, r)/|r| → +∞ as |r| → +∞. We call this behavior weakly coercive. These properties are relevant for potentials used in image processing and in models with fast diffusion potentials. Two cases are treated by completely different arguments, namely the case when j and j ∗ depend on t and x and the case when they depend only on t, in the duality L1 -L∞ . Chapter 3 approaches the study of the existence of solutions to nonlinear diffusion equations with a time-dependent nonlinearity whose potential is not coercive, of the form β(t, x, r) = a(t, x)sign(r −yc ), where sign is the multivalued function signum. This case requires very technical proofs in the duality (L∞ )' -L∞ , here the state y turning out to be a measure. The relevant example is related to the model of self-organized criticality (SOC) a paradoxical model in physics, also applied to other processes in biology, social sciences, economics. Chapter 4 focuses on the well-posedness of a nonlinear divergence parabolic equation yt − ∇ · β(t, x, ∇y) e f, with dynamic boundary conditions of Wentzell type. Here, the diffusive term β is derived from a potential j which depends on the gradient of the solution. The existence and uniqueness of a strong solution is obtained as the limit of a finite difference scheme in the time-dependent case solved by a duality approach, and via a semigroup approach combined with a duality approach in the time-invariant case. Besides the mathematical relevance for proving the well-posedness of nonlinear partial differential equations, this approach is useful in the study of inverse problems, optimal control, and optimization of processes. Thus, Chap. 5 presents an optimal control problem related to image denoising and restoration techniques, formulated with the purpose of denoising an initial blurred image. This is solved via some minimization problems for functionals involving appropriate conjugate functions j and j ∗ and the Legendre–Fenchel duality relations.

xiv

Introduction

The last Chap. 6 deals with an optimal control problem for a system governed by a transition phase model. The optimal control problem requires the control of a sharp interface between the two phases developing in the transition process modeled by the Penrose–Fife equations. The sharp interface is represented by the multivalued Heaviside function, and the use of the duality method relying on the Legendre– Fenchel relations perfectly fits this case. In these two last situations, as well as in more general optimal control problems governed by nonlinear partial differential equations, by this procedure, the original problem is transformed into an optimal control problem with a linear state equation which considerably simplifies the proof of the existence of minimizers and especially the determination of the first-order conditions of optimality. At the end, there is an appendix including some notation, definitions, and results of functional analysis, operator theory, and convex analysis often used in the book. We want to point out some important aspects of this duality technique based on Legendre–Fenchel relations, and as a matter of fact to underline the benefit of this method which provides elegant proofs for the existence of the solutions to nonlinear diffusion equations, under general assumptions. The first one is that it provides a general framework, in which solutions of a large class of partial differential equations can be identified as the minima of appropriate energy functionals, when the partial differential equations can be written as (CP ) with the operator A derived from a potential. In this case, a maximal monotone operator is involved, so that this theory brings together notions from three fundamental domains, namely maximal monotone operators, semigroups of contractions, convex functions, and their subdifferentials. A second aspect is that this approach is extremely useful because it allows the proof of the existence in some cases in which standard methods do not apply. More exactly, it can be applied to nonautonomous operators, that is time-depending A(t) and when the equation is accompanied by low (time) regular data f , y0 . In these cases, the classical semigroup approach cannot be applied, since the problems are not autonomous. Moreover, for a general nonlinearity with an insufficient regularity in time, the hypotheses required in [39] for proving the solution existence to nonautonomous evolution equations may be not fulfilled, and this theory cannot be followed in this case. We also refer, for example, to equations which may have a strong nonlinearity induced by a singular diffusion coefficient of Dirac type. Such problems cannot be treated by the Lions’ existence theorem in the variational nonautonomous case (see [56]), due to the lack of coercivity and again the saving way is the duality approach. Not lastly, this method enables a simpler numerical treatment of the equations by replacing the direct computation of the solution to (possibly singular) partial differential equations by the computation of the minimum of a convex functional with a linear associated state system. As (P ) is a convex optimization problem, the entire machinery of convex analysis may be involved to get efficient approximation algorithms.

Contents

1 Nonlinear Diffusion Equations with Slow and Fast Diffusion . . . . . . . . . . 1.1 A Continuous Potential with a Polynomial Growth . . . . . . . . . . . . . . . . . . . 1.1.1 Functional Framework and Preliminaries . . . . . . . . . . . . . . . . . . . . . 1.1.2 The Minimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Main Results for the Slow Diffusion Case . . . . . . . . . . . . . . . . . . . . 1.2 A Free Boundary Problem for a Special Fast Diffusion Case. . . . . . . . .

1 2 5 7 9 19

2

Weakly Coercive Nonlinear Diffusion Equations. . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Problem Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Functional Setting and the Minimization Problem . . . . . . . . . . . . . . . . . . . . 2.3 A Time and Space Dependent Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 A Time Dependent Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 34 36 39 50

3

Nonlinear Diffusion Equations with a Noncoercive Potential . . . . . . . . . . 3.1 Problem Statement and the Functional Framework . . . . . . . . . . . . . . . . . . . 3.2 The Minimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Main Results for the Noncoercive Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Existence in a Self-organized Criticality (SOC) Model . . . . . . . . . . . . . . .

67 67 70 72 81

4

Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Problem Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Strongly Coercive Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Weakly Coercive Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The Semigroup Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 The Obstacle Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 The Total Variation Wentzell Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 86 89 105 113 119 120

5

A Nonlinear Control Problem in Image Denoising . . . . . . . . . . . . . . . . . . . . . . . 123 5.1 Problem Presentation and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.2 Existence for the Minimization Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

xv

xvi

Contents

5.3 Approximating Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Optimality Conditions for the Approximating Problem . . . . . . 5.3.2 Numerical Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 A Potential with a Polynomial Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133 138 142 144

6

An Optimal Control Problem for a Phase Transition Model . . . . . . . . . . . 6.1 Presentation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Existence in the State System and Control Problem . . . . . . . . . . . . . . . . . . 6.3 Approximating Control Problem (Pε ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 The Second Approximating Control Problem (Pε,σ ) . . . . . . . . . 6.4.2 Optimality Conditions for (Pε,σ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Optimality Conditions for (Pε ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147 147 150 162 166 166 169 185

7

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Lp Spaces, Sobolev Spaces, and Vectorial Spaces . . . . . . . . . . . . . . . . . . . . 7.2 Operators in Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Convex Functions and Subdifferential Mappings . . . . . . . . . . . . . . . . . . . . .

189 190 196 200

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Acronyms

N R RN ||·||N |·| x·y ∂h ∇h, ∂x , hx N Σ ∇ ·v =

j =1

the set of all positive numbers the set of real numbers, the one-dimensional real space the N-dimensional Euclidian space the norm in RN the norm in the one-dimensional real space the dot product of vectors x, y ∈ RN the gradient of the function h ∂vj ∂xj

||·||X X' (x, y)X Ω f A∗ Lp (Ω) L∞ (Ω) ||·||p , ||·||∞ (·, ·)m,m' , m,m' Lp (0, T ; X) L∞ (0, T ; X) C([a, b]; X) W k,p (Ω) H k (Ω) H01 (Ω) AC([0, T ]; X) W 1,p (0, T ; X)

the divergence of the vector v the norm in a linear normed space X the dual of the normed space X the scalar product in the Hilbert space X an open bounded subset of RN the boundary ∂Ω of Ω the adjoint of the linear operator A the space of p-summable functions on Ω the space of essentially bounded functions on Ω the norm in the spaces Lp (Ω), L∞ (Ω) ' the pairing between Lm (Ω) and Lm (Ω), m1 + m1' = 1, m ∈ [1, ∞] the space of all p-summable functions from (0, T ) to the Banach space X the space of all essentially bounded functions from (0, T ) to X the space of all continuous functions from [a, b] to X Sobolev spaces, 1 ≤ p ≤ ∞, k ≥ 1 W k,2 (Ω) the Sobolev space {u ∈ H 1 (Ω); u|f = 0} the space of absolutely continuous functions from [a, b] to X the space {u ∈ AC([0, T ]; X); du/dt ∈ Lp (0, T ; X)} xvii

xviii

∂ϕ IK NK (u)

Acronyms

the subdifferential of the proper convex function ϕ : X → R = R ∪ {+∞} the indicator function of the set K the normal cone to K at u

Chapter 1

Nonlinear Diffusion Equations with Slow and Fast Diffusion

The first chapter aims at providing existence results for time-dependent nonlinear diffusion equations of the form .

∂y (t, x) − ∆β(t, x, y(t, x)) e f (t, x), ∂t

with initial and boundary conditions, by following a variational principle relying on the Legendre–Fenchel duality relations, in the case when the function .β(t, x, ·) is provided by a continuous potential .j (t, x, ·) having a polynomial growth. Generally, the function .β may be multivalued, fact that explains the symbol .e in the equation. This equation models various processes of diffusion of a physical quantity of density .y(t, x). The physical quantity having an initial non-homogeneous distribution, delivered by a source f or via the boundary conditions, tends in time to become uniform in the whole space domain by diffusing from the parts where it has a higher density to those with a lower density level. Typical examples include the diffusion of a substance in air or in other fluids, as for instance contaminants in surface waters, diffusion in porous materials like soils, heat transfer processes, and population dynamic diffusion. The function y stands for the concentration of the substance in the fluid, the moisture of the soil, the medium temperature, or the population density, respectively. This type of equation is also found in phase transitions or image processing. If .β(t, x, y) is differentiable with respect to .y, its derivative denoted .β ' (t, x, y) represents exactly the diffusion coefficient or the diffusivity. Generally, it is a nonnegative function. Its vanishing at a point induces a degenerate equation which is a singular case in the sense that the solution has less regularity. If .β ' is strictly positive, the equation is nondegenerate and the solution has more regularity. As previously said, here j is endowed with a growth of polynomial type with respect to the solution, with the exponent .m. The case when j has a polynomial growth with the exponent .m ≥ 2 characterizes a slow diffusion, the corresponding equation being the so-called porous media equation, while the case when .m ∈ (1, 2) © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Marinoschi, Dual Variational Approach to Nonlinear Diffusion Equations, PNLDE Subseries in Control 102, https://doi.org/10.1007/978-3-031-24583-1_1

1

2

1 Nonlinear Diffusion Equations with Slow and Fast Diffusion

is associated with a fast diffusion (see, e.g., [57]). The function j and its conjugate j ∗ enjoy the Legendre–Fenchel duality relations. The technique used to prove the existence consists in reducing the nonlinear problem to a convex optimization problem via the Legendre–Fenchel relations. A functional setting is appropriately ' chosen in the duality .Lm -.Lm in order to set up the minimization problem and to allow the proof of the equivalence between the nonlinear problem and the optimization one. In the case when the potential related to the diffusivity function is continuous and has a polynomial growth with .m ≥ 2, we shall prove that the optimization problem is equivalent to the original diffusion equation, meaning that a solution to one of them is the solution to the other. To this end, a series of preliminary results will be provided, namely the minimization problem has a solution (Theorem 1.4), the minimum is zero, and the minimizer is the unique solution to the diffusion equation (Theorem 1.5). This is the difficult implication, the converse one being obvious. This method allows the proof of the solution to degenerate or nondegenerate equations, with time dependent parameters and with initial data and nonhomogeneous terms belonging to more abstract spaces. The second case investigates a situation characterized by a singular diffusion coefficient .β ' (t, x, y), which blows up at a finite value .ys of the solution y. More exactly, this has a growth with the exponent .m − 2 for .m ∈ (1, 2) corresponding to a fast diffusion. Moreover, .β and its potential j have special properties that model a free boundary problem emerged from a real situation in water infiltration in soils. This describes a typical process of saturated–unsaturated water infiltration in porous media with the formation and advance of an interface between the saturated part (in which the medium pores are completely filled with water) and the unsaturated part. Theorems 1.8 and 1.10 show that the minimization problem with a constraint, introduced for this case in a more abstract framework of spaces of measures, involving the duality .L∞ -.(L∞ )' , has a solution which can be viewed as a generalized solution to the diffusion equation.

.

1.1 A Continuous Potential with a Polynomial Growth Let .Ω be an open bounded subset of .RN , .N ≥ 1, with the boundary .f sufficiently smooth, and let T be finite. We consider the nonlinear diffusion equation .

∂y (t, x) − ∆β(t, x, y(t, x)) e f, for (t, x) ∈ Q := (0, T ) × Ω, ∂t

(1.1)

with the initial condition y(0, x) = y0 , in Ω,

.

and boundary conditions of Robin type

(1.2)

1.1 A Continuous Potential with a Polynomial Growth

.

∂β(t, x, y) + αβ(t, x, y) e 0, on Σ = (0, T ) × f, ∂ν

3

(1.3)

or of Dirichlet type β(t, x, y) = 0, on Σ.

.

(1.4)

∂ The notation .ν stands for the unit outward normal to the boundary .f, . ∂ν represents the normal derivative to .f, and .α is a positive number. In this chapter, problem (1.1)–(1.2), with the condition (1.3) or (1.4), will be studied under the following hypotheses:

(H1 )

.

(H2 )

.

.

j (t, x, ·) : R → (−∞, ∞], a.e. (t, x) ∈ Q, (t, x) → j (t, x, r) is measurable on Q, for any r ∈ R.

j is a proper, convex, and lower semicontinuous (l.s.c. for short) function with respect to r such that ∂j (t, x, r) = β(t, x, r) a.e. (t, x) ∈ Q.

.

(H3 )

.

(1.5)

(1.6)

j has the polynomial growth .

C1 |r|m + C10 ≤ j (t, x, r) ≤ C2 |r|m + C20 , for any r ∈ R, a.e. (t, x) ∈ Q,

(1.7)

with .m > 1, .C1 , .C2 > 0. By (1.6), it follows that .β(t, x, ·) is a maximal monotone graph (possibly multivalued) on .R, a.e. .(t, x) ∈ Q (see Theorem 7.29 in the Appendix). The conjugate of .j, denoted by .j ∗ , is defined as j ∗ (t, x, ω) = sup (ωr − j (t, x, r)), a.e. on Q, r∈R

.

(1.8)

and it is proper, convex, and l.s.c. Moreover, the following two Legendre–Fenchel relations take place (see Proposition 7.37): j (t, x, r) + j ∗ (t, x, ω) ≥ rω, for any r, ω ∈ R, a.e. on Q,

(1.9)

j (t, x, r) + j ∗ (t, x, ω) = rω, if and only if ω ∈ ∂j (t, x, r), a.e. on Q.

(1.10)

.

.

We will show that problem (1.1)–(1.2), with (1.3) or with (1.4), respectively, reduces via the relations (1.9)–(1.10) to a certain minimization problem for a convex lower semicontinuous functional involving the functions j and .j ∗ and prove that they are equivalent. This chapter investigates two cases for the potential j . The first one takes into consideration that .j (t, x, ·) is continuous on .R and has a polynomial growth

4

1 Nonlinear Diffusion Equations with Slow and Fast Diffusion

with respect to .r ∈ R, with .m ≥ 2, while in the second case .j (t, x, ·) is related to a graph .β(t, x, ·) defined only on a subset of .R and has a particular polynomial growth. First, we provide a result describing some properties of the conjugate .j ∗ and of the subdifferentials of j and .j ∗ , which are implied by (1.7), for .m > 1. Lemma 1.1 Let (1.5), (1.6), and (1.7) hold. Then, the following relations take place: '

'

C3 |ω|m + C30 ≤ j ∗ (t, x, ω) ≤ C4 |ω|m + C40 ,

.

(1.11)

for any .ω ∈ R, a.e. .(t, x) ∈ Q, .

|η(t, x)| ≤ C5 |r|m−1 + C50 , η(t, x) ∈ β(t, x, r) a.e. on Q, .

|w (t, x)| ≤ C6 |ω|

m' −1

(1.12)

+ C60 , w (t, x) ∈ (β)−1 (t, x, ω) a.e. on Q, (1.13)

where .C3 , C4 , C5 , C6 > 0. Proof Let us consider that .j (t, x, r) ≤ C2 |r|m + C20 and write j ∗ (t, x, ω) ≥ rω − j (t, x, r), for any r ∈ R, a.e. on Q.

.

m'

We apply this relation for .r = ρ |ω| m sign .ω, with .ρ a constant such that .0 < ρ < 1 C2 − m−1 , and get '

'

'

j ∗ (t, x, ω) ≥ ρ |ω|m − C2 ρ m |ω|m − C20 = ρ(1 − ρ m−1 C2 ) |ω|m − C20

.

'

= C3 |ω|m + C30 , C3 > 0. For the other inequality, we start from j ∗ (t, x, ω) = sup (ωr − j (t, x, r)) ≤ sup (ωr − C1 |r|m − C10 ) r∈R r∈R ( ) 1 1 1 ' m' |ω| − C10 = C4 |ω|m + C40 , a.e. on Q. − = m 1 1 m−1 m m m−1 C1m−1

.

1

1

We took into account that the supremum is reached at .|r| = |ω| m−1 (mC1 )− m−1 . In order to prove (1.12), we write the definition of the subdifferential of j (see (7.31)) j (t, x, r) − j (t, x, θ ) ≤ η(t, x)(r − θ ), for any θ ∈ R, η(t, x) ∈ β(t, x, r),

.

η(t,x and set .θ (t, x) = λ |η(t,x| . We have

1.1 A Continuous Potential with a Polynomial Growth

5

j (t, x, r) + λ |η(t, x)| ≤ η(t, x)r + C2 λm + C20 ,

.

whence | | | | | | | | λ |η(t, x)| ≤ |η(t, x)| |r| + C2 λm + |C20 | + |C1 |r|m + C10 | .

.

(1.14)

Let .|r| > 1 and consider (1.14) for .λ = 2 |r| . It follows that .

|η(t, x)| ≤ (C2 + C1 ) |r|

m−1

+

| 0| | 0| |C | + |C | 2

1

|r|

≤ C5 |r|m−1 + C50 ,

'

and hence .η ∈ Lm (Q). Let .|r| ≤ 1 and consider (1.14) for .λ = 2. We get .|η(t, x)| ≤ C51 , a.e. on .Q. Finally, we deduce (1.13) by applying the above arguments for .j ∗ . n u Notation The norm on the spaces .Lm (Ω) will be denoted by .||·||m for .m ∈ [1, ∞] ' and the pairing between the spaces .Lm and the dual .Lm (see Theorem 7.1 in the Appendix) will be indicated by one of the following notations: { f (g) = m' ,m = (f, g)m' ,m =

f (x)g(x)dx

.

Ω

'

for .f ∈ Lm (Ω), .g ∈ Lm (Ω), .m ∈ [1, ∞]. In the subsequent part, where there is no danger of confusion, we { skip sometimes the writing of the function arguments in the integrands, that is, . Q f (t, x)dxdt is { simply written . Q f dxdt.

1.1.1 Functional Framework and Preliminaries First, we refer to (1.1) and (1.2) with the Robin boundary condition (1.3). In this case we consider V the Hilbert space .H 1 (Ω) endowed with the norm .

)1/2 ( ||φ||V = ||∇φ||2L2 (Ω) + α ||φ||2L2 (f) ,

which, as known, is equivalent to the standard Hilbertian norm on .H 1 (Ω), for .α > 0 (see [64], p. 20). The dual of .V is denoted .V ' . We define the operator AV : V → V '

.

by

6

1 Nonlinear Diffusion Equations with Slow and Fast Diffusion

{ .

V ' ,V =

{ ∇θ · ∇φdx + Ω

αθ φdσ, for any φ ∈ V .

(1.15)

f

It is easily seen that the operator .AV is monotone, continuous, and coercive, so that it is surjective (see Theorem 7.18 in the Appendix). The scalar product on .V ' is defined by / \ (θ, θ )V ' = θ, A−1 V θ

.

V ' ,V

, for θ, θ ∈ V ' .

This implies the relation .

||θ ||V ' = ||φ||V , where φ = A−1 V θ.

(1.16)

Let .m ∈ (1, ∞), and let .m' be its conjugate number, that is, . m1' + m1 = 1. For any .m > 1, we introduce the space } { ∂ψ 2,m + αψ = 0 on f .Xm := ψ ∈ W (Ω), ∂ν

(1.17)

and the operator Am : D(Am ) ⊂ Lm (Ω) → Lm (Ω), D(Am ) = Xm , Am ψ = −∆ψ.

.

(1.18)

The operator .Am is m-accretive on .Lm (Ω) (see [13], p. 107, Theorem 3.2) and coercive, hence surjective (see [13], p. 36, Corollary 2.2, or Theorem 7.18 in the Appendix). According to Lemma 7.21 in the Appendix, we may extend .Am to the dual space ' , as of .D(Am' ), namely to .(D(Am' ))' = (Xm' )' =: Xm ' -m ) ⊂ X' ' → X' ' , -m : D(A A m m

.

-m ) = Lm (Ω), D(A (1.19) < > m -m θ, ψ ' A = (θ, Am' ψ)m,m' for any θ ∈ L (Ω) and ψ ∈ Xm' . X ,X ' m'

m

-m is surjective and closed, so that it has a Still by Lemma 7.21, the operator .A continuous inverse. The lemma is applied with the choice .X = Lm (Ω), .X' = ' -m . Lm (Ω), .A0 = Am , .A∗0 = Am' , .A = A ' because for any element .g ∈ X ' We note that indeed the dual space of .Xm is .Xm m one can associate a linear and continuous functional .g : Xm → R by -−1' g, Am ψ)m' ,m for any ψ ∈ Xm = D(Am ). g(ψ) = (A m

.

1.1 A Continuous Potential with a Polynomial Growth

7

1.1.2 The Minimization Problem Let ' f ∈ Lm (0, T ; Xm ), y0 ∈ V ' .

(1.20)

.

Definition 1.2 We call a solution to (1.1)–(1.3) a pair .(y, η) such that ' ' y ∈ Lm (Q) ∩ W 1,m (0, T ; Xm ) ∩ C([0, T ]; Xm ),

.

'

η ∈ Lm (Q), η(t, x) ∈ ∂j (t, x, y(t, x)) a.e.(t, x) ∈ Q, which satisfies the equation { T/ .

0

\ { { T dy Xm' ,Xm dt (t), ψ(t) dt + ηAm ψdxdt = dt ' ,X Q 0 Xm m

(1.21)

for any .ψ ∈ Lm (0, T ; Xm ), and the initial condition .y(0, x) = y0 . This is equivalent to the abstract Cauchy problem

.

dy -m' β(y(t)) e f (t), a.e. t ∈ (0, T ) (t) + A dt y(0) = y0 ,

(1.22)

-m' is defined in (1.19). where the operator .A A solution to (1.22) (equivalently (1.21)) is a pair .(y, η) such that ' .y : [0, T ] → Xm is absolutely continuous on .[0, T ], and therefore a.e. differentiable -m ) a.e. .t ∈ (0, T ), .η(t, x) ∈ β(t, x, y(t, x)) a.e. on .Q, .y(t) on .(0, T ), .y(t) ∈ D(A satisfies the first equation (1.22) and .y(0) = y0 . We shall work with (1.21) as well as with (1.22) where it will be more appropriate. Now, we formulate the following minimization problem: Minimize J0 (y, w) for all (y, w) ∈ U0 ,

.

(P0 )

where '

J0 : Lm (Q) × Lm (Q) → (−∞, ∞],

.

J0 (y, w) =

.

⎧{ ( ) ⎪ ⎪ j (t, x, y(t, x)) + j ∗ (t, x, w(t, x)) − w(t, x)y(t, x) dxdt, ⎨ Q

⎪ ⎪ ⎩

if (y, w) ∈ U0 ,

+∞, otherwise, (1.23)

8

1 Nonlinear Diffusion Equations with Slow and Fast Diffusion

and the admissible set .U0 is defined as { ' ), w ∈ Lm' (Q), U0 = (y, w); y ∈ Lm (Q) ∩ W 1,m (0, T ; Xm

.

.

(1.24)

(y, w) satisfies (1.25) below} , dy -m' w(t) = f (t), a.e. t ∈ (0, T ), (t) + A dt y(0) = y0 .

(1.25)

It is clear that if the pair .(y, w) is in .U0 , then it satisfies {

T

0

.

/

dy (t), ψ(t) dt

\

{ T{ dt + wAm ψdxdt ' ,X Ω 0 Xm m { T Xm' ,Xm dt =

(1.26)

0

for all .ψ ∈ Lm (0, T ; Xm ) and .y(0) = y0 . We also observe that if .(y, w) ∈ U0 , it follows that {

{ j (t, x, y(t, x))dxdt < ∞,

.

Q

j ∗ (t, x, w(t, x))dxdt < ∞.

Q

This is obvious for j by (1.7) and also for .j ∗ by the property (1.11) implied still by (1.7) and proved in Lemma 1.1 before. In the subsequent part, we assume that the admissible set is nonempty, that is, U0 /= ∅,

.

(1.27)

and this implies that .J0 is proper. An example of an element staying in .U0 can -−1' f ) with .y0 ∈ Lm (Ω). Then, by (1.7), it follows that .j (·, ·, y0 (·)) ∈ be .(y0 , A m 1 -m' which is surjective, we have .A -−1' f (t) ∈ Lm' (Ω) L (Q). By the definition of .A m ' -−1' f ∈ Lm (Q),, and by (1.11), we deduce that a.e. .t ∈ (0, T ), whence .A m ∗ -−1' f (·, ·)) ∈ L1 (Q), so that .J0 (y0 , A -−1' f ) < ∞. Thus, .J0 cannot be .j (·, ·, A m m identically equal to .+∞, meaning that it is proper. We can interpret problem .(P0 ) as an optimal control problem with the state y ' in .Lm (Q) and with the controller w acting in .Lm (Q), subject to the constraint .(y, w) ∈ U0 . By applying a technique from the optimal control theory to solve this problem, one first proves that .(P0 ) has at least a solution. The necessary conditions that characterize an optimal pair .(y, w) are provided by determining the first order necessary conditions of optimality. The final purpose of the next proofs is to show that (1.1)–(1.3) and .(P0 ) are equivalent, that is, the solution to the first problem is the solution to the minimization problem and vice versa. It is clear that if .(y, η) is the unique solution to (1.1)–(1.3) (equivalently, the strong solution to (1.22)), then

1.1 A Continuous Potential with a Polynomial Growth

9

the minimum in .(P0 ) exists and it is zero, due to (1.10). We intend to prove the converse assertion, that is, the minimum in .(P0 ) exists and it is zero, and that the null minimizer .(y ∗ , w ∗ ) (at which .J0 (y ∗ , w ∗ ) = 0) is the unique solution to (1.1)– (1.3).

1.1.3 Main Results for the Slow Diffusion Case Let .m ≥ 2. We begin with a result providing an equivalent form for the functional J0 . More exactly, we prove that if .(y, w) ∈ U0 , then .J0 can be expressed as

.

{

(j (t, x, y(t, x)) + j ∗ (t, x, w(t, x)))dxdt

J0 (y, w) =

.

(1.28)

Q

} 1{ ||y(T )||2V ' − ||y0 ||2V ' 2 { T/ \ f (τ ), A−1 − m y(τ ) ' +

Xm ,Xm

0

dτ.

Let us point out that by this expression, it is simpler to see the properties of .J0 which are essential arguments in the further existence theorems. Also, this alternative expression of .J0 is often used in the calculations. Thus, we assert that .J0 is proper, convex, and l.s.c. The first property is implied by (1.27). The functions {

∗

ϕ(y) =

{

j ∗ (t, x, w(t, x))dxdt

j (t, x, y(t, x))dxdt and ϕ (w) =

.

Q

Q

are proper, convex, and l.s.c. (see Proposition 7.38 in the Appendix), and the sum of {T < > the norms .||y(T )||2V ' − 0 f (τ ), A−1 ' ,X dτ is also convex and continuous. m y(τ ) Xm m Now, we prove the following: Lemma 1.3 Let .m ≥ 2, and let .(y, w) be in .U0 . Then } { t{ 1{ ||y(t)||2V ' − ||y0 ||2V ' + ywdxdτ 2 { t/ . \0 Ω f (τ ), A−1 dτ, for all t ∈ [0, T ]. = m y(τ ) '

(1.29)

Xm ,Xm

0

'

' ), .w ∈ Lm (Q) Proof If .(y, w) ∈ U0 , it follows that .y ∈ Lm (Q) ∩ W 1,m (0, T ; Xm and the pair .(y, w) satisfies (1.25). Since .Am is surjective, for .y(t) ∈ Lm (Ω), there exists .ψ(t) ∈ Xm such that

Am ψ(t) = y(t) ∈ Lm (Ω) a.e. t ∈ (0, T ).

.

10

1 Nonlinear Diffusion Equations with Slow and Fast Diffusion

This implies that ψ(t) = A−1 m y(t) ∈ D(Am ) = Xm a.e. t ∈ (0, T ),

.

and by the elliptic regularity (according to the paper [1] by Agmon, Douglis, and Nirenberg), it follows that .

|| || || || −1 ||Am y(t)||

Xm

≤ C ||y(t)||Lm (Ω) a.e. t ∈ (0, T ),

(1.30)

with C a constant independent of .y(t). Therefore, m ψ = A−1 m y ∈ L (0, T ; D(Am )).

.

' ) and .A -m' is surjective, it follows that . dψ is given by Also, since .y ∈ W 1,m (0, T ; Xm dt

.

dψ -−1' dy ∈ Lm' (Q). =A m dt dt

Let us test (1.25) by .A−1 m y and integrate over .(0, t). We get { t/ .

0

\ { t{ dy (τ ), A−1 y(τ ) dτ + wydxdτ m dτ ' ,X 0 Ω X m m { t/ \ f (τ ), A−1 y(τ ) dτ. = m ' 0

(1.31)

Xm ,Xm

' ) as well. We compute the first term, by taking into account that .y ∈ Lm (0, T ; Xm We have \ { t/ dy (τ ), A−1 y(τ ) dτ m ' ,X 0 dτ Xm m \ |t { t / d | −1 y(τ ), (A y(τ )) . dτ (1.32) = Xm' ,Xm | − 0 dτ m \ m,m' 0 / { |t t dψ | Am ψ(τ ), (τ ) dτ. = Xm' ,Xm | − 0 dτ 0 m,m' '

Taking into account that .Xm ⊂ Lm (Ω) ⊂ Lm (Ω) if .m ≥ 2, we calculate .

Xm' ,Xm = m,m' = ||∇ψ(τ )||2L2 (Ω) + α ||ψ(τ )||2L2 (f) = ||ψ(τ )||2V = ||y(τ )||2V ' .

The last term on the right-hand side of (1.32) is written as

1.1 A Continuous Potential with a Polynomial Growth

{ t/ .

0

dψ Am ψ(τ ), (τ ) dτ

\

11

{ t/

\ dψ (τ ), ψ(τ ) dτ = dτ Am' dτ ' ,X 0 m,m' Xm m \ { t/ dy (τ ), A−1 y(τ ) dτ. = m ' ,X 0 dτ Xm m

By replacing the last two relations in (1.32), we get { t/ .

0

dy (τ ), A−1 m y(τ ) dτ

\ ' ,X Xm m

= ||y(t)||2V ' − ||y0 ||2V ' −

dτ { t/ 0

dy (τ ), A−1 m y(τ ) dτ

\ ' ,X Xm m

dτ,

whence { t/ .

0

dy (τ ), A−1 m y(τ ) dτ

\ ' ,X Xm m

dτ =

} 1{ ||y(t)||2V ' − ||y0 ||2V ' . 2

Finally, we come back to (1.31) and obtain for a.a. .t ∈ (0, T ) { t/ } { t{ \ 1{ ||y(t)||2V ' − ||y0 ||2V ' + f (τ ), A−1 ywdxdτ = dτ. m y(τ ) ' Xm ,Xm 2 0 Ω 0 (1.33) {t { {t < > Since .t → − 0 Ω ywdxdτ + 0 f (τ ), A−1 ' ,X dτ is continuous, we m y(τ ) Xm m conclude with (1.29), as claimed. u n .

Now, we are ready to show that .(P0 ) has a point of minimum. This will be proved for any dimension N when .m ≥ 2, which covers the slow diffusion case. Theorem 1.4 Let .m ≥ 2. Assume (1.7) and (1.20). Then, problem .(P0 ) has at least a solution .(y ∗ , w ∗ ). Proof We recall that we have already established that .J0 is proper, convex, and l.s.c. Let us denote .d := min{J0 (y, w); (y, w) ∈ U0 }, which is nonnegative because, as defined in (1.23), .J0 (y, w) ≥ 0, since the integrand is positive by (1.9). We take a minimizing sequence .(yn , wn )n≥1 , .(yn , wn ) ∈ U0 . Then, d ≤ J0 (yn , wn ) ≤ d +

.

1 , n

(1.34)

and .(yn , wn ) is the solution to

.

dyn -m' wn (t) = f (t), a.e. t ∈ (0, T ), (t) + A dt y(0) = y0 .

(1.35)

12

1 Nonlinear Diffusion Equations with Slow and Fast Diffusion

In (1.28), we apply Hölder’s inequality to the term {

/

T

.

0

f (t), A−1 m y(t)

\ ' ,X Xm m

{

T

dt ≤ 0

≤

||f (t)||Xm' ||y(t)||Lm (Ω) dt

C1 1 m' ||y||m Lm (Q) + ' m' /m ||f ||Lm' (0,T ;X' ) , m m m C1

whence it follows that C1 0 m' 0 ||y||m (1.36) Lm (Q) + C1 + C3 ||w||Lm' (Q) + C3 ' m { T 1 1 1 ' ||f (t)||m + ||y(T )||2V ' − ||y0 ||2V ' − ' dt, ' /m X m m 2 2 0 m' C1

J0 (y, w) ≥

.

where .m ≥ 2, .m' ∈ (1, 2]. By (1.28), (1.36), and (1.34), we deduce that .

||yn ||Lm (Q) ≤ C, ||wn ||Lm' (Q) ≤ C, ||yn (T )||V ' ≤ C,

with C denoting several constants independent of .n. By extracting subsequences denoted by the same subscript .n, we can write that yn → y ∗ weakly in Lm (0, T ; Lm (Ω)), as n → ∞, .

.

'

'

wn → w∗ weakly in Lm (0, T ; Lm (Ω)), as n → ∞, . yn (T ) → ξ weakly in V ' , as n → ∞.

(1.37) (1.38) (1.39)

-m' , we have By the definition of .A { .

0

T

< > -m' (wn − w ∗ )(t), ψ(t) ' A X

m ,Xm

{

T

dt = 0

( ) (wn − w∗ )(t), Am ψ(t) m' ,m dt

for any .ψ ∈ Lm (0, T ; Xm ) and so we deduce that ' -m' w ∗ weakly in Lm (0, T ; Xm -m' wn → A ), as n → ∞. A

.

(1.40)

Therefore, (1.35) yields .

dyn dy ∗ ' ), as n → ∞. → weakly in Lm (0, T ; Xm dt dt

(1.41)

Let us recall the Sobolev inequality, more precisely the Rellich–Kondrachov inequality (see Theorem 7.6 in the Appendix)

1.1 A Continuous Potential with a Polynomial Growth

13

W 1,2 (Ω) ⊂ C(Ω), for N = 1,

.

W 1,2 (Ω) ⊂ Lq (Ω), q ∈ [2, ∞), for N = 2, W 1,2 (Ω) ⊂ Lq (Ω), q ∈ [1, p ∗ ), p∗ =

2N ∈ (2, ∞), for N > 2, N −2

with compact injections. Since .m ≥ 2, it follows that .m' ∈ (1, 2] ⊂ [1, p∗ ) and so '

W 1,2 (Ω) ⊂ Lm (Ω) with compact injection, for any m' ∈ (1, 2].

.

(1.42)

This implies Lm (Ω) ⊂ V ' with compact injection, for m ≥ 2.

.

(1.43)

Consequently, the Aubin–Lions–Simon lemma (Lemma 7.9 in the Appendix) yields yn → y ∗ strongly in Lm (0, T ; V ' ), as n → ∞.

(1.44)

.

By (1.29) and (1.30) written for .yn , we get 2 . ||yn (t)|| ' V

≤

||y0 ||2V '

2 + ' m

{ 0

T

'

||wn (t)||mm'

L (Ω)

(1.45)

dt

{ 2 T ||yn (t)||m Lm (Ω) dt m 0 { T ||f (t)||Xm' ||yn (t)||Lm (Ω) dt +2

+

0

≤ C, for any t ∈ [0, T ]. ' , For .m ≥ 2, we still have that .Xm is embedded in .V , so .V ' is embedded in .Xm 2 implying that .||yn (t)||X' ≤ C for any .t ∈ [0, T ]. Therefore, this together with m equicontinuity of the sequence .(yn )n (whose derivative is bounded) implies by virtue of the Arzelà–Ascoli theorem (Theorem 7.11 in the Appendix) that ' yn (t) → y ∗ (t) strongly in Xm , as n → ∞, uniformly for t ∈ [0, T ].

.

(1.46)

By (1.39) and the limit uniqueness, we deduce that .y ∗ (T ) = ξ ∈ V ' . The last term is written as { .

0

T

/

f (t), A−1 m yn (t)

\ ' ,X Xm m

{

T

dt → 0

/

∗ f (t), A−1 m y (t)

\ ' ,X Xm m

dt, as n → ∞

14

1 Nonlinear Diffusion Equations with Slow and Fast Diffusion

so that we have all { ingredients to pass to the limit{ in (1.34). We recall that the functions .ϕ(y) = Q j (t, x, y)dxdt and .ϕ ∗ (w) = Q j ∗ (t, x, w)dxdt are weakly l.s.c. and deduce that d ≤ J0 (y ∗ , w ∗ ) ≤ lim inf J0 (yn , wn ) ≤ d.

.

n→∞

This confirms that .(y ∗ , w ∗ ) belongs to .D0 and realizes the minimum in .(P0 ). Also we notice that { { . j (t, x, y ∗ )dxdt < ∞, j ∗ (t, x, w ∗ )dxdt < ∞ Q

Q

because of (1.7) and (1.11), so that .(y ∗ , w ∗ ) ∈ U0 . Passing to the limit in the equivalent form of (1.35) {

T

.

0

/

dyn -m' wn (t), φ(t) (t) + A dt

\

{

' ,X Xm m

dt =

T

0

Xm' ,Xm dt,

(1.47)

Xm' ,Xm dt,

(1.48)

for any .φ ∈ Lm (0, T ; Xm ), we obtain T / dy ∗

{ .

0

-m' w ∗ (t), φ(t) (t) + A

dt

\

{

' ,X Xm m

T

dt = 0

for any .φ ∈ Lm (0, T ; Xm ). This shows that .(y ∗ , w ∗ ) is the solution to (1.25), and since it belongs to .U0 , it follows that .(P0 ) reaches its minimum on .U0 . u n Relying on Lemma 1.1, any pair .(y, w) ∈ D0 satisfies {

{

'

|w (t, x)|m dxdt < ∞,

.

Q

|η(t, x)|m dxdt < ∞,

(1.49)

Q

where .η(t, x) ∈ β(t, x, y(t, x)), .w (t, x) ∈ β −1 (t, x, w(t, x)) a.e. on .Q. It is clear now that if .(y, η) is a solution to (1.1)–(1.3) expressed by Definition 1.2 with .η(t, x) ∈ ∂j (t, x, y(t, x)), then it is a null minimizer in .(P0 ), that is, .J0 (y, w) = 0, by (1.10) and (1.23). The next theorem proves the main result asserting the equivalence between (1.1)–(1.3) and .(P0 ), namely that the minimum in .(P0 ) is zero and that it is reached at the solution to (1.1)–(1.3). Theorem 1.5 Let .f ∈ Lm (0, T ; Xm ) and .y0 ∈ V ' . If .(y ∗ , w ∗ ) is a solution to .(P0 ), then w ∗ (t, x) ∈ β(t, x, y ∗ (t, x)) a.e. (t, x) ∈ Q

.

and .(y ∗ , w ∗ ) is the unique solution to (1.1)–(1.3).

(1.50)

1.1 A Continuous Potential with a Polynomial Growth

15

Proof Let .(y ∗ , w ∗ ) be a point of minimum for .J0 . Then .(y ∗ , w ∗ ) satisfies (1.25) and J0 (y ∗ , w ∗ ) ≤ J0 (y, w) for any (y, w) ∈ U0 .

.

(1.51)

For .λ > 0, we introduce the variations y λ = y ∗ + λY, wλ = w ∗ + λW,

.

with .(Y, W ) regular enough, such that .(y λ , w λ ) ∈ U0 . In particular, we can choose ∞ λ λ .Y, .W ∈ C (Q), such that the pair .(y , w ) satisfies (1.25) .

dy λ -m' w λ (t) = f (t) a.e. t ∈ (0, T ), (t) + A dt y λ (0) = y0 .

Immediately, we get that the pair .(Y, W ) is the solution to the problem

.

dY (t) + Am W (t) = 0 a.e. t ∈ (0, T ), dt Y (0) = 0,

and from here we can deduce that { t .Y (t) = − Am W (s)ds, t ∈ [0, T ].

(1.52)

(1.53)

0

By replacing in (1.51) .(y, w) by .(y λ , w λ ) and taking into account the expression (1.28), we calculate J0 (y λ , w λ ) − J0 (y ∗ , w ∗ ) (1.54) { ( ) j (t, x, y λ ) − j (t, x, y ∗ ) + j ∗ (t, x, w λ ) − j ∗ (t, x, w ∗ ) dxdt =

.

Q

|| || || 1 || ||y λ (T )||2 ' − 1 ||y ∗ (T )||2 ' V V 2 2 { T/ \ λ ∗ − (y (t) − y (t)) f (t), A−1 m

+

' ,X Xm m

0

dt ≥ 0.

We define the directional derivative of j by j (t, x, y ∗ + λr) − j (t, x, y ∗ ) , for r ∈ R, a.e. (t, x) ∈ Q, λ→0 λ

j ' (t, x, y ∗ ; r) = lim

.

16

1 Nonlinear Diffusion Equations with Slow and Fast Diffusion

and assert that this exists because j is a proper, convex function, finite at any .r ∈ R thanks to (1.7), (see [21], p. 86). Moreover, by virtue of the Lebesgue dominated convergence theorem, we have { lim

.

λ→0 Q

j (t, x, y ∗ + λY ) − j (t, x, y ∗ ) dxdt = λ

{

j ' (t, x, y ∗ ; Y )dxdt,

Q

for .Y ∈ C ∞ (Q) and .y ∗ ∈ Lm (Q). Indeed, the left-hand side integrand is a.e. convergent to .j ' (t, x, y ∗ ; Y ) by the definition and | | | j (t, x, y ∗ + λY ) − j (t, x, y ∗ ) | | ≤ |η(t, x)| |Y (t, x)| < ∞, | . | | λ by the calculus for proving (1.12) in Lemma 1.1, where .η(t, x) ∈ β(t, x, y ∗ (t, x)) a.e. on .Q. We mention that for not overloading the text we skip the writing of the arguments t and x for some functions, e.g., for .y ∗ , Y and further for .w ∗ , W . A similar relation takes place for .j ∗ { .

lim

λ→0 Q

j ∗ (t, x, w ∗ + λW ) − j ∗ (t, x, w ∗ ) dxdt = λ

{

(j ∗ )' (t, x, w ∗ ; W )dxdt, Q

'

for .W ∈ C ∞ (Q) and .w ∗ ∈ Lm (Q). In (1.54), we divide by .λ > 0 and pass to the limit as .λ → 0. We obtain .

J0 (y λ , w λ ) − J0 (y ∗ , w ∗ ) λ→0 λ { = (j ' (t, x, y ∗ ; Y ) + (j ∗ )' (t, x, w ∗ ; W ))dxdt + (Y (T ), y ∗ (T ))V ' lim

{

Q T

− 0

/

f (t), A−1 m Y (t)

\ ' ,X Xm m

dt ≥ 0.

Changing .λ into .−λ and repeating all computations, we obtain the opposite inequality, so that we finally deduce that { .

Q

(j ' (t, x, y ∗ ; Y ) + (j ∗ )' (t, x, w ∗ ; W ))dxdt + (Y (T ), y ∗ (T ))V ' { T/ \ f (t), A−1 − dt = 0. m Y (t) ' 0

Xm ,Xm

Following the relation < > j ' (t, x, y ∗ ; Y ) ≥ η∗ , Y m' ,m for all η∗ (t, x) ∈ ∂j (t, x, y ∗ ) a.e. on Q

.

(1.55)

1.1 A Continuous Potential with a Polynomial Growth

17

(see [13], p. 53, Proposition 2.6), we have {

j ' (t, x, y ∗ ; Y )dxdt ≥

.

Q

{

η∗ Y dxdt for all η∗ (t, x) ∈ ∂j (t, x, y ∗ ) a.e. on Q Q

and the corresponding one for .(j ∗ )' {

∗ '

{

∗

w ∗ W dxdt for all w ∗ (t, x) ∈ ∂j ∗ (t, x, w ∗ )

(j ) (t, x, w ; W )dxdt ≥

.

Q

Q

a.e. on .Q, where .w ∗ (t, x) ∈ β −1 (t, x, w ∗ ) a.e. on Q (see [13], p. 6). Replacing in (1.55), we get {

T

{

∗

∗

{

∗

T

(η Y +w W )dxdt +(Y (T ), y (T )) −

.

0

V'

' ,X Xm m

0

Ω

/ \ f (t), A−1 m Y (t)

dt ≤ 0,

for any .Y, W ∈ C ∞ (Q). By changing .(Y, W ) into .(−Y, −W ), we obtain the opposite inequality, so finally we conclude that {

T

{

∗

∗

{

∗

T

(η Y +w W )dxdt +(Y (T ), y (T )) −

.

0

V'

0

Ω

/ \ f (t), A−1 m Y (t)

' ,X Xm m

dt = 0.

Using (1.53), we deduce {

T

( { t ) { η∗ − Am W (s)ds dxdt +

{

.

0

0

Ω

(

+ y ∗ (T ), −Am

{

)

T

W (t)dt 0

V'

T

{

w ∗ W dxdt

0

Ω

{

/ \ { t f (t), − W (s)ds

T

− 0

0

' ,X Xm m

dt = 0,

for any .W ∈ C ∞ (Q). By proceeding with some rearrangements, we obtain {

T 0

.

/

\ { t ∗ ∗ −Am' (η − w )(t), W (s)ds dt ' ,X 0 Xm / \ { Tm { T{ ∗ ∗ w W dxdt + y (T ), − W (t)dt + 0\ V ' ,V {0 T / Ω { t ∗ -m' w (t), f (t) − A + W (s)ds dt = 0, 0

0

' ,X Xm m

for any W ∈ C ∞ (Q).

By (1.25), the sum of the last two terms on the left-hand side becomes

(1.56)

18

1 Nonlinear Diffusion Equations with Slow and Fast Diffusion

/

{

∗

y (T ), −

.

\

T

W (t)dt

{ +

V ' ,V

0

T 0

{

T

=− 0

/

dy ∗ (t), dt

{

\

t

W (s)ds 0

< ∗ > y (t), W (t) X'

m ,Xm

' ,X Xm m

dt

dt.

Replacing this in (1.56) and changing the order of the integrals, we obtain {

T

/

.

-m' −A

0

{

T

\ (η − w )(s)ds + (w − y )(t), W (t) ∗

∗

∗

∗

' ,X Xm m

t

dt = 0,

(1.57)

for any .W ∈ C ∞ (Q) ⊂ Lm (0, T ; Xm ). This leads to .

-m' −A

{

T

(η∗ − w ∗ )(s)ds + (w ∗ − y ∗ )(t) = 0.

(1.58)

t

We denote p(t, x) = w ∗ (t, x) − y ∗ (t, x)

.

(1.59)

and see that .p ∈ Lm (0, T ; Lm (Ω)) by (1.49). Therefore, p(t, x) + y ∗ (t, x) ∈ β −1 (t, x, w ∗ (t, x)) a.e. on Q.

.

By (1.58), we obtain .

-m' (w ∗ − η∗ )(t) = 0, a.e. t ∈ (0, T ) − pt (t) + A

(1.60)

p(T , x) = 0

(1.61)

with .

' ). We multiply (1.60) by .A−1 p(t) and integrate over and so .pt ∈ Lm (0, T ; Xm m .(t, T ), obtaining

{ .

||p(t)||2V ' +

T

(w ∗ (s) − η∗ (s), p(s))m' ,m ds = 0.

t

Since .w ∗ (t, x) ∈ β(t, x, (p + y ∗ )(t, x)) and .η∗ (t, x) ∈ β(t, x, y ∗ (t, x)) a.e. on .Q, we finally deduce that .||p(t)||2V ' ≤ 0, whence .p(t) = 0 for any .t ∈ [0, T ], and hence y ∗ (t, x) = w ∗ (t, x) ∈ β −1 (t, x, w ∗ (t, x)), a.e. (t, x) ∈ Q.

.

(1.62)

This implies (1.50) and, consequently, that .(y ∗ , w ∗ ) is a solution to (1.1)–(1.3).

1.2 A Free Boundary Problem for a Special Fast Diffusion Case

19

Eventually, we prove that the solution is unique. Indeed, let us assume that there exists another solution .(y, w -) to (1.1)–(1.3) corresponding to the same data and write the equation satisfied by their difference .

d(y − y) -m' (η − (t) + A η)(t) = 0 a.e. t ∈ (0, T ), dt (y − y )(0) = 0,

where .η(t, x) ∈ β(t, x, y(t, x)), .η(t, x) ∈ β(t, x, y (t, x)) a.e. on Q and .(y, η) and (y, η) belong to .U0 . By multiplying the equation by .A−1 y )(t) and integrating m (y − over .(0, t), we obtain

.

1 ||(y − y )(t)||2V ' + . 2

{

t

η(t), y(t) − y (t)) dt = 0. (η(t) − -

0

y (t)||2V ' ≤ 0, whence .y(t) = By the maximal monotonicity of .β, we get .||y(t) − y (t) for all .t ∈ [0, T ]. u n Remark 1.6 In the case of the Dirichlet boundary condition .β(t, x, y) = 0 on .Σ, the previous results remain valid with the only modifications of the spaces .Xm and .V , which now become Xm := W 2,m (Ω) ∩ W01,m (Ω), V = H01 (Ω).

.

1.2 A Free Boundary Problem for a Special Fast Diffusion Case The case treated in this subsection refers to problem (1.1)–(1.3) in which .β has properties raising from a relevant case of the theory of water infiltration into a porous medium driven by a fast diffusion. The Robin boundary condition is chosen again because it is relevant for water infiltration in soils. We begin by giving a brief description of the physical model of water infiltration in a porous material, in particular in soil, with the formation of a free boundary. Let us denote the domain where the process evolves by .Ω and consider an elementary volume around a point .x ∈ Ω. The porosity of a soil is defined by the ratio between the volume of pores and the elementary volume around .x ∈ Ω. It may depend on the space variables, and it can also change in time. Here we assume it constant. The density of water in pores denoted by y represents the ration between the volume of water in pores and the elementary volume around .x ∈ Ω. The porous medium has a limit capacity of absorbing water up to the saturation value denoted .ys , which is equal to the porosity, so that it is obvious that .y ∈ [0, ys ]. The subset .Qu = {(t, x) ∈ Q = (0, T ) × Ω; .y(t, x) < ys } is called the unsaturated part and

20

1 Nonlinear Diffusion Equations with Slow and Fast Diffusion

the subset .Qs = {(t, x) ∈ Q; .y(t, x) = ys } in which pores are completely filled with water is the saturated part. These domains evolve in time and space as water infiltrates into the material and so their common boundary acts as a free boundary which changes in time. This process called a saturated–unsaturated process is characterized in the mathematical model by a coefficient of diffusion (denoted later by .β ' ) with particular properties leading to a fast diffusion type model (see [57]). In the framework of the theory presented in this subsection, both situations of a degenerate case .(β ' ≥ 0) and a nondegenerate case .(β ' > 0) can be treated. From the mathematical point of view, this problem may be seen as a variational inequality with constraints. Studies on the well-posedness of nonlinear diffusion equations involving time or space dependent subdifferentials with constraints produced an extremely rich literature. As a very few examples we refer the reader to the monographs [17, 56], the papers [15, 45, 52, 53], and the references indicated in these works. In this subsection, we treat this variational inequality by a dual variational method. In this case, we assume for the diffusion coefficient .β ' the following hypotheses: (H4 )

.

β ' : Q × [0, ys ) → (0, +∞), .(t, x) → β ' (t, x, r) is measurable on Q for any .r ∈ (0, ys ),

.

β ' (t, x, r) ≥ 0, for r ∈ [0, ys ), a.e. (t, x) ∈ Q.

.

(H5 )

.

β ' blows up at .r = ys ,

.

.

(H6 )

.

(1.63)

lim β ' (t, x, r) = +∞

(1.64)

β ' (t, x, s)ds = βs (t, x) < ∞,

(1.65)

r/ys

such that { .

lim

r/ys

r

0

where βs ∈ L∞ (Q), 0 < βs (t, x) ≤ βM a.e. (t, x) ∈ Q.

.

(1.66)

Since in these specific real problems the solution is nonnegative, the function .β ' is defined for a nonnegative variable. In order to allow the well-posedness of the problem, the function .β ' should be extended for negative arguments in a convenient way, by a technical mathematical artifice. Thus, we continuously extend .β ' at the left of 0 by the function β ' (t, x, r) = C1 |r|m−2 , for r < 0,

.

(1.67)

1.2 A Free Boundary Problem for a Special Fast Diffusion Case

21

where .C1 > 0, and .m > 2, odd. For instance, one can set .β ' (t, x, r) = C1 |r| . This is done if .β ' (t, x, r) ≥ 0. Under the assumption of a nondegenerate case when ' ' ' .β (t, x, r) ≥ ρ > 0, one extends .β (t, x, r) = β (t, x, 0) for .r < 0. ' By .β we denote the integral of .β from 0 to .r ≤ ys . At .r = ys , we extend the function .β up to .+∞ by defining .β(t, x, r) as the graph ⎧ ⎨ .

⎩

{

r

β(t, x, r) =

β ' (t, x, s)ds, for r < ys , a.e. (t, x) ∈ Q,

0

β(t, x, r) ∈ [βs (t, x), +∞),

(1.68)

for r = ys , a.e. (t, x) ∈ Q.

The extension of a nonlinear function to a multivalued one is common in the theory of nonlinear differential equations with discontinuous coefficients, as well as in the modeling of free boundary processes. The procedure of filling the jumps by using multivalued operators is necessary to ensure the well-posedness of the problem. It follows that .r → β(t, x, r) is continuous on .(−∞, ys ) and monotonically increasing, and obviously, .β(t, x, ·) is a maximal monotone graph on .(−∞, ys ] a.e. .(t, x) ∈ Q. Next, we define .j (t, x, ·) : (−∞, ys ] → (−∞, ∞] by j (t, x, r) =

.

⎧{ ⎨ ⎩

r

β(t, x, s)ds, r ≤ ys ,

0

+∞,

(1.69)

otherwise.

Therefore, .j (t, x, ·) is a proper, convex, and lower semicontinuous function on .R, a.e. .(t, x) ∈ Q, measurable on Q for each .r ∈ (−∞, ys ] (see Proposition 7.38 in the Appendix), satisfying ∂j (t, x, ·) = β(t, x, ·) a.e. (t, x) ∈ Q.

.

(1.70)

Also, we note that for any .r ∈ [0, ys ] and a.e. .(t, x) ∈ Q, we have { j (t, x, r) =

.

r

β(t, x, ξ )dξ ≤ βs (t, x)r ≤ βM r < ∞, for r ≤ ys .

(1.71)

0

By (1.67) and (1.71), we get that j (t, x, r) = C2 |r|m , for r ∈ (−∞, 0), a.e. (t, x) ∈ Q, m > 2, odd,

(1.72)

j (t, x, r) ≤ βM |r| , for r ∈ [0, ys ], a.e. (t, x) ∈ Q.

(1.73)

.

.

For the conjugate .j ∗ , it follows that j ∗ (t, x, ω) ≥ ωr − j (t, x, r) ≥ ωr − βM |r| , for r ∈ [0, ys ].

.

22

1 Nonlinear Diffusion Equations with Slow and Fast Diffusion

By setting .r := C3 sign .ω, we get C3 |ω| + C30 ≤ j ∗ (t, x, ω) for any ω ∈ R, a.e. (t, x) ∈ Q.

.

(1.74)

Hypotheses (1.64)–(1.65) together with the assumption that .β(t, x, ·) is a graph defined by (1.68) express exactly the situation of a fast diffusion with the formation of a free boundary (see [57]). The extension of .β ' (t, x, r) for .r < 0 is chosen such that to ensure the monotonicity of .β on .(−∞, ys ]. For example, a relevant coefficient of diffusion characterizing a fast diffusion has, e.g., the form β ' (t, x, r) = a(t, x)(ys −r)p−2 −b(t, x), for r ∈ [0, ys ) and p ∈ (1, 2),

.

(1.75)

where a and b are measurable functions, .a(t, x) ≥ a0 > 0, .b(t, x) ≤ ys a(t, x) a.e. (t, x) ∈ Q. The case .b(t, x) = ys a(t, x) when the diffusion coefficient vanishes at .r = 0 characterizes a degenerate situation, while .b(t, x) = 0 is associated with a nondegenerate situation. .

Preliminaries and the Minimization Problem In order to prove that (1.1)–(1.3) with .β previously defined has a solution, we shall formulate a minimization problem for an appropriate functional to this case and establish its equivalence with (1.1)–(1.3). Let us recall the definitions of the space .Xm and of the operator .Am (see (1.17) ' , .A for .m = ∞. For the sake of and (1.18)) and adapt the notation .Xm , .Xm m ' and write instead X simplicity, we shall drop the subscript .∞ from .X∞ and .X∞ ' and .X . Thus, ∞ ∞ A⎧ ∞ : D(A∞ ) ⊂ L (Ω) → L (Ω), A∞ ψ = −∆ψ, ⎫ ⎬ ⎨ n ∂ψ . + αψ = 0 on f . D(A∞ ) = ψ ∈ W 2,p (Ω); ∆ψ ∈ L∞ (Ω), ⎭ ⎩ ∂ν p≥2

For simplicity, we denote X := D(A∞ ).

.

We assume f ∈ L∞ (0, T ; L∞ (Ω)), y0 ∈ L∞ (Ω), |y0 (x)| ≤ ys a.e. on Ω.

.

(1.76)

Let us recall the definitions and results (7.41)–(7.45) in the Appendix, referring to the dual space .(L∞ (Q))' and to the convex functionals defined on .L∞ (Q), and define { ∞ .ϕ : L (Q) → (−∞, ∞], ϕ(y) = j (t, x, y(t, x))dxdt, (1.77) Q

1.2 A Free Boundary Problem for a Special Fast Diffusion Case

23

with the domain .D(ϕ) = {y ∈ L∞ (Q); .ϕ(y) < +∞} (see (7.44) in the Appendix). ∗ the indicator function of .D(ϕ) (see 7.33)) and its We denote by .ID(ϕ) and .ID(ϕ) conjugate (see 7.42), respectively. By (7.41), .w ∈ (L∞ (Q))' can be written as w = wa + ws , with wa ∈ L1 (Q), ws ∈ (L∞ (Q))' .

.

In this case, we introduce the functional J∞ : L∞ (Q) × (L∞ (Q))' → (−∞, ∞],

.

⎧{ 1 1 ⎪ ⎪ j (t, x, y(t, x))dxdt + ||y(T )||2V ' − ||y0 ||2V ' ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ Q { T ⎪ ⎪ ⎨ − (L∞ (Ω))' ,L∞ (Ω) dt .J∞ (y, w) = 0 { ⎪ ⎪ ⎪ ∗ ⎪ + j ∗ (t, x, wa (t, x))dxdt + ID(ϕ) (ws ) if (y, w) ∈ U∞ , ⎪ ⎪ ⎪ Q ⎪ ⎪ ⎩ +∞, otherwise, (1.78) where { U∞ = (y, w); y ∈ L∞ (Q), w ∈ (L∞ (Q))' , y(T ) ∈ V ' , { dy ∈ (L∞ (Q))' , j (t, x, y(t, x))dxdt < ∞, dt Q { . (1.79) ∗ ϕ ∗ (w) = j ∗ (t, x, wa (t, x))dxdt + ID(ϕ) (ws ) < ∞, Q } |y(t, x)| ≤ ys a.e. on Q, (y, w) verifies (1.80) below {

T

− .

0

/

dψ y(t), (t) dt

\ dt − L∞ (Ω),(L∞ (Ω))' { T (L∞ (Ω))' ,L∞ (Ω) dt, =

L∞ (Ω),(L∞ (Ω))'

+ (L∞ (Q))' ,L∞ (Q)

(1.80)

0

∞ ∞ for any .ψ ∈ L∞ (0, T ; X), with . dψ dt ∈ L (0, T ; L (Ω)) and .ψ(T ) = 0. Let us introduce the minimization problem

Minimize J∞ (y, w) for all (y, w) ∈ U∞ .

.

(P∞ )

Problem .(P∞ ) can be viewed as an optimal control problem with a state constraint |y(t, x)| ≤ ys and with the control w a measure. We shall show that .(P∞ ) has at least a solution and prove that a null minimizer .(y, w) provides a generalized solution to (1.1)–(1.3) in a sense that will be specified.

.

24

1 Nonlinear Diffusion Equations with Slow and Fast Diffusion

Lemma 1.7 The function .J∞ is proper, convex, and lower semicontinuous on L∞ (Q) × (L∞ (Q))' .

.

Proof Let .y ∈ L∞ (Q), such that .|y(t, x)| ≤ ys a.e. on .Q. By (1.72)–(1.73), it follows for any .y ∈ D(ϕ) { ϕ(y) =

(1.81)

j (t, x, y)dxdt

.

Q

{ =

{ {(t,x);y(t,x)∈[0,ys ]}

j (t, x, y)dxdt +

{(t,x);y(t,x)∈[−ys ,0)}

{

≤ βM

{(t,x);y(t,x)∈[0,ys ]}

j (t, x, y)dxdt

{ |y| dxdt + C2

{(t,x);y(t,x)∈[−ys ,0)}

|y|m dxdt

≤ βM ||y||L∞ (Q) meas(Q) + C2 ||y||m L∞ (Q) meas(Q) = C∞ , and hence .D(ϕ) = {y ∈ L∞ (Q); .|y(t, x)| ≤ ys a.e on .Q}. According to Theorem 7.39 in the Appendix, the conjugate of .ϕ is ϕ ∗ (w) =

{

.

Q

∗ j ∗ (t, x, wa (t, x))dxdt + ID(ϕ) (ws ), ∀ w ∈ (L∞ (Q))' ,

(1.82)

where .wa and .ws are the continuous and singular parts of .w ∈ (L∞ (Q))' (see ∗ (7.41)) and .ID(ϕ) (ws ) is the conjugate of the indicator function of .D(ϕ) (see (7.42)). Moreover, the functions .ϕ(y) and .ϕ ∗ (w) defined in (1.77) and (1.82) are proper, convex, and l.s.c. and it is obvious that .J∞ (y, w) has the same properties. Now, we show that .U∞ /= ∅. We choose the pair .(y0 , w) where .|y n0 (x)|2,p≤ ys ∞ (Q), it follows that .w(t) ∈ and .w(t) := A−1 (f (t)). Since . f ∈ L W (Ω) ∞ p≥2

∞ a.e. { on .(0, T ) and so .w ∈ L (Q). Then, .j (t, x, y0 )∞≤ βM |y0 | , whence .ϕ(y0 ) = Q j (t, x, y0 )dxdt < ∞ by (1.81). Since .w ∈ L (Q), it has the singular part ∗ equal to .0. This implies that .ID(ϕ) (0) = 0. Therefore, recalling (1.11), we have ∗ ∗ .j (t, x, w(t, x)) < ∞ and so .ϕ (w) < ∞, that is, .J∞ (y0 , w) < ∞. This means that .(y0 , w) ∈ U∞ . Equivalently, .J∞ is proper. u n

Otherwise, one can show that the level set of .J∞ is closed. Consequently to (1.81), we still have ϕ ∗ (w) =

.

≥

sup (w(y) − ϕ(y)) ≥

y∈L∞ (Q)

sup

y∈L∞ (Q), ||y||L∞ (Q) ≤ys

for any .w ∈ (L∞ (Q))' .

sup

y∈L∞ (Q), ||y||L∞ (Q) ≤ys

(w(y) − ϕ(y))

w(y) − C∞ = ys ||w||(L∞ (Q))' − C∞ ,

(1.83)

1.2 A Free Boundary Problem for a Special Fast Diffusion Case

25

Theorem 1.8 Problem .(P∞ ) has at least a solution. Proof By hypotheses (1.72) and (1.83), we easily see that .J∞ is lower bounded by .

1 ||y0 ||2V ' − C∞ − ys T ||f ||L∞ (Q) 2

−

so that the infimum exists, .d := inf(y,w)∈U∞ J∞ (y, w). Let .(yn , wn ) ∈ U∞ be a minimizing sequence for .J∞ . Then, we can write {

1 1 j (t, x, yn )dxdt + ϕ ∗ (wn ) + ||yn (T )||2V ' − ||y0 ||2V ' 2 2 Q { T 1 (L∞ (Ω))' ,L∞ (Ω) dt ≤ d + . − n 0

d≤ .

(1.84)

The pair .(yn , wn ) satisfies {

T

− .

0

/

dψ yn (t), (t) dt

\ L∞ (Ω),(L∞ (Ω)){'

+ (L∞ (Q))' ,L∞ (Q) =

dt − L∞ (Ω),(L∞ (Ω))'

0

T

(1.85) (L∞ (Ω))' ,L∞ (Ω) dt

∞ ∞ for any .ψ ∈ L∞ (0, T ; X), . dψ dt ∈ L (0, T ; L (Ω)), .ψ(T ) = 0. By (1.72) and (1.83), we have .

d + ys ||wn ||(L∞ (Q))' − C∞ − ys T ||f ||L∞ (Q) { 1 ≤ j (t, x, yn )dxdt + ϕ ∗ (wn ) + ||yn (T )||2V ' 2 Q ≤

1 1 ||y0 ||2V ' + ys T ||f ||L∞ (Q) + d + , 2 n

whence it follows that .

||yn ||L∞ (Q) ≤ ys , ||wn ||(L∞ (Q))' ≤ C, ||yn (T )||V ' ≤ C

with C a positive constant independent of .n. We deduce that on a subsequence .

yn → y weak* in L∞ (Q), as n → ∞, yn (T ) → ξ weakly in V ' , as n → ∞.

(1.86)

Since .(yn , wn ) ∈ U∞ , then .|yn (t, x)| ≤ ys a.e. on Q and it follows that |y(t, x)| ≤ ys a.e. on .Q.

.

26

1 Nonlinear Diffusion Equations with Slow and Fast Diffusion

We remark that .(wn )n is bounded in .(L∞ (Q))' and assert that .(wn )n is weak* compact in .(L∞ (Q))' . This assertion is pointed out in the proof of Corollary 2B in [70]. Another argument can be found in [25], and we adapt it below. We denote by .M(Q) the dual space of the separable space .C(Q), that is, the space of bounded Radon measures on .Q. Let us consider the linear operator .w : C(Q) → L∞ (Q), - , which maps a continuous function into the corresponding class of .wz = w - (of all functions a.e. equal). Its adjoint .w ∗ : (L∞ (Q))' → M(Q) is equivalence .w defined by .(w ∗ μ)(z) := μ(wz) for any .z ∈ C(Q). If .(μn )n is bounded in .(L∞ (Q))' and also in .M(Q), then .(w ∗ μn )n is bounded in .M(Q), and using Corollary 7.8 in the Appendix, it follows that .(w ∗ μn )n is weak* sequentially compact in .M(Q). Therefore, it follows that .(μn )n is weak* sequentially compact in .M(Q). Passing to - ∈ L∞ (Q), the limit in .μn (wz) = (w ∗ μn )(z), we get .μ(wz) = (w ∗ μ)(z) for any .w which is of the form .wz with .z ∈ C(Q). Since .w is an endomorphism from .C(Q) to .L∞ (Q), we conclude that .(μn )n is weak* sequentially compact in .(L∞ (Q))' . Therefore, one can extract a subsequence such that .wn → w weak* in ∞ ' .(L (Q)) , that is, .

(L∞ (Q))' ,L∞ (Q) → w(υ), as n → ∞, for any υ ∈ L∞ (Q)

(1.87)

and so .

(L∞ (Q))' ,L∞ (Q) → w(A∞ ψ), as n → ∞,

(1.88)

for any .ψ ∈ L∞ (0, T ; X). Passing to the limit in (1.85), we obtain that .(y, w) verifies (1.80). It remains to prove that .ξ = y(T ). To this end, we make a parenthesis for proving first that the function { .zn (t) := yn (t, x)φ(x)dx, a.e. t ∈ (0, T ) (1.89) Ω

is with bounded variation on .[0, T ]. For this, we shall need the kernel theorem of L. Schwartz (see [72]) extended here to functionals on .L∞ ((0, T ) × Ω). This theorem asserts that if ∞ ' t ∈ L(L∞ (Ω); (L∞ (0, T ))' ) and .w ∈ (L ((0, T ) × Ω)) , then there are .w x ∞ ∞ ' .w ∈ L(L (0, T ); (L (Ω)) ) such that w(θ ζ ) = w t (θ )(ζ ) = w x (ζ )(θ ), for all θ ∈ L∞ (Ω), ζ ∈ L∞ (0, T ).

.

The first equality can still be written .

/< \ > (L∞ (Q))' ,L∞ (Q) = w t , θ (L∞ (Ω))' ,L∞ (Ω) , ζ

for all .ζ ∈ L∞ (0, T ) and .θ ∈ L∞ (Ω).

(L∞ (0,T ))' ,L∞ (0,T )

(1.90)

1.2 A Free Boundary Problem for a Special Fast Diffusion Case

27

In particular, for .ψ(t, x) = φ(x)ζ (t) with .φ ∈ X, .ζ ∈ W 1,∞ (0, T ) and .ζ (T ) = 0, Eq. (1.85) yields { .

0

T

/

d dt { +

\ yn (t)φdx, ζ (t)

{ Ω

/< \ > wnt , A∞ φ (L∞ (Ω))' ,L∞ (Ω) , ζ (t)

T 0

{

0

{

d for all .φ ∈ X, where . dt implies

.

d dt

{ Ω

dt

(L∞ (0,T ))' ,L∞ (0,T )

dt

< > (L∞ (Ω))' ,L∞ (Ω) , ζ (t) (L∞ (0,T ))' ,L∞ (0,T ) dt,

T

=

(L∞ (0,T ))' ,L∞ (0,T )

Ω yn (t)φdx

is considered in the sense of distributions. This

< > yn (t)φdx + wnt (t), A∞ φ (L∞ (Ω))' ,L∞ (Ω) = (L∞ (Ω))' ,L∞ (Ω) , (1.91)

for all .φ ∈ X, whence we deduce that d . dt

{

yn (t)φdx ∈ (L∞ (0, T ))' for all φ ∈ X

Ω

|| { || || d || || . yn (t, x)φ(x)dx || || dt || Ω

(L∞ (0,T ))'

≤ C,

independently on n. It turns out that .t → zn (t) is with bounded variation on .[0, T ]. By the Helly theorem (see Theorem 7.13 in the Appendix), it follows that { yn (t)φdx → z(t) as n → ∞, for all t ∈ [0, T ].

.

Ω

Going back to (1.86), we write {

T

{

{

0

T

yn vdxdt →

.

0

Ω

{

yvdxdt, as n → ∞, for any v ∈ L∞ (0, T ; X),

Ω

which, for the particular choice .v = φζ, with .φ ∈ X, .ζ ∈ L∞ (0, T ), yields {

{ yn (t)φdx →

.

Ω

y(t)φdx, as n → ∞, for any φ ∈ X, a.e. t ∈ (0, T ). Ω

{ Therefore, .z(t) = Ω y(t, x)φ(x)dx, a.e. .t ∈ (0, T ). We recall that .(y, w) verifies (1.80), and by the previous reasoning, it turns out that z is with bounded variation, so that the previous relation extends to all .t ∈{ [0, T ]. Now, .yn (t) ∈ L∞ (Ω) ⊂ X' (since .X ⊂ L1 (Ω)), so that we can write . Ω yn (t)φdx = X' ,X and conclude that

28

1 Nonlinear Diffusion Equations with Slow and Fast Diffusion

yn (T ) → y(T ) weakly in X' , as n → ∞.

.

Moreover, by (1.90), we have / .

/ →

dyn , φζ dt d dt

\

/ (L∞ (Q))' ,L∞ (Q)

=

d dt

\

{ yn (t)φdx, ζ Ω

\

{ y(t)φdx, ζ

/ (L∞ (0,T ))' ,L∞ (0,T )

Ω

=

dy , φζ dt

(L∞ (0,T ))' ,L∞ (0,T )

(1.92)

\ (L∞ (Q))' ,L∞ (Q)

,

∞ ' and hence . dy dt ∈ L (Q)) . Actually, we have obtained that .(y, w) ∈ U∞ . By passing to the limit in (1.84) and using the weakly lower semicontinuity of the functions ∗ .y → j (t, x, y) and .w → ϕ (w), we obtain

{ d= .

1 1 j (t, x, y)dxdt + ϕ ∗ (w) + ||y(T )||2V ' − ||y0 ||2V ' 2 2 Q { T (L∞ (Q))' ,L∞ (Q) dt. − 0

This certifies that .(y, w) realizes the minimum in .(P∞ ) which ends the proof.

u n

Definition 1.9 A solution to .(P∞ ) is called a generalized solution to (1.1)–(1.3). Now, we prove that a null minimizer in .(P∞ ) is a generalized solution to (1.1)– (1.3). Theorem 1.10 Let the pair .(y, w) ∈ U∞ be a null minimizer in .(P∞ ), i.e., .

min(P∞ ) = J∞ (y, w) = 0,

and assume in addition that .

1 1 ||y(T )||2V ' − ||y0 ||2V ' − 2 2

{ 0

T

(L∞ (Q))' ,L∞ (Q) dt ≥ −w(y).

(1.93)

Then wa (t, x) ∈ β(t, x, y(t, x)) a.e. (t, x) ∈ Q,

(1.94)

ws ∈ ND(ϕ) (y),

(1.95)

.

.

where .ND(ϕ) (y) ⊂ (L∞ (Q))' is the normal cone to .D(ϕ) at .y. Proof Let .(y, w) be the null minimizer in .(P∞ ). Then

1.2 A Free Boundary Problem for a Special Fast Diffusion Case

{ J∞ (y, w) =

{ j (t, x, y(t, x))dxdt +

.

Q

Q

1 1 + ||y(T )||2V ' − ||y0 ||2V ' − 2 2

29

∗ j ∗ (t, x, wa (t, x))dxdt + ID(ϕ) (ws )

{ 0

T

(L∞ (Q))' ,L∞ (Q) dt = 0.

By (1.93), we have { ∗ j (t, x, y(t, x))dxdt + j ∗ (t, x, wa (t, x))dxdt + ID(ϕ) (ws ) Q Q { − wa ydxdt − ws (y) ≤ J∞ (y, w) = 0.

{ .

(1.96)

Q

We remark that (1.93) can take place, for example, when w is more regular. ∗ ∗ By the definition of .ID(ϕ) (ws ) (see (7.42)), it follows that .ID(ϕ) (ws ) ≥ ws (y). Since by the Legendre–Fenchel relation {

{

{

j (t, x, y(t, x))dxdt +

.

Q

j (t, x, wa (t, x))dxdt − Q

wa ydxdt ≥ 0, Q

we see that the left-hand side in (1.96) is nonnegative and so it follows to be zero { .

{ ∗ j (t, x, y(t, x))dxdt + j ∗ (t, x, wa (t, x))dxdt + ID(ϕ) (ws ) Q Q { − wa ydxdt − ws (y) = 0. Q

This implies that {

(j (t, x, y) + j ∗ (t, x, wa ) − wa y)dxdt = 0

.

Q

and .

∗ − ws (y) + ID(ϕ) (ws ) = 0.

First we deduce that .j (t, x, y) + j ∗ (t, x, wa ) − wa y = 0 a.e. on Q and so wa (t, x) ∈ β(t, x, y(t, x)) a.e. (t, x) ∈ Q.

.

∗ From the second equation, .ws (y) = ID(ϕ) (ws ) = {ws (ψ); .ψ ∈ D(ϕ)}, we get that ∗ .ws belongs to the normal cone to .D(ϕ) at .y. Indeed, by the definition of .I D(ϕ) (ws ) = supz∈D(ϕ) {ws (z)}, we have .ws (y) ≥ ws (z) for any .z ∈ D(ϕ), whence .ws (y−z) ≥ 0 for any .z, which implies (1.95), as claimed. u n

30

1 Nonlinear Diffusion Equations with Slow and Fast Diffusion

We give an interpretation of the result in Theorem 1.10 in relation with the physical significance of this case, actually explaining the meaning of a generalized solution. We shall see that the diffusion equation splits in a continuous component and a singular component and this result has a physical interpretation of a simultaneous development of unsaturated and saturated flow (roughly speaking, a phase change phenomenon). Equation (1.80) can be still written in the sense of distributions as .

dy (ψ) + wa (A∞ ψ) + ws (A∞ ψ) e f (ψ), dt

for any .ψ ∈ W 1,∞ (0, T ; L∞ (Ω)) ∩ L∞ (0, T ; X), .ψ(T ) = 0. ∞ ' We note that . dy dt ∈ (L (Q)) and so we write it as the sum between a continuous and a singular component, denoted by the subscripts a and s .

dy = dt

(

dy dt

)

( +

a

dy dt

) . s

Separating the equation corresponding to the singular contribution, we obtain ( .

dy dt

) (ψ) + ws (A∞ ψ) e 0,

(1.97)

s

so that for the continuous part we have ( .

dy dt

) (ψ) + wa (A∞ ψ) e f (ψ), wa (t, x) ∈ β(t, x, y(t, x)) a.e. (t, x) ∈ Q, a

(1.98) for any .ψ ∈ W 1,∞ (0, T ; L∞ (Ω)) ∩ L∞ (0, T ; X), .ψ(T ) = 0. Let .y ∈ int .Qu := {y ∈ L∞ (Q); ess sup y(t, x) < ys a.e. on Q}, the interior (t,x)∈Q

of .Qu . The normal cone to .Qu at y reduces to .{0}, implying by (1.95) that .ws = ∞ .∂Qu = {y ∈ L (Q); 0. It turns out that the support of .ws is on the ( boundary ) dy ess. sup y = ys }. Then, by (1.97), we get . dt = 0 on .Q. We deduce that s (t,x)∈Q ( ) ( ) = dy only the first component . dy dt dt a remains and (1.98) reads in the sense of distributions ( ) dy . − ∆β(t, x, y) = f, a.e. on Qu = {(t, x); y(t, x) < ys }, (1.99) dt a with the initial condition .y(0) = y0 and the Robin condition for .β(t, x, y). This equation describes the flow in the unsaturated domain .Qu . When .y ∈ ∂Qu , then there exists a subset .Qs of Q where .y(t, x) = ys , with meas.(Qs ) > 0. Then, on the one hand, we have equation (1.99), and on the other

1.2 A Free Boundary Problem for a Special Fast Diffusion Case

31

hand we have a singular contribution (1.97). Taking into account that the support of ws is in the set .Qs = {(t, x); .y(t, x) = ys }, we can rewrite (1.97) as

.

( .

dy dt

) − ∆ws e 0 on Qs ,

(1.100)

s

where .ws ∈ ND(ϕ) (y) and .∆ws are considered in the sense of distributions. Here, (1.99) expresses the saturation process. We also mention that if .J∞ is strictly convex, then the solution to .(P∞ ) is unique and this implies that the generalized solution is unique, as well. This happens, e.g., when .β ' (t, x, r) ≥ ρ > 0, that is, in the nondegenerate case for the diffusion coefficient.

Chapter 2

Weakly Coercive Nonlinear Diffusion Equations

In this chapter we address the existence of solutions to nonlinear diffusion equations (N DEs) of type . ∂y ∂t (t, x) − ∆β(t, x, y(t, x)) e f (t, x), with a weaker assumption for the potential than that considered in Chap. 1, more precisely with a monotonically increasing time and space depending nonlinearity .β(t, x, ·) provided by a potential .j (t, x, ·) having a weak coercivity property. For this reason, we call this .(N DE) weakly coercive. These hypotheses may apply to .j (t, x, r) of polynomial type with respect to .r, with the exponent .m ∈ (1, 2) characterizing a fast diffusion. Also, the results in this chapter provide existence under very general conditions on the nonlinear function .β with more complicated forms, as for instance:

.

β(t, x, r) = sign(r) log(|r| + a(t, x)), a(t, x) ≥ a0 > 0,

.

β(t, x, r) = sign(r) exp(a(t, x)r 2 ), a(t, x) ≥ a0 > 0, where sign is the multivalued function signum. Relevant examples of mathematical models with .β having such expressions are related to image denoising processes. The purpose is to show by a dual formulation that the solution of the nonlinear equation can be obtained as the null minimizer of an appropriate minimization problem for a convex functional involving the potential of the nonlinearity and its conjugate .j ∗ . With respect to the treatment of the cases which assumed a polynomial behavior of j in Chap. 1, the present case requires a sharp analysis in the .L1 -space on which both j and .j ∗ are defined and uses different arguments and techniques in the proofs. An optimization problem .(P ) is introduced in an appropriate functional framework involving the duality .L1 -.L∞ . The existence of at least a solution to .(P ) when j depends on time and space is proved in Theorem 2.3, this being viewed as a generalized solution associated with the nonlinear problem .(N DE). The uniqueness is deduced under the assumption of the strict convexity of .j. Moreover, when a boundedness state constraint is included in the admissible set, Theorem 2.5 shows that the null minimization solution is the unique solution to .(N DE).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Marinoschi, Dual Variational Approach to Nonlinear Diffusion Equations, PNLDE Subseries in Control 102, https://doi.org/10.1007/978-3-031-24583-1_2

33

34

2 Weakly Coercive Nonlinear Diffusion Equations

In the case when j does not depend on x but only on t and has the same behavior at .|r| large, Theorem 2.7 proves that the solution to the minimization problem is the unique solution to .(N DE) without assuming the previous additional state constraint. This is based on Lemma 2.6, which plays an essential role in the proof of this result. We mention that stochastic porous media equations were studied under similar assumptions in [24], by a different method.

2.1 Problem Presentation We consider the problem .

∂y − ∆β(t, x, y) e f, in Q := (0, T ) × Ω, ∂t

∂β(t, x, y) + αβ(t, x, y) e 0, on Σ := (0, T ) × f, ∂ν y(0, x) = y0 , in Ω,

(2.1)

where the domain .Ω is an open bounded subset of .RN , .N ≤ 3, having the boundary ∂ 1 .f of class .C , T is finite, . ∂ν denotes the outward normal derivative on .f, and .α is positive. As in the first chapter, .β is derived from a potential .j. We assume that the function j : Q × R → (−∞, ∞]

.

and its conjugate j ∗ : Q × R → (−∞, ∞]

.

have the properties: (H1 ) (H2 )

(t, x) → j (t, x, r) is measurable on .Q, for all .r ∈ R j (t, x, ·) is proper, convex, and continuous function, a.e. .(t, x) ∈ Q, such that

.

.

.

.

∂j (t, x, r) = β(t, x, r) for all r ∈ R, a.e. (t, x) ∈ Q,

(2.2)

j (t, x, r) → ∞, as |r| → ∞, uniformly for (t, x) ∈ Q, |r|

(2.3)

j ∗ (t, x, ω) → ∞, as |ω| → ∞, uniformly for (t, x) ∈ Q, |ω|

(2.4)

.

.

.

j (·, ·, 0) ∈ L∞ (Q), j ∗ (·, ·, 0) ∈ L∞ (Q),

.

(2.5)

2.1 Problem Presentation

.

35

(∃) ξ, η ∈ L∞ (Q) such that ξ ∈ ∂j (t, x, 0), η ∈ ∂j ∗ (t, x, 0) a.e. (t, x) ∈ Q,

(2.6)

where we recall that .j ∗ is defined by j ∗ (t, x, ω) = sup (ωr − j (t, x, r)), a.e. (t, x) ∈ Q. r∈R

(2.7)

.

By (2.2), it follows that .β is a maximal monotone graph (it may be multivalued) on .R, a.e. on .Q. Relations (2.3)–(2.4) are equivalent with the boundedness of −1 (t, x, ·) and .β(t, x, ·) on bounded subsets, uniformly a.e. .(t, x) ∈ Q. This .(β) means that for any .M > 0, there exist .YM and .WM , independent on t and .x, such that { } −1 . sup |r| ; r ∈ β (t, x, ω), |ω| ≤ M ≤ WM , . (2.8) sup {|ω| ; ω ∈ β(t, x, r), |r| ≤ M} ≤ YM

(2.9)

(see Proposition 7.35 in the Appendix). In particular, when j does not depend on t and x, relations (2.3)–(2.4) express that D(∂j (r)) = R(∂j (r)) = R, D(∂j ∗ (r)) = R(∂j ∗ (r)) = R.

.

We also recall that ∂j ∗ (t, x, ·) = (∂j (t, x, ·))−1 a.e. on Q.

.

We call weakly coercive the functions j and .j ∗ with the properties (2.3)–(2.4). The functions j and .j ∗ are bounded below by affine functions j (t, x, r) ≥ k1 (t, x)r + k2 (t, x), j ∗ (t, x, ω) ≥ k3 (t, x)ω + k4 (t, x)

.

(2.10)

for any .r, ω ∈ R (see Proposition 7.27 in the Appendix) where we assume that ki ∈ L∞ (Q), i = 1, ..., 4.

.

(2.11)

Moreover, .j ∗ is proper, convex, and l.s.c. and j and .j ∗ satisfy the Legendre–Fenchel relations (1.9)–(1.10) (see Proposition 7.37 in the Appendix). A potential j and its conjugate .j ∗ with the properties (2.3) and (2.4) appropriately model image denoising processes, as we shall see in a further chapter. The purpose is to prove the existence of a (unique) solution to (2.1) by a variational technique involving the duality Legendre–Fenchel relations.

36

2 Weakly Coercive Nonlinear Diffusion Equations

2.2 Functional Setting and the Minimization Problem We resume the definitions of some spaces and operators already introduced in Sect. 1.1. Thus, V is the Hilbert space .H 1 (Ω) with the norm .

)1/2 ( ||φ||V = ||∆φ||2L2 (Ω) + α ||φ||2L2 (f)

and .V ' is its dual with the scalar product / \ (θ, θ )V ' = θ, A−1 θ V

.

(2.12)

V ' ,V ,

where .AV : V → V ' is the operator { .

V ' ,V =

{ ∇ψ · ∇φdx + Ω

αψφdσ, for any φ ∈ V .

(2.13)

f

We recall the space } { ∂ψ 2,2 + αψ = 0 on f , .X2 := ψ ∈ W (Ω); ∂ν with .X2' its dual, and the operator A2 : D(A2 ) ⊂ L2 (Ω) → L2 (Ω), D(A2 ) := X2 , A2 ψ = −∆ψ.

.

(2.14)

The operator .A2 is m-accretive on .L2 (Ω), coercive, and surjective. Its extension to L2 (Ω) is

.

-2 : L2 (Ω) ⊂ X2' → X2' , A

.

defined by < > -2 θ, ψ ' . A = X ,X 2

2

{ θ A2 ψdx, ∀θ ∈ L2 (Ω), ∀ψ ∈ X2 ,

(2.15)

Ω

and it is surjective on .X2' , as shown in Lemma 7.21 in the Appendix. We note that .AV is the extension of .A2 defined by (2.14) to .V ' . With respect to the case treated in Chap. 1, the weakly coercive case will be set up in a new functional framework. We recall the definition from Chap. 1 of the space .X∞ (denoted for simplicity .X) X :=

.

⎧ ⎨ ⎩

ψ∈

n p≥2

W 2,p (Ω); ∆ψ ∈ L∞ (Ω),

⎫ ⎬

∂ψ + αψ = 0 on f ⎭ ∂ν

(2.16)

2.2 Functional Setting and the Minimization Problem

37

and of the operator A∞ : D(A∞ ) ⊂ L∞ (Ω) → L∞ (Ω), D(A∞ ) := X, A∞ ψ = −∆ψ.

.

(2.17)

We introduce the operator A : D(A) ⊂ L1 (Ω) → L1 (Ω), Aψ = −∆ψ, } { ∂ψ . + αψ = 0 on f , D(A) = ψ ∈ W 1,1 (Ω); ∆ψ ∈ L1 (Ω), ∂ν

(2.18)

which is m-accretive on .L1 (Ω), as proved in [31]. To study the problem corresponding to the weakly coercive case, we extend A to 1 .L (Ω) in the following way: - := L1 (Ω), - : D(A) - ⊂ X' → X' , D(A) A { > .< - ψ ' = Aθ, θ A∞ ψdx, ∀θ ∈ L1 (Ω), ∀ψ ∈ X. X ,X

(2.19)

Ω

Definition 2.1 Let .f ∈ L∞ (0, T ; L2 (Ω)) and .y0 ∈ V ' . We call a solution to (2.1) a pair .(y, w) y ∈ L1 (Q) ∩ W 1,1 (0, T ; X' ) ∩ C([0, T ]; X' ),

.

η ∈ L1 (Q), η(t, x) ∈ β(t, x, y(t, x)) a.e. (t, x) ∈ Q, which satisfies the equation {

T

/

.

0

dy (t), ψ(t) dt

\ X' ,X

{

{

dt + Q

ηA∞ ψdxdt =

T 0

X' ,X dt

(2.20)

for any .ψ ∈ L∞ (0, T ; X), and the initial condition .y(0) = y0 . With these considerations, we can introduce the minimization problem Minimize J (y, w),

.

(P )

(y,w)∈U

where ⎧{ ( ) 1 . . j (t, x, y(t, x)) + j ∗ (t, x, w(t, x)) dxdt + ||y(T )||2V ' . . 2 ⎨ Q { T{ 1 .J (y, w) = −1 2 − ||y0 ||V ' − yA2 f dxdt if (y, w) ∈ U, . . . 2 Ω 0 . ⎩ +∞, otherwise, (2.21)

38

2 Weakly Coercive Nonlinear Diffusion Equations

and { U = (y, w); y ∈ L1 (Q) ∩ W 1,1 (0, T ; X' ), y(T ) ∈ V ' , w ∈ L1 (Q), . j (·, ·, y(·, ·)) ∈ L1 (Q), j ∗ (·, ·, w(·, ·)) ∈ L1 (Q), } (y, w) satisfies (2.23) below , dy (t) + Aw(t) = f (t) a.e. t ∈ (0, T ), . dt y(0) = y0 .

(2.22)

(2.23)

' Here, . dy dt is viewed in the sense of .X -valued distributions on .(0, T ). −1 −1 By .A2 f , we actually mean .A2 f (t), where .f (t) ∈ L2 (Ω). By the existence theory of elliptic boundary value problems, it follows that ∞ A−1 2 f (t) ∈ X2 ⊂ L (Ω), a.e. t ∈ (0, T )

.

(see [1]), so the last term in the expression of J makes sense. The minimization problem .(P ) can be treated as an optimal control problem with the state .y, required to be in .L1 (Q), controlled by the controller .w ∈ L1 (Q) such that to reach the minimum of the functional .J. In the subsequent part, we assume that the admissible set U is nonempty, that is, U /= ∅,

(2.24)

.

and this implies that the functional J is proper. For example, relation { t (2.24) can take place because the pair .(y(t), 0) ∈ U, where .y(t) = y0 + 0 f (s)ds with .y0 ∈ L∞ (Ω). Indeed, .y ∈ L∞ (Q) and ∗ ∞ 1 .j (·, ·, 0) ∈ L (Q) ⊂ L (Q) by (2.5). Moreover, by the definition of the subdifferential (Definition 7.28 in the Appendix), we have j (t, x, y(t, x)) − j (t, x, 0) ≤ η(t, x)y(t, x), η(t, x) ∈ β(t, x, y(t, x)) a.e. on Q.

.

Owing to (2.5) and (2.9), it follows that {

{

{ j (t, x, y)dxdt ≤

.

Q

so that .J (y, 0) < ∞.

j (t, x, 0)dxdt + YM Q

0

T

{ y(t, x)dxdt < ∞, Ω

2.3 A Time and Space Dependent Potential

39

2.3 A Time and Space Dependent Potential In this part we focus on the case when j and .j ∗ depend on t and .x. We assume .(H1 )–.(H2 ). Lemma 2.2 The function J is convex and lower semicontinuous on .L1 (Q) × L1 (Q). Proof We see that J is convex because each term is convex. For the convexity of the first two terms, we recall Proposition 7.38 in the Appendix. It remains to prove the semicontinuity property. To this end, we prove that the level set Sλ (J ) = {(y, w) ∈ L1 (Q) × L1 (Q); J (y, w) ≤ λ}

.

is closed in .L1 (Q) × L1 (Q), for .λ > 0. Let .(yn , wn ) ∈ Sλ (J ) such that yn → y strongly in L1 (Q), wn → w strongly in L1 (Q), as n → ∞.

.

(2.25)

Since .J (yn , wn ) ≤ λ, it follows that .(yn , wn ) ∈ U, and it is the solution to dyn - n (t) = f (t), a.e. t ∈ (0, T ), (t) + Aw . dt yn (0) = y0

(2.26)

and {

(j (t, x, yn (t, x)) + j ∗ (t, x, wn (t, x)))dxdt

J (yn , wn ) =

.

(2.27)

Q

} 1{ ||yn (T )||2V ' − ||y0 ||2V ' − 2

+

{ Q

yn A−1 2 f dxdt ≤ λ.

The convergences in (2.25) imply that {

T

.

< > - n (t), ψ(t) Aw

dt = X' ,X

0

{ Q

{ wn A∞ ψdxdt →

Q

wA∞ ψdxdt, as n → ∞,

for any .ψ ∈ L∞ (0, T ; X) and { .

Q

yn A−1 2 f dxdt →

By (2.26), we can write

{ Q

yA−1 2 f dxdt, as n → ∞.

40

2 Weakly Coercive Nonlinear Diffusion Equations

{

/

T

.

0

dyn (t), ψ(t) dt

\

{

X' ,X

{

dt = − Q

wn A∞ ψdxdt +

f ψdxdt,

(2.28)

Q

for any .ψ ∈ L∞ (0, T ; X), and we deduce that .

dyn dy → weakly in L1 (0, T ; X' ) as n → ∞, dt dt

meaning that .yn is absolutely continuous on .[0, T ] with values in .X' . Also, by (2.26), we have {

t

yn (t) = y0 +

.

{

t

f (s)ds −

0

- n (s)ds, for t ∈ [0, T ]. Aw

(2.29)

0

From here, we get { .

Ω

yn (t)φdx = V ' ,V + {

f (s)φdxds 0

X' ,X

0

(2.30)

Ω

< > - n (s), φ Aw

t

−

{ t{

ds,

for any .φ ∈ X and .t ∈ [0, T ]. Passing to the limit, we obtain { l(t) = lim

.

n→∞ Ω

{

t

− 0

yn (t)φdx = V ' ,V +

{ t{ f (s)φdxds 0

Ω

< > Aw(s), φ X' ,X ds.

By multiplying by .ϕ0 ∈ L∞ (0, T ) and integrating over .(0, T ), we get that {

T

(2.31)

ϕ0 (t)l(t)dt

.

0

{

T

= 0

( ) { t{ { t < > V ' ,V + Aw(s), φ X' ,X ds ϕ0 (t)dt. f (s)φdxds − 0

0

Ω

Now, we multiply (2.29) by .ϕ0 (t)φ(x) and integrate over .(0, T ) × Ω. We have {

{

T

ϕ0 φyn dxdt =

.

Q

0

( ) { { t V ' ,V + f (s)φdsdx ϕ0 (t)dt (2.32)

{

T

− 0

{ 0

Ω 0

t

< > - n (s), φ Aw

X' ,X

ϕ0 (t)dsdt.

2.3 A Time and Space Dependent Potential

41

By the strong convergence .yn → y in .L1 (Q), we deduce {

{ ϕ0 φydxdt =

.

T

0

Q

( ) { { t V ' ,V + f (s)φdsdx ϕ0 (t)dt (2.33) Ω 0

{

T

−

{

0

< > Aw(s), φ X' ,X ϕ0 (t)dsdt.

t 0

Comparing (2.31) and (2.33), we deduce that {

T

.

{

ϕ0 φydxdt for all ϕ0 ∈ L∞ (0, T ),

ϕ0 (t)l(t)dt =

0

Q

whence {

{ l(t) = lim

.

n→∞ Ω

yn (t)φdx =

y(t)φdx for all φ ∈ X, t ∈ [0, T ]. Ω

This yields yn (t) → y(t) weakly in X' as n → ∞, for all t ∈ [0, T ]

.

(2.34)

and, therefore, yn (T ) → y(T ), yn (0) → y(0) = y0 weakly in X' , as n → ∞.

.

Letting .n → ∞ in (2.28), we obtain {

T

.

0

/

\ { T{ dy (t), ψ(t) dt + wA∞ ψ(t)dxdt dt Ω 0 X' ,X { T X' ,X dt, = 0

which proves that .(y, w) is a solution to (2.23). By (2.27) and (2.10), we can write that { Q .

1 (k1 yn + k2 + k3 wn + k4 )dxdt + ||yn (T )||2V ' 2 { 1 ≤ (j (t, x, yn (t, x)) + j ∗ (t, x, wn (t, x)))dxdt + ||yn (T )||2V ' 2 Q || || 1 || −1 || 2 ≤ ||y0 ||V ' + ||yn ||L1 (Q) ||A2 f || ∞ + λ ≤ C, L (Q) 2

whence, using (2.25), we get

(2.35)

42

2 Weakly Coercive Nonlinear Diffusion Equations

.

( ) 1 ||yn (T )||2V ' ≤ C ||y||L1 (Q) + ||w||L1 (Q) + 1 + λ = C1 2

with C and .C1 constants. It follows that .yn (T ) → ξ weakly in .V ' as .n → ∞. As seen earlier, .yn (T ) → y(T ) weakly in .X' , and by the uniqueness of the limit, we get .ξ = y(T ) ∈ V ' . The function { ϕ : L (Q) → R, ϕ(z) =

.

1

j (t, x, z(t, x))dxdt Q

is proper, convex, and l.s.c. (see Proposition 7.38 in the Appendix). Then, { ϕ(y) ≤ lim inf ϕ(yn ) = lim inf

.

n→∞

n→∞

j (t, x, yn (t, x))dxdt < ∞.

(2.36)

Q

{ Also, . Q j ∗ (t, x, w(t, x))dxdt < ∞, and thus we have shown that .(y, w) ∈ U. By passing to the limit in (2.27) as .n → ∞, we obtain by the lower semicontinuity property of the integrand that {

(j (t, x, y(t, x)) + j ∗ (t, x, w(t, x)))dxdt +

.

Q

−

1 ||y0 ||2V ' − 2

{ Q

1 ||y(T )||2V ' 2

yA−1 2 f dxdt ≤ lim inf J (yn , wn ) ≤ λ, n→∞

which means that .(y, w) ∈ Sλ (J ). This ends the proof. Theorem 2.3 Problem .(P ) has at least a the solution of .(P ) is unique.

solution .(y ∗ , w ∗ ).

u n If j is strictly convex,

Proof By (2.10), we note that if .(y, w) ∈ U, then J (y, w) ≥ − ||k1 ||∞ ||y||L1 (Q) − ||k2 ||∞ − ||k3 ||∞ ||w||L1 (Q) − ||k4 ||∞ || || 1 || || − ||y0 ||2V ' − ||y||L1 (Q) ||A−1 2 f ||L∞ (Q) . 2

.

Let us write d :=

.

inf

(y,w)∈U

J (y, w).

We assume first that .d > −∞ and we shall show later that this is the only possibility. Let us consider a minimizing sequence .(yn , wn ) ∈ U , such that d ≤ J (yn , wn ) ≤ d +

.

where the pair .(yn , wn ) satisfies (2.26).

1 , n

(2.37)

2.3 A Time and Space Dependent Potential

43

By (2.3)–(2.4), for any .M > 0, there exist .CM and .DM such that j (t, x, r) > M |r| as |r| > CM ,

.

and j ∗ (t, x, ω) > M |ω| as |ω| > DM .

.

Then, (2.37) can be written as follows: { .

{ {(t,x);|yn (t,x)|≤CM }

j (t, x, yn (t, x))dxdt

{ +

{(t,x);|wn (t,x)|≤DM }

+M

{(t,x);|yn (t,x)|>CM }

j ∗ (t, x, wn (t, x))dxdt + M

1 1 1 + ||yn (T )||2V ' − ||y0 ||2V ' ≤ d + 2 2 n { || || ({ || || |y | + ||A−1 dxdt + f || n 2 ∞

{(t,x);|yn (t,x|)≤CM }

|yn | dxdt

{ {(t,x);|wn (t,x)|>DM }

{(t,x);|yn (t,x)|>CM }

|wn | dxdt

) |yn | dxdt .

Denoting .

|| || || −1 || ||A2 f ||

L∞ (Q)

=: f∞

and taking M large enough such that .M > f∞ , it follows that { I0 := (M − f∞ )

.

{ +M

{(t,x);|yn (t,x)|>CM }

{(t,x);|wn (t,x)|>DM }

|yn | dxdt

|wn | dxdt +

1 ||yn (T )||2V ' 2

{ 1 ||y0 ||2V ' + f∞ CM meas(Q) + |j (t, x, yn (t, x))| dxdt 2 {(t,x);|yn (t,x)|≤CM } { | | ∗ |j (t, x, wn (t, x))| dxdt + d + 1. + ≤

{(t,x);|wn (t,x)|≤DM }

Next, recalling (2.10), we denote j ∗ (t, x, ω) = j ∗ (t, x, ω) − k3 ω − k4 , j (t, x, r) = j (t, x, r) − k1 r − k2 , -

.

44

2 Weakly Coercive Nonlinear Diffusion Equations

and replacing j and .j ∗ in the previous estimation for .I0 , we have I0 ≤

.

1 ||y0 ||2V ' + f∞ CM meas(Q) 2 { | | |j (t, x, yn (t, x))| dxdt + { + { + { +

{(t,x);|yn (t,x)|≤CM }

{(t,x);|wn (t,x)|≤DM }

{(t,x);|yn (t,x)|≤CM }

| | ∗ |j (t, x, wn (t, x))| dxdt + d + 1

|k1 yn + k2 | dxdt

{(t,x);|wn (t,x)|≤DM }

|k3 wn + k4 | dxdt.

Recalling (2.2), (2.9), and (2.8), we can write 1 j (t, x, yn (t, x)) ≤ |j (t, x, 0)| + |ηn (t, x)| |yn (t, x)| ≤ YM

.

on .{(t, x); |yn (t, x)| ≤ CM }, | | 1 j ∗ (t, x, wn (t, x)) ≤ |j ∗ (t, x, 0)| + |εn (t, x)| |y(t, x)| ≤ WM

.

on .{(t, x); |wn (t, x)| ≤ DM }, where ηn (t, x) ∈ β(t, x, yn ) and εn (t, x) ∈ (β)−1 (t, x, wn ) a.e. on Q.

.

Then, 1 0≤j (t, x, yn (t, x)) ≤ YM + ||k1 ||∞ CM + ||k2 ||∞

.

(2.38)

on .{(t, x); |yn (t, x)| ≤ CM }, 1 0≤j ∗ (t, x, wn (t, x)) ≤ WM + ||k3 ||∞ DM + ||k4 ||∞

.

on .{(t, x); |wn (t, x)| ≤ DM }, and we deduce that { |yn | dxdt (M − f∞ ) {(t,x);|y { n (t,x)|>CM } . 1 |wn | dxdt + ||yn (T )||2V ' ≤ C + d. +M 2 {(t,x);|wn (t,x)|>DM }

(2.39)

(2.40)

2.3 A Time and Space Dependent Potential

45

This yields .

||yn ||L1 (Q) ≤ C, ||wn ||L1 (Q) ≤ C, ||yn (T )||V ' ≤ C.

(2.41)

Here, C stands for several constants independent on n. From (2.37), we get { In :=

{

j ∗ (t, x, wn (t, x))dxdt ≤ C.

j (t, x, yn (t, x))dxdt +

.

Q

(2.42)

Q

We continue by proving that each term is bounded, i.e., {

{

j ∗ (t, x, wn (t, x))dxdt ≤ C2 .

j (t, x, yn (t, x))dxdt ≤ C1 ,

.

Q

(2.43)

Q

We write { In =

.

{(t,x);|yn (t,x)|≤M}

j (t, x, yn (t, x))dxdt

{

+ { + { +

{(t,x);|yn (t,x)|>M}

{(t,x);|wn (t,x)|≤M}

{(t,x);|wn (t,x)|>M}

j (t, x, yn (t, x))dxdt j ∗ (t, x, wn (t, x))dxdt j ∗ (t, x, wn (t, x))dxdt ≤ C.

Therefore, using (2.38) and (2.39), we have { {(t,x);|yn (t,x)|>M} {

.

+ ≤

j (t, x, yn (t, x))dxdt

j ∗ (t, x, wn (t, x))dxdt {(t,x);|wn (t,x)|>M} 1 1 C + YM meas(Q) + WM meas(Q) = C3 .

Since .j (t, x, yn (t, x)) ≥ k1 (t, x)yn (t, x) + k2 (t, x), it follows that { .

{(t,x);|wn (t,x)|>M}

j ∗ (t, x, wn (t, x))dxdt ≤ C4 ,

whence {

j ∗ (t, x, wn (t, x))dxdt ≤ C1 .

.

Q

(2.44)

46

2 Weakly Coercive Nonlinear Diffusion Equations

Finally, (2.42) yields { j (t, x, yn (t, x))dxdt ≤ C2 ,

.

(2.45)

Q

with .C1 and .C2 independent of .n. Next, we shall show that the sequences .(yn )n and .(wn )n are weakly compact in 1 .L (Q). { To this end, we have to show that the integrals . A |wn | dxdt, with .A ⊂ Q, are {equi-absolutely continuous, meaning that for every .ε > 0 there exists .δ such that . A |wn | dxdt < ε whenever meas.(A) < δ. Here, meas is the Lebesgue measure of .A. Let .Mε > 2Cε 2 , where .C2 is the constant in (2.43), and let .RM be such that ∗ j (t,x,wn ) . ≥ Mε for .|wn | > RM , by (2.4). If .δ < 2RεM , we can write |wn | {

{ |wn | dxdt ≤

.

A

{

{(t,x);|wn (t,x)|>RM }

≤ Mε−1

{

|wn | dxdt +

{(t,x);|wn (t,x)|≤RM }

|wn | dxdt

j ∗ (t, x, wn (t, x))dxdy + RM δ < ε.

Q

Hence, by the Dunford–Pettis theorem (see Theorem 7.3 in the Appendix), it follows that .(wn )n is weakly compact in .L1 (Q). In a similar way, we proceed for showing the weakly compactness of the sequence .(yn )n . Selecting some subsequences, we obtain .

yn → y ∗ weakly in L1 (Q), wn → w ∗ weakly in L1 (Q) as n → ∞, - ∗ weakly in L1 (0, T ; X' ), as n → ∞, - n → Aw Aw

by (2.19). This implies by (2.26) that .

dy ∗ dyn → weakly in L1 (0, T ; X' ) as n → ∞. dt dt

Passing to the limit in (2.28), we get that .(y ∗ , w ∗ ) verifies (2.20) {

/

T

.

0

dy ∗ (t), ψ(t) dt

\ X' ,X

{ dt + Q

w ∗ A∞ ψ(t)dxdt =

{ 0

T

X' ,X dt

or, equivalently, (2.23). Next, we show that yn (T ) → y ∗ (T ) and yn (0) → y ∗ (0) = y0 weakly in V ' , as n → ∞,

.

in a similar way as in Lemma 2.2.

2.3 A Time and Space Dependent Potential

47

Finally, by passing to the limit in (2.37), on the basis of the weakly lower semicontinuity of the convex functional J on .L1 (Q) × L1 (Q), we obtain J (y ∗ , w ∗ ) = d.

.

In conclusion, we have got that .y ∗ ∈ L1 (Q), .w ∗ ∈ L1 (Q), .y ∗ (T ) ∈ V ' , and ∗ ∗ .(y , w ) satisfies (2.23). By (2.43) and the weakly lower semicontinuity property, we get {

j (t, x, y ∗ (t, x))dxdt < ∞,

.

Q

{

j ∗ (t, x, w ∗ (t, x))dxdt < ∞.

Q

With these relations, we have ended the proof that .(y ∗ , w ∗ ) belongs to U and that it is a solution to .(P ). Going back to the beginning, we recall that up to now we have assumed that .d > −∞. Indeed, otherwise, for every K real positive, there exists .nK , such that for every .n ≥ nK we have .J (yn , wn ) < −K. Following the computations in the same way as before, we arrive at the inequality (2.40), which reads now {

{ (M − f∞ )

.

{(t,x);|yn (t,x)|>CM }

|yn | dxdt + M +

{(t,x);|wn (t,x)|>DM }

|wn | dxdt

1 ||yn (T )||2V ' ≤ C − K. 2

Since C is a fixed constant, this may imply .C − K < 0, for K large enough, and this leads to a contradiction. So, the only possibility is .d > −∞, already treated. Eventually, let us prove the solution uniqueness when j is strictly convex. Let .(y1 , w1 ) and .(y2 , w2 ) be two different solutions to .(P ). We have ( .

J

) y1 + y2 w1 + w2 , 2 2 ( ) )) { ( ( y1 + y2 w1 + w2 j t, x, (t, x) + j ∗ t, x, (t, x) dxdt = 2 2 Q || ||2 { || y1 + y2 1 || y1 + y2 −1 1 2 || || + || (T )|| − ||y0 ||V ' − A2 f dxdt. 2 2 2 2 Q V'

Now, we recall the inequality .

|| ||2 ||2 ||2 || ||2 || || || || || || || || || 1 || || y1 (T ) + y2 (T ) || ≤ || y1 (T ) || + || y2 (T ) || − 1 || y1 (T ) − y2 (T ) || . || || || || || || || 2 2 2 2 2 2 2 2 ||V ' V' V' V'

48

2 Weakly Coercive Nonlinear Diffusion Equations

The, using the strictly convexity of j and the convexity of .j ∗ , we can write ( J

) y1 + y2 w1 + w2 . , 2 2 2 {{ ( ) 1Σ j (t, x, yi ) + j ∗ (t, x, wi ) − yi A−1 f dxdt < 2 2 Q

(2.46)

i=1

|| ||2 } || 1 1 || y1 − y2 1 || + ||yi (T )||2V ' − ||y0 ||2V ' − || (T ) || || ' 2 2 2 2 V || ||2 || 1 1 || y1 − y2 (T )|| = (J (y1 , w1 ) + J (y2 , w2 )) − || || || ' . 2 2 2 V Let .d := inf(y,w)∈U J (y, w). Since both .(y1 , w1 ) and .(y2 , w2 ) are solutions to .(P ), it follows that .d = J (yj , wj ), .j = 1, 2. Then ( d≤J

.

y1 + y2 w1 + w2 , 2 2

) N = 1,

.

W 1,q (Ω) ⊂ Lp (Ω), p ∈ [1, ∞), if q = N = 1, W 1,q (Ω) ⊂ Lp (Ω), p ∈ [1, q ∗ ),

1 1 1 = − , if N = 2, 3. q∗ q N

52

2 Weakly Coercive Nonlinear Diffusion Equations

Therefore, .q ∗ =

Nq N −q

≥ 2 and it follows that W 1,q (Ω) ⊂ L2 (Ω), for N ≤ 3.

.

(2.55)

This is the motivation for choosing .N ≤ 3. Then, wε ∈ L1 (0, T ; L2 (Ω)).

.

Next, we deduce that yε ∈ L1 (0, T ; L2 (Ω)) ∩ L∞ (0, T ; V ),

.

by a similar argument as for .wε , since .y ∈ L1 (Q) ∩ L∞ (0, T ; V ' ) and .yε (t) = (I + εA∆ )−1 y(t). Eventually, fε ∈ L∞ (0, T ;

n

.

W 2,p (Ω)) ⊂ L∞ (0, T ; L∞ (Ω)),

p≥2

by the elliptic regularity. Moreover, A is m-accretive on .L1 (Ω), and it follows by the second relation in (2.53) for .A∆ = A that wε (t) → w(t) strongly in L1 (Ω), a.e. t ∈ (0, T )

.

and .

||wε (t)||L1 (Ω) ≤ ||w(t)||L1 (Ω) , a.e. t ∈ [0, T ]

(see again [31]). For a later use, we assert that wε → w strongly in L1 (Q), as ε → 0.

.

Here there is the argument. We denote { κε (t) =

(wε (t, x) − w(t, x))dx, a.e. t ∈ (0, T )

.

Ω

and have kε (t) → 0 a.e. t ∈ (0, T ).

.

Moreover, { .

{

|κε (t)| ≤

|wε (t)|dx + Ω

Ω

|w(t)|dx ≤ 2 ||w(t)||L1 (Ω) .

2.4 A Time Dependent Potential

53

Therefore, by the Lebesgue dominated convergence theorem, we deduce that κε → κ strongly in L1 (0, T ) as ε → 0,

.

and this implies wε → w strongly in L1 (Q), as ε → 0.

.

(2.56)

Similarly, we have that yε → y strongly in L1 (Q), as ε → 0.

.

(2.57)

Finally, fε → f, strongly in L∞ (0, T ; L2 (Ω)), as ε → 0.

.

(2.58)

By the first relation in (2.53), we still have that (I + εAV )−1 y(t) → y(t) strongly in V ' for any t ∈ [0, T ].

.

We also observe that {

T

.

0

< > - ε (t), ψ(t) ' dt = Aw X ,X

{

{ Q

wε A∞ ψdxdt →

Q

wA∞ ψdxdt as ε → 0,

for any .ψ ∈ L∞ (0, T ; X), and by (2.52) .

dy dyε → weakly in L1 (0, T ; X' ) as ε → 0. dt dt

Passing to the limit in (2.52) tested at .ψ ∈ L∞ (0, T ; X), {

T

.

0

/

\ { { T dyε X' ,X dt, (t), ψ(t) dt + wε A∞ ψdxdt = dt Q 0 X' ,X

we check that .(y, w) indeed satisfies (2.23). Next, we assert that {

{ j (t, yε (t, x))dxdt ≤

.

Q

j (t, y(t, x))dxdt.

(2.59)

Q

Indeed, let us introduce the Yosida approximation of .β (see (7.22)), βλ (t, r) =

.

1 (1 − (1 + λβ(t, ·))−1 )r, a.e. t, for all r ∈ R and λ > 0. λ

(2.60)

54

2 Weakly Coercive Nonlinear Diffusion Equations

We have .βλ (t, r) =

∂jλ ∂r (t, r),

where .jλ is the Moreau approximation of j

{

|r − s|2 .jλ (t, r) = inf + j (t, s) 2λ s∈R

} a.e. t, for all r ∈ R,

(2.61)

which can be still written as jλ (t, r) =

.

|2 1 || | |(1 + λβ(t, ·))−1 r − r | + j (t, (1 + λβ(t, ·))−1 r)). 2λ

(2.62)

The function .jλ is convex and continuous and satisfies jλ (t, r) ≤ j (t, r), for all r ∈ R, λ > 0, lim jλ (t, r) = j (t, r), for all r ∈ R

.

(2.63)

λ→0

(see Theorem 7.34 in the Appendix). We have, using the first relation in (2.53) for A∆ = AV , that

.

{ .

Q

{ { T jλ (t, yε (t, x))dxdt ≤ jλ (t, y(t, x))dxdt−ε V ' ,V dt. 0

Q

Since for any .z ∈ V one has by the Green formula that { .

− V ' ,V = −

{ αβλ (t, z)zdσ − f

Ω

∂βλ (t, z) |∇z|2 dx ≤ 0, ∂z

we obtain {

{ jλ (t, yε (t, x))dxdt ≤

.

Q

jλ (t, y(t, x))dxdt.

(2.64)

Q

We remark that the calculus of .− V ' ,V before does not apply if j would depend on x and that is why this case is run out by this proof. Let us assume first that j would be nonnegative. By (2.63), jλ (t, y(t, x)) ≤ j (t, y(t, x))

.

and .

lim jλ (t, y(t, x)) = j (t, y(t, x)) a.e. on Q.

λ→0

(2.65)

Then, by the Lebesgue dominated convergence theorem, we get { .

lim

λ→0 Q

{ jλ (t, y(t, x))dxdt =

j (t, y(t, x))dxdt, for any y fixed. Q

(2.66)

2.4 A Time Dependent Potential

55

Passing to the limit in (2.64) as .λ → 0, we obtain that {

{ j (t, yε (t, x))dxdt ≤

.

Q

j (t, y(t, x))dxdt, for all ε > 0.

(2.67)

Q

Since, in general, j is not necessarily nonnegative, we use (2.10) and write in (2.64) { Q .

(jλ (t, yε (t, x)) − k1 (t)(1 + λβ(t, ·))−1 yε (t, x) − k2 (t))dxdt { ≤ (jλ (t, y(t, x)) − k1 (t)(1 + λβ(t, ·))−1 y(t, x) − k2 (t))dxdt {Q + k1 (t)((1 + λβ(t, ·))−1 y(t, x) − (1 + λβ(t, ·))−1 yε (t, x))dxdt. Q

Then, on the basis of (2.62) and (2.10), the integrand on the left-hand side and the first integrand on the right-hand side are nonnegative and we can pass to the limit as .λ → 0 and apply the Lebesgue dominated convergence theorem. We get { Q .

(j (t, yε (t, x)) − k1 (t)yε (t, x) − k2 (t))dxdt { ≤ (j (t, y(t, x)) − k1 (t)y(t, x) − k2 (t))dxdt {Q + k1 (t)(y(t, x) − yε (t, x))dxdt, Q

which implies (2.59) as claimed. A similar relation to (2.59) takes place for .j ∗ {

j ∗ (t, wε (t, x))dxdt ≤

.

Q

{

j ∗ (t, w(t, x))dxdt,

(2.68)

Q

and by the last two inequalities, we infer that j (·, yε (·, ·)) ∈ L1 (Q), j ∗ (·, wε (·, ·)) ∈ L1 (Q),

.

for all .ε > 0. By the same argument as for yw, we deduce that yε wε ∈ L1 (Q).

.

We test (2.52) by .A−1 2 yε (t) (defined by (2.14)) and integrate over .(0, T ). Since −1 1 ∞ 1 2 .yε ∈ L (Q) ∩ L (0, T ; V ), .wε ∈ L (0, T ; L (Ω)), and .A 2 yε (t) ∈ X2 , a.e. .t ∈ (0, T ), we have by (2.15) { .

0

T

/ \ -2 wε (t), A−1 yε (t) A 2

X2' ,X2

{ dt =

yε wε dxdt. Q

56

2 Weakly Coercive Nonlinear Diffusion Equations

Then, by a few computations, we deduce by (2.52) that { .

−

yε wε dxdt = Q

1 1 ||yε (T )||2V ' − ||y0 ||2V ' − 2 2

{ Q

yε A−1 2 fε dxdt.

(2.69)

Recalling (2.53), we have (I + εAV )−1 y(t) → y(t) strongly in V ' for any t ∈ [0, T ],

.

and passing to the limit in (2.69) as .ε → 0, we obtain ( { ) { 1 1 2 2 − . lim yε wε dxdt = ||y(T )||V ' − ||y0 ||V ' − yA−1 2 f dxdt. ε→0 2 2 Q Q

(2.70)

Moreover, by the strongly convergence of .(yε )ε and .(wε )ε , (2.57), and (2.56), we get yε → y a.e. in Q, wε → w a.e. in Q, as ε → 0,

.

which implies that yε wε → yw a.e. in Q, as ε → 0.

.

The functions j and .j ∗ are continuous and so .

j (t, yε (t, x)) → j (t, y(t, x)), j ∗ (t, wε (t, x)) → j ∗ (t, w(t, x)), a.e. on Q, as ε → 0.

Now, by (2.59) and (2.68), we have { ( ) j (t, yε (t, x)) + j ∗ (t, wε (t, x)) − yε wε dxdt Q { . ( ) ≤ j (t, y(t, x)) + j ∗ (t, w(t, x)) − yε wε dxdt, Q

and we apply Fatou’s lemma because .j (t, yε ) + j ∗ (t, wε ) − yε wε ≥ 0. We get, using (2.59) and (2.68), that { Q

.

(

) j (t, y(t, x)) + j ∗ (t, w(t, x)) − yw dxdt { ( ) ≤ lim inf j (t, yε (t, x)) + j ∗ (t, wε (t, x)) − yε wε dxdt ε→0 {Q ( ) ≤ lim sup j (t, x, y(t, x)) + j ∗ (t, x, w(t, x)) − yε wε dxdt { ε→0 Q { ( ) ≤ j (t, y(t, x)) + j ∗ (t, w(t, x)) dxdt − lim yε wε dxdt, Q

ε→0 Q

2.4 A Time Dependent Potential

57

whence, by using (2.70), we see that { { 1 1 2 2 .− ywdxdt ≤ ||y(T )||V ' − ||y0 ||V ' − yA−1 2 f dxdt. 2 2 Q Q

(2.71)

We continue the proof by relying on the same arguments, starting this time with Fatou’s lemma applied to the positive function .j (t, x, −yε ) + j ∗ (t, x, wε ) + yε wε . By similar computations, we get { { 1 1 2 2 .− ywdxdt ≥ ||y(T )||V ' − ||y0 ||V ' − yA−1 2 f dxdt, 2 2 Q Q u n

which together with (2.71) implies (2.51). -2 given in Sect. 2.1. We recall the notations .X2 , .X2' , .A2 , and .A

Now, we are ready to prove the main result of this section which shows that a null minimizer in .(P ) provides a unique solution to (2.1). Theorem 2.7 Under the assumptions .(H1 )–.(H2 ) and (2.49), problem .(P ) has a solution .(y ∗ , w ∗ ) such that .y ∗ ∈ L∞ (0, T ; V ' ). This solution is actually a null minimizer in .(P ), that is, J (y ∗ , w ∗ ) =

.

inf

(y,w)∈U

J (y, w) = 0,

(2.72)

and also it is a solution to (2.1) with the property j (·, y(·, ·)), j ∗ (·, β(·, y(·, ·))) ∈ L1 (Q).

.

The solution is unique in the class of solutions .y ∈ L2 (Q) or if j is strictly convex. Proof Let us introduce the approximating problem ∂y − ∆βλ (t, y) = f in Q, ∂t (t, y) ∂β λ . = αβλ (t, y) on Σ, − ∂ν y(0, x) = y0 in Ω,

(2.73)

where .βλ is the Yosida approximation of .β. Let .σ be positive and consider the approximating problem indexed upon .σ ∂y − ∆(βλ (t, y) + σy) = f in Q, ∂t (t, y) + σy) ∂(β λ . = α(βλ (t, y) + σy) on Σ, − ∂ν y(0, x) = y0 in Ω.

(2.74)

58

2 Weakly Coercive Nonlinear Diffusion Equations

The potential of .βλ (t, r) + σ r is jλ,σ (t, r) = jλ (t, r) +

.

σ 2 r , 2

(2.75)

where .jλ is the Moreau regularization of .j. By a simple computation using (2.61), (2.10), and (2.5), we get that σ 2 |r| + k1 r + k2 − 2λk12 ≤ jλ,σ (t, r) ≤ j (t, 0) + |r|2 . 2

(

) 1 2 +σ . 2λ

(2.76)

Hence .jλ,σ satisfies hypothesis (1.7) in Chap. 1 with .m = 2, so that we can refer to the results regarding the minimization problem .(P0 ) for .J0 defined in (1.23). By Theorem 1.4, problem .(P0 ) has a solution .(yλ,σ , wλ,σ ) ∈ U0 , .

yλ,σ ∈ L2 (Q) ∩ W 1,2 (0, T ; X2' ), yλ,σ (T ) ∈ V ' , wλ,σ = βλ (t, yλ,σ ) + σyλ,σ ∈ L2 (Q),

and satisfies (1.25), namely, dyλ,σ -2 wλ,σ (t) = f (t) a.e. t ∈ (0, T ), (t) + A dt y(0) = y0 ,

(2.77)

yλ,σ wλ,σ dxdτ } { t{ ||2 1 {|| 2 || || yλ,σ (t) V ' − ||y0 ||V ' − (A−1 = 2 yλ,σ )f dxdτ, 2 0 Ω

(2.78)

.

and − .

{ t{ 0

Ω

for all .t ∈ [0, T ], see Lemma 1.3 in Chap. 1. Next, by Theorem 1.5, this solution turns out to be the null minimizer in .(P0 ), i.e.,

.

J0 (y { λ,σ , wλ,σ ) ( ) ∗ = jλ,σ (t, yλ,σ (t, x)) + jλ,σ (t, wλ,σ (t, x)) − yλ,σ wλ,σ dxdt = 0.

(2.79)

Q

Taking into account (2.78) and (2.79), we still can write { .

∗ (jλ,σ (t, yλ,σ (t, x)) + jλ,σ (t, wλ,σ (t, x)))dxdt Q { } { T{ ||2 1 || + ||yλ,σ (T )||V ' − ||y0 ||2V ' = yλ,σ A−1 2 f dxdt. 2 Ω 0

(2.80)

2.4 A Time Dependent Potential

59

We note that .

∗ (t, ω) jλ,σ

|ω|

→ ∞ as |ω| → ∞,

(2.81)

uniformly in .λ and .σ. This happens due to (2.9) because by setting ηλ, σ (t, r) = ∂jλ,σ (t, r), ηλ,σ (t, r) = βλ (t, r) + σ r = β(t, (1 + λβ(t, ·))−1 r) + σ r,

.

then .ηλ,σ is bounded on bounded subsets .|r| ≤ M, uniformly in .λ and .σ, for .λ and σ small (smaller than 1, for example). We also note that

.

{ jλ,σ (t, yλ,σ (t, x))dxdt ≥ jλ (t, yλ,σ (t, x))dxdt Q { Q { |2 1 || | = j (t, (1 + λβ(t, ·))−1 yλ,σ )dxdt + |yλ,σ − (1 + λβ(t, ·))−1 yλ,σ | dxdt 2λ Q Q

{ .

and {

T

{

.

0

Ω

yλ,σ A−1 2 f dxdt {

+ Q

{

Q

+ Q

Q

(1 + λβ(t, ·))−1 yλ,σ A−1 2 f dxdt

≤ {

{ ( ) yλ,σ − (1 + λβ(t, ·))−1 yλ,σ A−1 = 2 f dxdt

|2 1 || | |yλ,σ − (1 + λβ(t, ·))−1 yλ,σ | dxdt + 2λ 2λ

{ Q

2 (A−1 2 f ) dxdt

(1 + λβ(t, ·))−1 yλ,σ A−1 2 f dxdt.

Plugging these in (2.80), we get after some algebra that { j (t, (1 + λβ(t, ·))

.

Q

−1

{ yλ,σ )dxdt + Q

∗ jλ,σ (t, wλ,σ (t, x)))dxdt

} || 1 {|| ||yλ,σ (T )||2 ' − ||y0 ||2 ' (2.82) V V 2 { { 2 f dxdt + 2λ (A−1 ≤ (1 + λβ(t, ·))−1 yλ,σ A−1 2 2 f ) dxdt.

+

Q

Q

Furthermore, we set (1 + λβ(t, ·))−1 yλ,σ = zλ,σ

.

and argue as in Theorem 2.3 to deduce by the Dunford–Pettis theorem that .(zλ,σ )σ and .(wλ,σ )σ are weakly compact in .L1 (Q). Recalling (2.41), we also get

60

2 Weakly Coercive Nonlinear Diffusion Equations

|| || ||yλ,σ (T )||

.

V'

≤C

(2.83)

independently on .σ and .λ. Taking into account that .wλ,σ = βλ (yλ,σ ) + σyλ,σ , Eq. (2.78) yields { t{

|| 1 || ||yλ,σ (t)||2 ' + V 2

.

=

0

1 ||y0 ||2V ' + 2

Ω

2 (βλ (τ, yλ,σ )yλ,σ + σyλ,σ )dxdτ

{ t/ 0

yλ,σ (τ ), A−1 2 f (τ )

\ V ' ,V

dτ

{ t || { ||2 || 1 t || || −1 || ||yλ,σ (τ )||2 ' dτ, ||A2 f (τ )|| dτ + V V 2 0 0

1 1 = ||y0 ||2V ' + 2 2

for all .t ∈ [0, T ]. Because .βλ (t, r)r ≥ 0, for all .r ∈ R, and in virtue of the Gronwall lemma, we deduce that .

|yλ,σ ||L∞ (0,T ;V ' ) ≤ C, √ σ ||yλ,σ ||L2 (Q) ≤ C,

(2.84)

and {

{ j (t, zλ,σ (t, x))dxdt ≤ C,

.

Q

Q

∗ jλ,σ (t, wλ,σ (t, x))dxdt ≤ C

(2.85)

independently on .σ and .λ. For getting (2.85), we recall the arguments leading to (2.44) and (2.45). ∗ imply that Then, (2.80) and relation (2.10) for .jλ,σ { jλ,σ (t, yλ,σ (t, x))dxdt ≤ C

.

(2.86)

Q

independently on .σ and .λ. Following the proof of Theorem 2.3, we deduce on a subsequence, denoted still by .σ, that zλ,σ → wλ,σ → √ σ yλ,σ → yλ,σ → . yλ,σ (T ) → Awλ → dyλ,σ → dt By (2.86) and (2.62), we have

zλ wλ ζλ yλ ξ Awλ dyλ dt

weakly in L1 (Q), weakly in L1 (Q), weakly in L2 (Q), weak* in L∞ (0, T ; V ' ), weakly in V ' , weakly in L1 (0, T ; X' ), weakly in L1 (0, T ; X' ).

2.4 A Time Dependent Potential

{ .

Q

61

{ |2 1 || | −1 − (1 + λβ(t, ·)) y dxdt ≤ jλ,σ (t, yλ,σ (t, x))dxdt ≤ C, |yλ,σ λ,σ | 2λ Q

) √ ( whence, denoting .χλ,σ = yλ,σ − zλ,σ / 2λ, we see that .(χλ,σ )σ is bounded in 2 2 .L (Q) and, on a subsequence, .χλ,σ → χλ weakly in .L (Q), as .σ → 0. Then yλ,σ − zλ,σ →

.

√

2λχλ weakly in L1 (Q), as σ → 0,

where .||χλ ||L1 (Q) ≤ C. Since .zλ,σ → zλ weakly in .L1 (Q), it follows that .(yλ,σ )σ is bounded in .L1 (Q), so it converges weakly, and by the limit uniqueness, we have yλ,σ → yλ weakly in L1 (Q), as σ → 0.

.

We also have yλ = zλ +

.

√ 2λχλ a.e. on Q.

(2.87)

By the Arzelà–Ascoli theorem (since .V ' is compact in .X' because X is compact in .V ), it follows that yλ,σ (t) → yλ (t) in X' , uniformly in t ∈ [0, T ], as σ → 0,

.

so .ξ = yλ (T ) and .yλ (0) = y0 . Passing to the limit in (2.77), we get that .(yλ , wλ ) satisfies

.

dyλ - λ (t) = f (t) a.e. t ∈ (0, T ), (t) + Aw dt y(0) = y0 .

(2.88)

Passing to the limit in (2.82) as .σ → 0, using the weakly lower semicontinuity property, we get { .

} 1{ ||yλ (T )||2V ' − ||y0 ||2V ' (j (t, zλ (t, x)) + jλ∗ (t, wλ (t, x)))dxdt + 2 Q { { −1 −1 2 yλ A2 f dxdt − 2λ (A2 f ) dxdt ≤ 0. − Q

(2.89)

Q

We repeat again the arguments developed in Theorem 2.3 and deduce by the Dunford–Pettis theorem that .(zλ )λ and .(wλ )λ are weakly compact in .L1 (Q). It still follows that { { . ||zλ (T )||V ' ≤ C, j (t, zλ (t, x))dxdt ≤ C, jλ∗ (t, wλ (t, x))dxdt ≤ C Q

Q

(2.90)

62

2 Weakly Coercive Nonlinear Diffusion Equations

independently on .λ (recall (2.41), (2.44), and (2.45)).. Passing to the limit in (2.84) as .σ → 0, we get .

||yλ ||L∞ (0,T ;V ' ) ≤ C,

where C are several constants independent on .λ. Then, proceeding along with the proof of Theorem 2.3, we obtain from (2.89), by selecting a subsequence denoted still by .λ, that zλ wλ yλ . y (T ) λ Awλ dyλ dt

→ → → → →

z∗ w∗ y∗ y ∗ (T ) Aw ∗

weakly in L1 (Q), weakly in L1 (Q), weak* in L∞ (0, T ; V ' ), weakly in V ' , weakly in L1 (0, T ; X' ),

→

dy ∗ dt

weakly in L1 (0, T ; X' ).

By (2.87), we get that .z∗ = y ∗ a.e. on Q, and by (2.90), we obtain .

|| || ∗ ||y (T )||

{

j (t, y ∗ (t, x))dxdt ≤ C,

≤ C,

V'

Q

{

j ∗ (t, w ∗ (t, x))dxdt ≤ C. Q

(2.91) The first inequality is obvious. For the second (if .j (t, r) ≥ 0), Fatou’s lemma yields {

{

∗

j (t, y (t, x))dxdt = .

Q

lim inf j (t, zλ (t, x))dxdt

Q λ→0 {

≤ lim inf λ→0

j (t, zλ (t, x))dxdt ≤ C. Q

If j is not positive, we use again (2.10), and denoting j (t, r) = j (t, r) − k1 (t)r − k2 (t) ≥ 0,

.

we write { .

j (t, y ∗ (t, x))dxdt ≤ lim inf λ→0

Q

{

j (t, zλ (t, x))dxdt, Q

whence we get { j (t, y ∗ (t, x))dxdt − (k1 y ∗ + k2 )dxdt Q Q { { ≤ lim inf j (t, zλ (t, x))dxdt − (k1 y ∗ + k2 )dxdt,

{ .

λ→0

Q

Q

(2.92)

2.4 A Time Dependent Potential

63

i.e., (2.92). In what concerns the third inequality in (2.91), we can write by (2.62) and (2.90) { ( .

Q

) |2 1 || | |wλ − (1 + λβ −1 (t, ·))−1 wλ | + j ∗ (t, (1 + β −1 (t, ·))−1 wλ ) dxdt 2λ { = Q

jλ∗ (t, wλ (t, x))dxdt ≤ C.

(2.93) Recalling

that .j ∗ (t, x, ω) {

≥ k3 (t, x)ω + k4 (t, x), we get that

(1 + β −1 (t, ·))−1 wλ dxdt ≤ C + k4 meas(Q),

k3

.

Q

( ) and hence . (1 + β −1 (t, ·))−1 wλ λ is bounded in .L1 (Q). Then { Q

.

|2 1 || | |wλ − (1 + λβ −1 (t, ·))−1 wλ | dxdt 2λ {

(k3 (1 + β −1 (t, ·))−1 wλ + k4 )dxdt ≤ C1 .

≤C+ Q

It follows that (1 + λβ −1 (t, ·))−1 wλ → w ∗ weakly in L1 (Q), as λ → 0.

.

Then, we pass to the limit in (2.93) as .λ → 0 (if .j ∗ is nonnegative). Otherwise, we use again (2.10) for .j ∗ . Passing to the limit in (2.88) and (2.89) as .λ → 0, we get dy ∗ - ∗ (t) = f (t) a.e. t ∈ (0, T ), (t) + Aw . dt y(0) = y0 ,

(2.94)

and { .

Q

(j (t, y ∗ (t, x)) + j ∗ (t, w ∗ (t, x)))dxdt } { ||2 1 {|| ∗ 2 || || y (T ) V ' − ||y0 ||V ' − + y ∗ A−1 2 f dxdt ≤ 0. 2 Q

(2.95)

We have got that .(y ∗ , w ∗ ) ∈ U, .y ∗ ∈ L∞ (0, T ; V ' ) and so, by Lemma 2.6, it follows that .y ∗ w ∗ ∈ L1 (Q). Replacing the sum of the last two terms on the righthand side in (2.95) by (2.51), we get

64

2 Weakly Coercive Nonlinear Diffusion Equations

{

(j (t, y ∗ (t, x)) + j ∗ (t, w ∗ (t, x)) − y ∗ (t, x)w ∗ (t, x))dxdt ≤ 0.

.

Q

Recalling the Legendre–Fenchel relation {

(j (t, y ∗ (t, x)) + j ∗ (t, w ∗ (t, x)) − y ∗ (t, x)w ∗ (t, x))dxdt ≥ 0,

.

Q

we obtain { . (j (t, y ∗ (t, x)) + j ∗ (t, w ∗ (t, x)) − y ∗ (t, x)w ∗ (t, x))dxdt = 0,

(2.96)

Q

which, eventually, implies that j (t, y ∗ (t, x)) + j ∗ (t, w ∗ (t, x)) − y ∗ (t, x)w ∗ (t, x) = 0 a.e. on Q.

.

Therefore, we conclude that .w ∗ (t, x) ∈ β(t, y ∗ (t, x)) a.e. on .Q, by the second Legendre–Fenchel relation. On the other hand, due again to (2.51) in Lemma 2.6, relation (2.96) means in fact that .(y ∗ , w ∗ ) realizes the minimum in .(P ), as claimed in (2.72). If j is strictly convex, the solution is unique as shown in Theorem 2.3. Otherwise, if .y ∈ L2 (Q), the uniqueness follows directly by (2.1) using the monotonicity of .β. Indeed, let .(y,.η) and .(y, η) be two solutions to (2.1) corresponding to the same data, where .

η(t, x) ∈ β(t, y(t, x)), η(t, x) ∈ β(t, y (t, x)) a.e. on Q, (y, η), (y, η) ∈ U, y, y ∈ L∞ (0, T ; V ' ) ∩ L2 (Q),

and j (·, y), j (·, y ), j ∗ (·, η), j ∗ (·, η) ∈ L1 (Q).

.

We write the equations satisfied by their difference .

d(y − y) - −(t) + A(η η)(t) = 0 a.e. t ∈ (0, T ), dt (y − y )(0) = 0,

multiply the equation by .A−1 y )(t) ∈ L∞ (Ω), integrate over .(0, t), and obtain 2 (y − .

1 ||(y − y )(t)||2V ' + 2

{ t{ 0

Ω

η)(y − y ) dxdt = 0. (η − -

2.4 A Time Dependent Potential

65

We note that the second integral is well-defined because by Lemma 2.6, ηy, ηy, ηy, ηy ∈ L1 (Q).

.

y (t)||2V ' ≤ 0, whence But .β(t, r) is maximal monotone, and hence we get .||y(t) − .y(t) = y (t) for all .t ∈ [0, T ]. u n We remark now that if .(y ∗ , η∗ ) is the solution to (2.1), with .η∗ (t, x) ∈ β(t, y ∗ (t, x)) a.e. on Q and .y ∗ ∈ L∞ (0, T ; V ' ), then it is a unique solution to .(P ). Indeed, relying on Lemma 2.6, replacing (2.51) in (2.21), we obtain .J (y ∗ , η∗ ) = 0 and so the minimum of .J is realized at .(y ∗ , η∗ ). So, we conclude that (2.1) is equivalent to the minimization problem .(P ). Concerning possible applications, we refer to the equation .

∂(m(t, x)y) − ∆β0 (y) e f ∂t

with initial and boundary conditions. This can be associated with various physical models as for example: fluid diffusion in saturated–unsaturated deformable porous media with the porosity m time and space dependent, or to absorption–desorption processes in saturated porous media in which m is the absorption–desorption rate of the fluid by the solid (see appropriate problems in [41, 57]). This nonautonomous problem was treated in [41] by a semigroup approach under strong regularity conditions for .m. Now, it can be reduced by a change of variable .my → y to an equation of the form (2.1) and treated in the framework of the theory of this chapter.

Chapter 3

Nonlinear Diffusion Equations with a Noncoercive Potential

This chapter is devoted to the study of the existence of solutions to nonlinear diffusion equations .(N DE) with a time dependent nonlinearity whose potential is not coercive. We discuss about equations that may have a strong nonlinearity represented for example by a singular diffusion coefficient of Dirac type. Such problems cannot be treated by the Lions theorem in the variational nonautonomous case (see [56]), due to the lack of coercivity. Also, the classical semigroup approach cannot be applied, since the problems are not autonomous. Moreover, for a general nonlinearity without a particular regularity in time, the hypotheses required in [39] for proving the solution existence to nonautonomous evolution equations are not fulfilled, and so this theory cannot be followed in this case. The .(N DE) is replaced by a minimization problem for an appropriate convex functional, settled in the duality .(L∞ )' -.L∞ , involving thus spaces of measures. In some situations to be specified, this provides a generalized solution to the nonlinear equation, see Theorem 3.5, following a conditional result based on a relation between the solution and the nonlinearity, presented in Lemma 3.4. Then, it is shown in Theorem 3.7 that a null minimizer for the functional is a weak solution to the nonlinear equation. The existence of a solution to the minimization problem without using this conditional hypothesis remains an open problem. In particular, this model describes a selforganized criticality (SOC) process. In order to justify this, we present in the last part of the chapter a short description of a SOC physical model, by proposing an extension of the classical model to a more general dynamics, for which we give an existence result.

3.1 Problem Statement and the Functional Framework Let us consider a domain .Ω, an open and bounded subset of .RN , for .N ≤ 3, and .t ∈ (0, T ), with T finite. We investigate the model © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Marinoschi, Dual Variational Approach to Nonlinear Diffusion Equations, PNLDE Subseries in Control 102, https://doi.org/10.1007/978-3-031-24583-1_3

67

68

3 Nonlinear Diffusion Equations with a Noncoercive Potential

.

∂y (t, x) − ∆w(t, x) = f (t, x), in Q := (0, T ) × Ω, . ∂t

(3.1)

w(t, x) ∈ β(t, x, y(t, x)), a.e. on Q, w(t, x) = 0, on Σ := (0, T ) × F, . y(0, x) = y0 , in Ω,

(3.2) (3.3)

where .F denotes the boundary of .Ω, assumed to be sufficiently smooth (for instance of class .C 2 ), and .β : Q × R → R is provided by a potential .j. The potential .j : Q × R → R = R ∪ {+∞} is assumed to be a proper, convex, and lower semicontinuous (l.s.c.) function, having the properties: .

(t, x) → j (t, x, r) is measurable on Q for all r ∈ R, .

(3.4)

∂j (t, x, r) = β(t, x, r) a.e. (t, x) ∈ Q, ∀r ∈ R, .

(3.5)

C1 |r| + C10 ≤ j (t, x, r) ≤ C2 |r| + C20 ,

(3.6)

with .Ci , .Ci0 .(i = 1, 2) constants and .C1 , .C2 positive. By (3.5), it follows that .β(t, x, ·) is maximal monotone (generally multivalued) on .R, for a.e. .(t, x) ∈ Q. A main example for j satisfying property (3.6) is j (t, x, r) = a(t, x) |r − yc | , β(t, x, r) = a(t, x)sign(r − yc ), for r ∈ R, (3.7) with .a ∈ L∞ (Q), .a(t, x) ≥ a0 > 0, a.e. on Q, and .yc positive, where sign is the signum graph .

⎧ if r > 0, ⎨ 1, .sign(r) := [−1, 1], if r = 0 ⎩ −1, if r < 0. In this case, Eq. (3.1) can be assigned to a self-organized criticality model, with the diffusion coefficient the Dirac function .δyc . This will be discussed in the last section of this chapter. Other examples may include functions of the form j (t, x, r) = a(t, x) |r| + b(t, x)

.

xp , xp + C

with C a constant, for some appropriate p and .b ∈ L∞ (Q). Let .j ∗ : Q × R → R be the conjugate of .j, defined by j ∗ (t, x, ω) := sup (ωr − j (t, x, r)). r∈R

.

(3.8)

3.1 Problem Statement and the Functional Framework

69

This is a proper, convex, l.s.c. function, and j and .j ∗ satisfy the Legendre–Fenchel relations j (t, x, r) + j ∗ (t, x, ω) ≥ rω, for all r, ω ∈ R, .

.

j (t, x, r) + j ∗ (t, x, ω) = rω iff ω ∈ j (t, x, r).

(3.9) (3.10)

Next, we present the functional setting appropriate for this problem. Let Y be a Banach space, and let .B : D(B) ⊂ Y → Y. We recall (see relation (7.21) in the Appendix) that B is called accretive on Y provided that .

||(I + λB)u − (I + λB)u||Y ≥ ||u − u||Y , for all u, u ∈ Y and λ > 0,

(3.11)

and it is called m-accretive iff in addition .R(I + B) = Y. Here, I is the identity operator on .Y, and .D(B) and .R(B) are the domain and range of .B, respectively. We introduce some spaces associated to the Dirichlet boundary conditions chosen to accompany the nonlinear equation in this chapter. 1,p By .W0 (Ω), we denote the Sobolev space .W 1,p (Ω) of the functions with vanishing traces on .F. We denote by V the Hilbert space .H01 (Ω) with the norm .

||ψ||V = ||∇ψ||L2 (Ω) .

Its dual .V ' = H −1 (Ω) is endowed with the scalar product .

/ \ ( ) −1 θ, θ V ' := θ (A−1 V θ ) = θ, AV θ

V ' ,V

, for θ, θ ∈ V ' ,

(3.12)

where .AV : V → V ' , { .

V ' ,V :=

∇z(x) · ∇ψ(x)dx, z ∈ V , for all ψ ∈ V .

(3.13)

Ω

Often we shall use the operator .A2 defined by A.2 : D(A2 ) ⊂ L2 (Ω) → L2 (Ω), D(A2 ) = W 2,2 (Ω) ∩ H01 (Ω), A2 ψ = −∆ψ. (3.14) This operator is m-accretive on .L2 (Ω), coercive, and surjective. To treat the problem (3.1)–(3.3) under the hypotheses (3.4)–(3.6), let us define .

B1 : D(B1 ) ⊂ L1 (Ω) → L1 (Ω), B1 ψ = −∆ψ, X := D(B1 ) = {ψ ∈ W01,1 (Ω); ∆ψ ∈ L1 (Ω)}.

(3.15)

According to the results given in [31], the operator .B1 is m-accretive on .L1 (Ω). Moreover, we have

70

3 Nonlinear Diffusion Equations with a Noncoercive Potential 1,q

X = D(B1 ) ⊂ W0 (Ω), 1 ≤ q
C1 ,

(3.22)

− C20 + I[−C2 ,C2 ] (ω) ≤ j ∗ (t, x, ω) ≤ −C10 + I[−C1 ,C1 ] (ω),

(3.23)

and .

where .I[−Ci ,Ci ] is the indicator function of the set .[−Ci , Ci ], and .Ci , .Ci0 are those in (3.6), .i = 1, 2. By (3.23), the effective domain of .ϕ ∗ is D(ϕ ∗ ) = {w ∈ L∞ (Q); ||w||∞ ≤ C2 }.

.

Since .ϕ is the conjugate of .ϕ ∗ , we have ϕ(y) = .

=

sup

(y(w) − ϕ ∗ (w)) ≥

sup

(y(w)) + C4

w∈L∞ (Q),||w||∞ ≤C2 w∈L∞ (Q),||w||∞ ≤C2

sup

w∈L∞ (Q),||w||∞ ≤C2

(y(w) + C4 )

= C2 ||y||(L∞ (Q))' + C4 , for all y ∈ w ∈ L∞ (Q), (3.24) where .C4 = C10 meas.(Q).

3.3 Main Results for the Noncoercive Case

73

For the moment, we are going to treat an intermediate problem. In view of (3.21), we introduce the functional J- : (L∞ (Q))' × L∞ (Q) → R,

.

⎧ 1 1 . −1 . ⎨ ϕ(y) + ϕ ∗ (w) + ||y(T )||2V ' − ||y0 ||2V ' − y(A2 f ), 2 2 -(y, w) := .J if (y, w) ∈ U, . . ⎩ +∞, otherwise,

(3.25)

and the minimization problem Minimize {J-(y, w); (y, w) ∈ U }.

.

-) (P

Here, .A2 is the operator defined in (3.14), with .D(A2 ) ⊂ C(Ω). If .f ∈ C([0, T ]; L2 (Ω)), then by elliptic regularity (see [1]), A−1 2 f (t) ∈ D(A2 ) ⊂ C(Ω) for all t ∈ [0, T ],

.

and so ∞ A−1 2 f ∈ C([0, T ]; D(A2 )) ⊂ C([0, T ]; C(Ω)) ⊂ L (Q).

.

Thus, the term .y(A−1 2 f ) in (3.25) makes sense. -) has at least a First, we give a characterization for .J-, and then we prove that .(P solution. Lemma 3.2 Assume that .y0 ∈ L2 (Ω) and .f ∈ C([0, T ]; L2 (Ω)). Then, the functional .J- is proper, convex, and l.s.c. on .(L∞ (Q))' × L∞ (Q). Proof The convexity follows from (3.25), for the first two terms applying Theorem 7.39 in the Appendix. We easily see that of .J- is not ( the domain ) empty, {t so that it is proper. A pair belonging to U is . y(t) = y0 + 0 f (s)ds, 0 , where y ∈ C([0, T ]; L2 (Ω)). Then, .j ∗ (t, x, 0) is bounded by (3.23) and

.

{ (

{ j (t, x, y(t, x))dxdt ≤

.

Q

Q

| | | |) { t | | | | | C1 |y0 + f (s, x)ds || + |C20 | dxdt < ∞. 0

Thus, it follows that .J-(y, 0) < ∞. Next, according to Definition 7.23 in the Appendix, we show that for each .λ > 0, the level set Sλ (J-) = {(y, w) ∈ (L∞ (Q))' × L∞ (Q); J-(y, w) ≤ λ} is closed,

.

74

3 Nonlinear Diffusion Equations with a Noncoercive Potential

which implies that .J-is lower semicontinuous. Further, it is weakly l.s.c. too, because it is convex (see the comment after Proposition 7.24 in the Appendix). Let .(yn , wn )n ∈ (L∞ (Q))' × L∞ (Q) be such that yn → y strongly in (L∞ (Q))' , wn → w strongly in L∞ (Q), as n → ∞,

.

and J-(yn , wn ) ≤ λ.

(3.26)

.

By (3.24), we have that .

1 ||yn (T )||2V ' (3.27) 2 | | || || 1 | | || || ≤ |C4 | + |C20 | meas(Q) + ||y0 ||2V ' + ||yn ||(L∞ (Q))' ||A−1 f || + λ ≤ C, 2 ∞ 2

C2 ||yn ||(L∞ (Q))' +

where C will denote several positive constants. It follows that .(yn (T ))n lies in a bounded subset of .V ' , and we can extract a subsequence (still denoted by .n ) such that yn (T ) → ξ weakly in V ' , as n → ∞.

.

By (3.17), we write that {

t

yn (t) = y0 +

.

(f (s) − Bwn (s))ds,

(3.28)

0

and so, we have .

||yn (t) − y(t)||X'

||{ t || || || || = || (Bw(t) − Bwn (t))dt || || 0

X'

≤ C ||w − wn ||∞ .

It follows that yn (t) → y(t) strongly in X' , as n → ∞ for all t ∈ [0, T ].

.

By the uniqueness of the limit, we get that .ξ = y(T ). Passing to the limit in (3.26) and taking into account that all terms are l.s.c. and convex, we get .J-(y, w) ≤ λ, i.e., the level set is closed. This means that .J- is l.s.c. u n || || || || Proposition 3.3 Let .f ∈ C([0, T ]; L2 (Ω)), such that .||A−1 2 f ||∞ < C2 , and let 2 -) has at least one solution. .y0 ∈ L (Ω). Then, the minimization problem .(P

3.3 Main Results for the Noncoercive Case

75

Proof We note { t that the admissible set U is not empty because it contains .(y, 0) with y(t) = y0 + 0 f (s)ds. By (3.24), we have

.

.C2

||y||(L∞ (Q))' + (C10 − C20 )meas(Q) −

|| || 1 || || ||y0 ||2V ' − ||y||(L∞ (Q))' ||A−1 f || ≤ J-(y, w), 2 ∞ 2

for all .(y, w) ∈ U. Then, .J- is bounded from below 0

.(C1

− C20 )meas(Q) −

1 ||y0 ||2V ' ≤ J-(y, w), 2

and so its infimum exists. Let it be .d = inf(y,w)∈U J-(y, w). We take a minimizing sequence .(yn , wn ) ∈ U such that .d

≤ ϕ(yn ) + ϕ ∗ (wn ) +

1 1 1 ||yn (T )||2V ' − ||y0 ||2V ' − yn (A−1 2 f) ≤ d + . 2 2 n

(3.29)

We deduce (as before by (3.27), with .λ replaced by .d + 1/n) that .(yn )n is bounded in .(L∞ (Q))' and .(yn (T ))n is bounded in .V ' . Therefore, selecting a subsequence (denoted still by the subscript .n ), we get .yn (T

) → ξ weakly in V ' , as n → ∞.

We also deduce that .(yn )n is weak* compact in .(L∞ (Q))' , so that on a generalized subsequence (denoted still be the subscript .n ), we have .yn

→ y weak* in (L∞ (Q))' , as n → ∞.

The argument is the same with that used in Chap. 1, Theorem 1.8, to prove the weak* compactness of the sequence .(wn )n . Because .||wn ||∞ ≤ C2 , we can select a subsequence such that .wn

→ w weak* in L∞ (Q), as n → ∞,

which implies that .Bwn

→ Bw weak* in L∞ (0, T ; X ' ), as n → ∞.

By (3.28) and (3.17), we still get .yn

→ y weak* in L∞ (0, T ; X ' ), a.e. t ∈ (0, T ), as n → ∞, .

dy dyn → weak* in L∞ (0, T ; X' ), as n → ∞, dt dt

76

3 Nonlinear Diffusion Equations with a Noncoercive Potential

meaning that y is absolutely continuous from .[0, T ] in .X' . Again by (3.28), we can deduce that → y(t) weakly in X ' , for all t ∈ [0, T ], as n → ∞.

.yn (t)

Therefore, we obtain that .yn (0) → y0 weakly in .V ' and that by the limit uniqueness .ξ = y(T ). Now, we pass to the limit in (3.29) on the basis of the weakly lower semicontinuity of .J- and get .d ≤ J-(y, w) ≤ d, i.e., .J- attains its minimum at a point .(y, w) ∈ U . u n -), and this relies on Now, we assert that there is a connection between .(P ) and .(P the representation formula (3.21) for .y(w). We shall show in the next lemma that indeed, there is at least a case in which this holds true, as for example, if .(y, w) ∈ M(Q) × C(Q), where .M(Q), the space of bounded Radon measures on .Q, is the dual of .C(Q). Lemma 3.4 Let .f ∈ C([0, T ]; L2 (Ω)), .y0 ∈ L2 (Ω), and .(y, w) ∈ U such that .(y, w) ∈ M(Q) × C(Q). Then (3.21) takes place. Proof Let .(y, w) ∈ M(Q) × C(Q), let .ε be positive, and set yε (t) := (I + εA∆ )−1 y(t), for any t ∈ [0, T ],

.

wε (t) := (I + εA∆ )−1 w(t), for any t ∈ [0, T ],

.

fε (t) := (I + εA∆ )−1 f (t), for any t ∈ [0, T ],

.

yε (0) := (I + εA∆ )−1 y0 ,

.

where, for not overloading the notation, we designate by .A∆ the operator .−∆ defined on several spaces, with Dirichlet boundary conditions. More exactly, in the last two relations, .A∆ = A2 (defined in (3.14)), while in the second relation, .A∆ denotes the operator .−∆ : C 2 (Ω) ∩ C0 (Ω) ⊂ C(Ω) → C(Ω). All these operators are m-accretive. -2 , the extension of the operator .A2 to the In the first equation, .A∆ represents .A space .(D(A2 ))' , with the domain .L2 (Ω), defined by .

< > -2 z, ψ A

(D(A2 ))' ,D(A2 )

:= z(A2 ψ), for any ψ ∈ D(A2 ), z ∈ L2 (Ω)

-2 is surjective (see (see the definition (1.19) in Chap. 1). We recall that the operator .A Lemma 7.21 in the Appendix). We note that .X' ⊂ (D(A2 ))' , and so it follows that -2 )−1 y(t) ∈ L2 (Ω), for all t ∈ [0, T ]. yε (t) = (I + εA

.

3.3 Main Results for the Noncoercive Case

77

1,q

More exactly, .yε (t) ∈ W0 (Ω) ⊂ L2 (Ω) for .N ≤ 3. Next, fε (t) := (I + εA∆ )−1 f (t) ∈ D(A2 ) ⊂ C(Ω), for all t ∈ [0, T ],

.

and wε (t) ∈ C 2 (Ω) ∩ C0 (Ω), for all t ∈ [0, T ],

.

because we assumed .w(t) ∈ C(Ω) for all .t ∈ [0, T ]. We apply the operator .(I + εA∆ )−1 to (3.17) and get dyε (t) + Bwε (t) = fε (t), a.e. t ∈ (0, T ), . dt yε (0) = (I + εA2 )−1 y0 .

(3.30)

ε We see that .Bwε (t) ∈ C(Ω), and consequently, . dy dt (t) ∈ C(Ω). Moreover,

(I + εA2 )−1 y0 ∈ D(A2 ).

.

We test the first equation in (3.30) by .A−1 2 (yε (t)) and write {

{ .

dyε (t)A−1 yε (t)wε (t)dx + 2 (yε (t))dx { Ω Ω dt = yε (t)A−1 2 fε (t)dx, t ∈ [0, T ].

(3.31)

Ω

By a few computations after the integration with respect to .t ∈ [0, T ], we obtain −y(I + εA∆ )−2 w

.

=

(3.32)

||2 1 1 || || || ||yε (T )||2V ' − ||(I + εA2 )−1 y0 || ' − y((I + εA∆ )−2 A−1 2 f ). V 2 2

The first term on the left-hand side is given by the relation .

( ) (I + εA∆ )−1 y(t), (I + εA∆ )−1 w(t)

L2 (Ω)

= y(I + εA∆ )−2 w.

This is obtained by density, after being proved first for an approximation of .y(t) by a sequence .(yn (t))n ⊂ L2 (Ω). By (3.28), we have { .

Since

||yε (t) − y(t)||X' ≤

t 0

||B(wε (s) − w(s))ds||X' ≤ C ||wε − w||∞ .

78

3 Nonlinear Diffusion Equations with a Noncoercive Potential

wε = (I + εA∆ )−1 w → w strongly in C(Q), as ε → 0,

.

and .A∆ is m-accretive on .C(Ω), it follows that yε (t) → y(t) strongly in X' , as ε → 0, for all t ∈ [0, T ].

.

Also, (I + εA∆ )−2 w → w strongly in C(Q), as ε → 0,

.

and we deduce that y(I + εA∆ )−2 w → y(w), as ε → 0.

.

Moreover, since .A−1 2 f ∈ C(Q), we have −1 (I + εA∆ )−2 A−1 2 f → A2 f strongly in C(Q), as ε → 0,

.

and so −1 y((I + εA∆ )−2 A−1 2 f ) → y(A2 f ), as ε → 0.

.

We also take into account that yε (T ) = (I + εA∆ )−1 y(T ) → y(T ) strongly in V ' , as ε → 0,

.

by the m-accretivity of the restriction of .A∆ on .V ' (which here is actually .AV defined in (3.13)), and -2 )−1 y0 → y0 in L2 (Ω), strongly as ε → 0. yε (0) = (I + εA

.

u n

By passing to the limit in (3.32), we get (3.21). We remark that (3.21) cannot be obtained if .(y, w) ∈ argument is that if .w ∈ L∞ (Q), (I + εA∆ )−1 w(t) ∈

n

.

(L∞ (Q))'

× L∞ (Q).

The

W 2,m (Ω) ∩ L∞ (Ω)

m≥2

and (I + εA∆ )−2 w → w weak* in L∞ (Q), as ε → 0,

.

which does not allow the convergence in measure. Due to this fact, the proof of the existence in problem .(P ), corresponding to the set U where .(y, w) ∈ (L∞ (Q))' ×

3.3 Main Results for the Noncoercive Case

79

L∞ (Q), remains open for now. However, problem .(P ) may have a solution .(y, w) ∈ M(Q) × C(Q). -). If .(y, w) ∈ M(Q) × C(Q), then it Theorem 3.5 Let .(y, w) be a solution to .(P is a variational solution to (3.1)–(3.3). The proof follows immediately by Lemma 3.4 and (3.21) because we have 1 1 J-(y, w) = ϕ(y) + ϕ ∗ (w) + ||y(T )||2V ' − ||y0 ||2V ' − y(A−1 2 f) 2 2 = ϕ(y) + ϕ ∗ (w) − y(w) = J (y, w).

.

It turns out that .(y, w) is a solution to .(P ), and the conclusion follows according to Definition 3.1. We now give a definition of a weak solution to (3.1)–(3.3) and investigate if this can be obtained as a minimizer in .(P ). We recall that .y ∈ (L∞ (Q))' is decomposed as .y = ya + ys . Definition 3.6 Let .f ∈ C([0, T ]; L2 (Ω)), .y0 ∈ V ' . We call a weak solution to (3.1)–(3.3) a pair .(y, w) such that y ∈ (L∞ (Q))' ∩ W 1,∞ (0, T ; X' ), w ∈ L∞ (Q),

.

w(t, x) ∈ β(t, x, ya (t, x)) a.e. on Q,

(3.33)

ys ∈ ND(ϕ ∗ ) (w),

(3.34)

.

.

and .(y, w) satisfies (3.17). In (3.34), .ND(ϕ ∗ ) (w) is the normal cone (in .(L∞ (Q))' ) to .D(ϕ ∗ ) at .w, defined by ND(ϕ ∗ ) (w) := {μ ∈ (L∞ (Q))' ; μ(w − w) ≥ 0, for all w ∈ D(ϕ ∗ )},

.

and .D(ϕ ∗ ) is the effective domain of .ϕ ∗ (see (3.20)). Theorem 3.7 Let .(y, w) be a solution to .(P ) that minimizes J to 0, i.e., .J (y, w) = 0. Then, .(y, w) is a weak solution to (3.1)–(3.3). Moreover, (3.1)–(3.3) has at most one solution .y ∈ L1 (Q), .w ∈ L∞ (Q). Proof The argument is the following: if .(y, w) is a null minimizer for .J, then, recalling the definition of .ϕ(y), we have { .

Q

∗ j (t, x, ya (t, x))dxdt + ID(ϕ ∗ ) (ys ) { + j ∗ (t, x, w(t, x))dxdt − ya (w) − ys (w) = 0. Q

(3.35)

80

3 Nonlinear Diffusion Equations with a Noncoercive Potential

By definition, ∗ ID(ϕ ∗ ) (ys ) =

.

sup (ys (ψ))

ψ∈D(ϕ ∗ )

whence ∗ ID(ϕ ∗ ) (ys ) − ys (w) =

.

sup (ys (ψ)) − ys (w) ≥ 0.

ψ∈D(ϕ ∗ )

Moreover, (3.35) implies that the singular part vanishes ∗ ID(ϕ ∗ ) (ys ) − ys (w) = 0

.

(3.36)

and also the .L1 -part { .

(j (t, x, ya (t, x)) + j ∗ (t, x, w(t, x)) − ya (t, x)w(t, x))dxdt = 0.

Q

From the second equality, in which the integrand is nonnegative by the Legendre– Fenchel relation, we get that it is zero j (t, x, ya (t, x)) + j ∗ (t, x, w(t, x)) − ya (t, x)w(t, x) = 0, a.e. on Q,

.

and consequently, w(t, x) ∈ ∂j (t, x, ya (t, x)), a.e. on Q.

.

From relation (3.36), we can write ys (w) ≥ ys (ψ) for all ψ ∈ D(ϕ ∗ ),

.

so that .ys ∈ ∂ID(ϕ ∗ ) (w) = ND(ϕ ∗ ) (w). In conclusion, according to Definition 3.6, .(y, w) is a weak solution to the nonlinear diffusion problem. Let us assume now that .y ∈ L1 (Q). Then, y satisfies (3.17) with .w ∈ β(t, x, y(t, x)) a.e. on .Q, and the uniqueness follows by the monotonicity of .β. n u As seen at the beginning of Sect. 3.3, D(ϕ ∗ ) = {w ∈ L∞ (Q); −C2 ≤ w(t, x) ≤ C2 a.e. on Q}.

.

We deduce by (3.34) that the state y has an absolutely continuous part .ya ∈ L1 (Q) if .w ∈ (−C2 , C2 ). If w reaches the bounds .±C2 , then the state has a singular part .ys .

3.4 Existence in a Self-organized Criticality (SOC) Model

81

3.4 Existence in a Self-organized Criticality (SOC) Model Self-organized criticality (SOC) is a property of certain dynamical systems of having a critical point as an attractor toward which the system spontaneously emerges. In equilibrium systems, such as phase transitions for example, a critical state, specifying the conditions as temperature or pressure, at which the phase boundary ceases to exist, is reached only by tuning a control parameter precisely. In some non-equilibrium systems, at least a part of the system structure selforganizes without explicit constraints from outside the system, but from the internal ones resulting by the interactions between the components. The organization can evolve in either time or space, can maintain a stable form, or can exhibit transient phenomena. The basic example is the Bak–Tang–Wiesenfeld model, or the sand-pile model (see [9, 82]). The sand-pile model describes the dynamics of the sand slowly dropped onto a plane surface forming a pile. The slope builds up as grains of sand are randomly placed onto the pile, until the slope exceeds a specific threshold value .yc , at which the site collapses, producing avalanches and transferring sand into the adjacent sites. Since the sand pile adjusts itself the shape of its sides in order to arrive at a certain critical equilibrium state, the criticality is called self-organized. Other examples of self-organizing systems are the forest-fire model (see [10]), and the Bak–Sneppen model (see [8]) describing the coevolution between interacting species. Self-organization is put into evidence in various system dynamics, such as crack formation, non-conservative neuronal network flows, cloud formation, economy, and epidemics (see, e.g., [51]). The model can be formalized as a cellular automaton (see, e.g., [11, 14]) in which the dynamics of the variable .y(t) = (yij (t))M i,j =1 is followed in the 2D region .Ω, which is discretized as a .M × M square matrix. The variable .yij (t) representing the height of the sand column is assumed to have only positive integer values. It is assigned to each site .(i, j ), and its dynamics is expressed by the equation yij (t + 1) → yij (t) − Zijkl aij (t), for (k, l) ∈ Fij ,

.

(3.37)

where .Fij = {(i + 1, j ), (i, j + 1), (i − 1, j ), (i, j − 1)} is the set of all 4 nearest neighbors of .(i, j ). Here, .aij (t) is a positive integer (it equals 1 in the other sand-pile models) and

Zijkl

.

⎧ ⎨ 4, if i = k, j = l, = −1, if (k, l) ∈ Fij , ⎩ 0, if (k, l) ∈ / Fij .

(3.38)

A site .(i, j ) becomes “activated” when .yij (t) ≥ yc . The dynamics of this site described by the law (3.37) shows that the site loses a number of sand grains equal to .4aij (t), which move to nearest neighbors in the interval of time .(t, t +1). So, a small avalanche forms and a new configuration of the sand pile occurs. The transition from .y(t) to .y(t + 1) can be written as

82

3 Nonlinear Diffusion Equations with a Noncoercive Potential

yij (t + 1) − yij (t) = −Zij aij (t)H (yij (t) − yc ), i, j = 1, . . . , M,

.

(3.39)

where H is the Heaviside graph, ⎧ if r > 0, ⎨ 1, .H (r) := [0, 1], if r = 0, ⎩ 0, if r < 0, and .Zij = (Zijkl )(k,l)∈Fij . This means that the transfer is done in an activated site (i, j ), where .yij (t) ≥ yc , and in the rest .yij (t) remains unchanged. We note that M .Z = (Zij ) i,j =1 is the second-order difference operator in the spatial domain, .

Zij (Yij ) = Yi+1,j + Yi−1,j + Yi,j +1 + Yi,j −1 − 4Yi,j for all i, j = 1, . . . , M,

.

and so one can still write (3.39) as y(t + 1) − y(t) = −Za(t)H (y(t) − yc ).

.

The threshold condition makes it a nonlinear diffusion operator. This equation is the discrete form of .

∂y (t) = ∆(a(t, x)H (y(t) − yc )) in Q = (0, T ) × Ω. ∂t

(3.40)

Moreover, a Dirichlet boundary condition y − yc = 0 on Σ = (0, T ) × F

.

(3.41)

should be added for the consistency of the model. We also consider an initial condition y(0) = y0 ≥ yc a.e. in Ω.

.

(3.42)

Now, we discuss the existence to (3.40). First, we specify that the previous results are true if in particular, .β(t, x, ·) is of the form (3.7). Proposition 3.8 Under the hypotheses of Theorem 3.5, problem (3.40)–(3.42) has a variational solution. Proof Let us introduce .u = y − yc . The problem becomes .

∂u (t) = ∆(a(t, x)H (u)), in Q, ∂t u = 0,

.

on Σ,

(3.43) (3.44)

3.4 Existence in a Self-organized Criticality (SOC) Model

u(0) = u0 ,

.

in Ω.

83

(3.45)

To connect with the previous results in Sect. 3.3, we must have the solution -), let .f ≡ 0 and .y0 ≥ 0. Let us replace positiveness. In the previous problem .(P .(P ) by a slightly modified problem, that is, -}, Minimize {J-(y, w); (y, w) ∈ U

.

-) (P

where - := {(y, w) ∈ U ; y(t) ≥ 0 for all t ∈ [0, T ]}. U

.

If .y(t) ∈ X' , .y(t) ≥ 0 means that .X' ,X ≥ 0 for all .ψ ∈ X, .ψ ≥ 0 and for all .t. -) has at The functional .J- defined on .U1 is proper, convex, and l.s.c., and .(P -) in least one solution. The proofs are led in the same way as for problem .(P Proposition 3.3. Moreover, .y(t) ≥ 0 for all .t ∈ [0, T ]. Indeed, let us take a minimizing sequence .yn (t) ≥ 0. Then, for all .ψ ∈ X, .ψ ≥ 0, we have by the weak convergence of .yn (t) in .X' that 0 ≤ lim X' ,X = X' ,X for all t ∈ [0, T ],

.

n→∞

-. and so .y(t) ∈ U In system (3.43)–(3.45), .f = 0 and .u0 ≥ 0. This implies that we can apply -) has at least one solution .u(t) ≥ 0, Proposition 3.3 to establish that problem .(P -. Under the so that .sign(u) reduces to .H (u) and .J reaches its minimum on .U hypotheses of Theorem 3.5, it follows that this solution is a variational solution to (3.43)–(3.45), and so .y(t) = u(t) + yc ≥ yc is a variational solution to (3.40)– (3.43). u n The interpretation of the result is that if the initial configuration of the sand pile has all sites activated, then they remain activated for all further times and the structure evolves as a self-organized system.

Chapter 4

Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary Conditions

In this chapter, we investigate the well-posedness of a nonlinear divergence parabolic equation .yt − ∇ · β(t, x, ∇y) e f, with dynamic boundary conditions of Wentzell type. Here, the diffusive term .β is derived from a potential j that depends on the gradient of the solution. In the time dependent case, the existence and uniqueness of a strong solution is obtained as the limit of a finite diffeernce scheme, solved by a duality approach, while in the time-invariant case the approach is via a semigroup technique combined with the duality method. Compared with previous existence theory for this problem, here we have two new aspects: the first is the generality of the nonlinearity .β assumed to be discontinuous and consequently extended to a multivalued function by filling the jumps. The second is the constructive approach based on a finite difference scheme, which permits to treat the time dependent case, too. Two situations for the potential j of .β will be discussed, besides the fact that it is assumed proper, convex, and l.s.c. Thus, the assumption of a polynomial growth for j characterizes a strongly coercive case. This is approached by a variational principle involving an appropriate minimization problem formulated for the system obtained by a finite time discretization of the equations. It is proved that the time-discretized scheme system has a unique solution and that it is stable in Proposition 4.2. This allows the proof of the existence of a weak solution as the strong limit of the h-discretized solution, with h the time step. This result is used in Theorem 4.4, together with some further arguments, to show that this solution is strong and is unique. The situation when j is continuous, weakly coercive, and exhibits a symmetry at infinity is treated in Sect. 4.3. The latter case that provides in Theorem 4.8 a strong solution in the Sobolev space .W 1,1 is of particular interest in image processing with observation on the boundary (see, e.g., [16, 23]). In Sect. 4.4, we present an alternative semigroup approach to the existence theory when the potential j is time independent.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Marinoschi, Dual Variational Approach to Nonlinear Diffusion Equations, PNLDE Subseries in Control 102, https://doi.org/10.1007/978-3-031-24583-1_4

85

86

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . .

4.1 Problem Presentation Let .Ω ⊂ RN , .N ≥ 1, be an open bounded regular domain, of class .C 2 for instance, and consider a nonlinear parabolic equation coupled with a dynamic boundary condition of reaction–diffusion type yt − ∇ · β(t, x, ∇y) e f, in Q := (0, T ) × Ω, .

(4.1)

β(t, x, ∇y) · ν + yt e g, on Σ := (0, T ) × |, .

(4.2)

.

y(0) = y0 , in Ω,

(4.3)

where .t ∈ (0, T ), .T < ∞,(.x ∈ Ω, .| is the ) boundary of .Ω, .ν is the outward normal ∂y ∂y , and .∇y = , . . . , to .|, .yt := ∂y ∂t ∂x1 ∂xN is the gradient of .y. In the following, we denote by .|·|N the Euclidian norm in .RN , by .|·| the norm in ∂u ∂u N .R, and by .u · v the scalar product of .u, v ∈ R . By .∇ · u = ∂x1 + . . . + ∂xN , it is denoted the divergence of the vector .u. Along the chapter, we shall discuss cases related to various properties of the potential j , such that some possible combinations of the following hypotheses will be used: (H1 )

.

j : Q×RN → (−∞, ∞] is continuous on .Q×RN , and for each .(t, x) ∈ Q, it is convex with respect to r,

.

∂j (t, x, r) = β(t, x, r), for any r ∈ RN , t ∈ [0, T ], x ∈ Ω,

(4.4)

there is ξ0 ∈ C(Q; RN ) with ∇ · ξ0 ∈ L2 (Ω), such that ξ0 (t, x) ∈ β(t, x, 0), ∀(t, x) ∈ Q.

(4.5)

.

.

(H2 )

.

(Strong coercivity hypothesis) There exist .Ci , Ci0 ∈ R, .i = 1, 2, .C1 , C2 > 0, such that p

.

(H3 )

.

p

C1 |r|N + C10 ≤ j (t, x, r) ≤ C2 |r|N + C20 , ∀(t, x) ∈ Q, for 1 < p < ∞.

(4.6)

(Weak coercivity hypothesis) The functions j and .j ∗ satisfy j (t, x, r) = +∞, uniformly with respect to t, x, |r|N →∞ |r|N

(4.7)

j ∗ (t, x, ω) = +∞, uniformly with respect to t, x, |ω|N →∞ |ω|N

(4.8)

.

.

lim

lim

where .j ∗ : Q × RN → R is the conjugate of .j, defined by

4.1 Problem Presentation

87

j ∗ (t, x, ω) = sup (ω · r − j (t, x, r)), for all ω ∈ RN , ∀(t, x) ∈ Q, N r∈R (4.9) and .∂j (t, x, ·) denotes the subdifferential of .j (t, x, ·), that is, .

∂j (t, x, r)={w ∈ RN ; j (t, x, r)−j (t, x, r) ≤ w · (r−r), ∀r ∈ RN }. (4.10) (Symmetry at infinity) There exist .γ1 , .γ2 such that .

(H4 )

.

j (t, x, r) ≤ γ1 j (t, x, −r) + γ2 , γ1 > 0, γ2 ≥ 0.

.

(H5 )

.

(4.11)

(Regularity in .t) There exists .L > 0 such that .

j (t, x, r) ≤ j (s, x, r) + L |t − s| j (t, x, r), ∀t, s ∈ [0, T ], x ∈ Ω, r ∈ RN .

(4.12)

By (4.5), we see that, by redefining .j (t, x, r) as .j (t, x, r) − j (t, x, 0), we may assume without loss of generality that j (t, x, 0) = 0, j (t, x, r) ≥ 0, j ∗ (t, x, r) ≥ 0 for all (t, x) ∈ Q, r, ω ∈ RN . (4.13) We note that (4.7) and (4.8) are equivalent to .

.

{ } sup |r|N ; r ∈ β −1 (t, x, ω), |ω|N ≤ M ≤ WM , . { } sup |ω|N ; ω ∈ β(t, x, r), |r|N ≤ M ≤ YM ,

(4.14) (4.15)

respectively, where M, .WM , .YM are positive constants (see, e.g., [13], p. 6–9). Let us present some examples. The strongly coercivity hypothesis .(H2 ) includes, for instance, the situation β : Q × RN → RN ,

.

β(t, x, r1 , . . . , rN ) = (β1 (t, x, r1 ), . . . , βN (t, x, rN )), βi (t, x, ri ) = ∂ji (t, x, ri ), i = 1, . . . , N, where .ji : Q × R → R are convex functions, continuous with respect to .ri and continuous with respect to .(t, x) ∈ Q, and .j : Q × RN → R is given by j (t, x, r1 , . . . , rN ) = j1 (t, x, r1 ) + . . . + jN (t, x, rN ).

.

Another example refers to .ji of the form

88 .ji (t, x, r)

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . . p

= αi (t, x) |r|N + κi (t, x) log(|r|N + 1) + δi (t, x) · r + δi1 (t, x), i = 1, . . . , N,

for .αi , .κi , .δi1 ∈ C 1 (Q), .κi ≥ 0, .δi ∈ C 1 (Q; RN ). In particular, for .κi = δi = δi1 = 0, this leads to the parabolic equation with a non-isotropic p-Laplacian N Σ ∂ .yt − ∂xi

(

i=1

yt +

N Σ i=1

(

) | | | ∂y |p−2 ∂y | αi (t, x) || = f, in Q, ∂xi | ∂xi

(4.16)

) | | | ∂y |p−2 ∂y | αi (t, x) || · νi = g, on Σ. ∂xi | ∂xi

Instead of (4.16), one can consider a general model for a diffusion process in a fractured medium, described by the parabolic problem N Σ ∂ .yt − ∂xi

(

( αi (t, x)H

i=1

yt +

N Σ i=1

( αi (t, x)H

(

) | )| | ∂y |p−2 ∂y ∂y | | − ri | e f, in Q, ∂xi ∂xi | ∂xi

(4.17)

) | )| | ∂y |p−2 ∂y ∂y | − ri || · νi e g, on Σ, ∂xi ∂xi | ∂xi

where .ri ∈ R, and H is the Heaviside multivalued function ⎧ ⎪ r < 0, ⎨ H (r) = 0, . H (r) = [0, 1], r = 0, ⎪ ⎩ H (r) = 1, r > 1.

(4.18)

In fact, a discontinuous nondecreasing function .r → β(t, x, r) becomes a maximal monotone multivalued function by filling the jumps at the discontinuity points .ri , that is, by taking .β(t, x, ri ) = [β(t, x, ri − 0), β(t, x, ri + 0], and this is the natural way of treating Eq. (4.1) with a discontinuous .β(t, x, ·). Problem (4.1)–(4.3) extends the classical Wentzell boundary condition and models various phenomena in mathematical physics, and in particular, diffusion and reaction–diffusion processes, phase transition, and image restoring with observation { on the boundary. If we view .E(y) = Q j (t, x, ∇y)dxdt as the energy of the system, then hypothesis .(H2 ) describes diffusion processes with coercive and differentiable energy, while .(H3 ) refers to systems with .W 1,1 regular energy. For various interpretations and treatment of the dynamic boundary conditions (4.2), we refer, e.g., to the works [35, 42, 43, 71]. In [81], there are studied equations of the form .−∇ · (|∇u|p−2 ∇u) + |u|p−2 u + α1 (x, u) = f in .Ω, with the Wentzell boundary condition .−∇ · (|∇| u|p−2 ∇| u) + |∇u|p−2 ∂u/∂ν + |u|p−2 u + α2 (x, u) = g on .|. Previously, in [46], there were studied problems with Wentzell boundary

4.2 The Strongly Coercive Case

89

conditions of the form .ut − ∇ · (a(|∇u|2 )∇u) + f (u) = h1 (x) (where a is a given nonnegative function), with the boundary condition .ut +b(x)a(|∇u|2 )∂u/∂ν+g(u) = h2 (x).

4.2 The Strongly Coercive Case Let p and .p' be two conjugate numbers, for .1 ≤ p ≤ ∞. We denote .W 1,p (Ω) the Sobolev spaces with the standard norm and .H 1 (Ω) := W 1,2 (Ω). Let us recall the definition given in the Appendix regarding the fractional Sobolev spaces .W s,p (Ω), see (7.7)–(7.8), and the trace theorem (7.9), saying that the operator 1,r (Ω) onto .W 1− 1r ,r (|), where .γ (z) is the trace of z .z → γ (z) is surjective from .W on .| (see also [28], p. 315). Everywhere in the following, the gradient operator .∇, as well as the divergence .∇·, is considered in the sense of distributions on .Ω. We consider that .β is derived from a potential j that satisfies hypotheses .(H1 ), .(H2 ), .(H5 ). For .p > 1, we define the space .U

= {z ∈ L2 (Ω); ∇z ∈ (Lp (Ω))N , γ (z) ∈ L2 (|)},

endowed with the natural norm .||z||U = ||z||L2 (Ω) + ||∇z||Lp (Ω) + ||γ (z)||L2 (|) . By ∗ the Sobolev embeddings (Theorem 7.6 in the Appendix), .W 1,2 (Ω) ⊂ Lp (Ω), if ∗ = 2N , .W 1,2 (Ω) ⊂ Lq (Ω), if .N = 2, for any .q ∈ [2, +∞), .N > 2, where .p N−2 1,2 ∞ .W (Ω) ⊂ L (Ω), if .N = 1, with continuous injections, we conclude that if .z ∈ U, we have { p∗ , if N > 2, p > p∗ > 2, 1,p .z ∈ W (Ω), p = (4.19) p, otherwise. In particular, if .p ≥ 2, it follows that .U ⊂ H 1 (Ω) with a dense and continuous embedding. Let us retain that under assumption (4.6), one can easily deduce that . |ξ |

≤ C3 |r|p−1 + C30 , for any ξ ∈ β(t, x, r),

(4.20)

where .C3 and .C30 are positive constants. We also assume that .y0

∈ U, f ∈ L2 (Q), g ∈ L2 (Σ).

(4.21)

The first results are intended to prove the existence and stability of the solution to the time-discretized system.

90

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . .

We consider an equidistant partition .0 = t0 ≤ t1 ≤ t2 ≤ . . . ≤ tn = T of the interval .[0, T ], with .ti = ih for .i = 1, . . . , n, .h = T /n,, and the finite sequences h h .{f }i=1,...,n , .{g }i=1,...,n , defined by the time averages i i 1 = h

h .fi

{

ih

f (s)ds, (i−1)h

gih

1 = h

{

ih

g(s)ds,

(4.22)

(i−1)h

whence we see that .fih ∈ L2 (Ω), .gih ∈ L2 (|). We consider the time-discretized system .

h − yh yi+1 i

h

h h ) e fi+1 , in Ω, i = 0, . . . , n − 1, . − ∇ · β(ti+1 , x, ∇yi+1

h β(ti+1 , x, ∇yi+1 )·ν+

h − yh yi+1 i

h

h , on |, . e gi+1

(4.23) (4.24)

y0h = y0 , in Ω.

(4.25)

Definition 4.1 We call a weak solution to the time-discretized system (4.23)–(4.24), a set of functions .{yih }i=1,...,n , .yih ∈ U, which satisfies (for each .i = 1, . . . , n − 1) {

{ .

Ω

h yi+1 ψdx + h

{

Ω

{

= Ω

yih ψdx

{

+ |

h ηi+1 · ∇ψdx +

{

yih ψdσ

+h Ω

|

h yi+1 ψdσ

(4.26) {

h fi+1 ψdx

+h |

h gi+1 ψdσ, for any ψ ∈ U,

h , such that .ηh (x) ∈ β(t h for some measurable function .ηi+1 i+1 , x, ∇yi+1 (x)), a.e. .x ∈ i+1 Ω. h In the third integral on the left-hand side in (4.26), we understand by .yi+1 the h h trace of .yi+1 ∈ U on .| . If .yi is a solution, it follows by (4.6) and (4.20) that .j (ti , ·, ∇yi

h

'

) ∈ L1 (Ω), ηih ∈ (Lp (Ω))N , i = 0, . . . , n.

(4.27)

The next Proposition 4.2 is concerned with the stability of the finite difference scheme (4.23)–(4.24). Proposition 4.2 Let us assume (4.21), and let .j (0, ·, ∇y0 ) ∈ L1 (Ω). System (4.23)– (4.24) has a unique weak solution satisfying .

|| || || h || ||yi ||

L2 (Ω)

|| || || || ||γ (yih )||

L2 (|)

≤ C, i = 1, . . . , n, . ≤ C, i = 1, . . . , n,

(4.28) .

(4.29)

4.2 The Strongly Coercive Case

|| || h ||p ||∇yi+1 || p

≤ C, m = 1, . . . , n, .

(4.30)

||2 − yih || || || || 2 h

≤ C, m = 1, . . . , n, .

(4.31)

|| || h h ||2 || γ (yi+1 ) − γ (yi ) || || || || || 2 h

≤ C, m = 1, . . . , n, .

(4.32)

h

m−1 Σ ||

L (Ω)

i=0

||

h

m−1 h Σ || || yi+1 i=0

h

91

|| ||

L (Ω)

m−1 Σ || i=0

h

L (|)

m−1 Σ{ i=0

Ω

h j (ti+1 , x, ∇yi+1 )dx ≤ C, i = 1, . . . , n,

(4.33)

where C is a positive constant, independent of h. Proof Let us fix .t ∈ [0, T ], .w1 ∈ L2 (Ω), .w2 ∈ L2 (|) and consider the intermediate problem .u

− h∇ · β(t, x, ∇u) e w1 in Ω,

(4.34)

u + hβ(t, x, ∇u) · ν e w2 on |. By .U ' , we denote the dual space of .U. We define .b ∈ U ' as {

{

:=

.b(ψ)

w1 (x)ψ(x)dx + Ω

w2 (σ )ψ(σ )dσ, for all ψ ∈ U,

(4.35)

|

and note that . |b(ψ)|

≤ ||w1 ||L2 (Ω) ||ψ||L2 (Ω) + ||w2 ||L2 (|) ||ψ||L2 (|) for all ψ ∈ U.

(4.36)

Let .u ∈ U. Taking into account (4.6) and (4.20), we have .j (t, ·, ∇u)

'

∈ L1 (Ω), η ∈ (Lp (Ω))N ,

for t fixed and all measurable sections .η(x) of .β(t, x, ∇u(x)). We define a weak solution to (4.34) a function .u ∈ U, such that there is .η ∈ ' (Lp (Ω))N , .η(x) ∈ β(t, x, ∇u(x)) a.e. .x ∈ Ω, and {

{ (uψ + hη · ∇ψ)dx +

.

Ω

uψdσ = b(ψ), for all ψ ∈ U.

(4.37)

|

In order to prove that (4.34) has a solution, it is appropriate to use a variational argument, i.e., we show that a solution to this equation is retrieved as a solution to the minimization problem

92

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . . .Minimize

{ϕ(u); u ∈ U } ,

(4.38)

where .ϕ : L2 (Ω) → R ∪ {+∞} is defined by .ϕ(u)

=

{ { 1

2 Ωu

2 dx

+∞,

+h

{

Ω j (t, x, ∇u)dx

+

{

1 2 |

u2 dσ − b(u), if u ∈ U, otherwise.

(4.39) It is easily seen that .ϕ is proper, strictly convex, and lower semicontinuous on .L2 (Ω). By (4.6) and (4.36), we have .ϕ(u)

1 1 p ||u||2L2 (Ω) + hC1 ||∇u||Lp (Ω) + hC10 + ||u||2L2 (|) − |b(u)| 2 2 1 1 p ≥ ||u||2L2 (Ω) + hC1 ||∇u||Lp (Ω) + ||u||2L2 (|) 4 4 ≥

(4.40)

+hC10 − 4 ||w1 ||2L2 (Ω) − 4 ||w2 ||2L2 (|) , ∀u ∈ U. Let us denote by .d = infu∈U ϕ(u), and let us consider a minimizing sequence .(un )n≥1 , ∈ U. Then,

.un

.d

≤ ϕ(un ) =

1 2

{

{

Ω

u2n dx + h

j (t, x, ∇un )dx + Ω

1 2

{ |

u2n dσ − b(un ) ≤ d +

1 . n

(4.41)

By (4.40), it follows that . ||un || 2 L (Ω)

2

p

+ h ||∇un ||Lp (Ω) + ||un ||2L2 (|) ≤ C, ∀n ∈ N,

where C is a positive constant independent of .n. By selecting a subsequence .(n → ∞), we have un → u weakly in L2 (Ω), . γ (un ) → χ weakly in L2 (|), as n → ∞, ∇un → ξ weakly in (Lp (Ω))N , as n → ∞. It follows that .ξ = ∇u, a.e. in .Ω, and so .u ∈ W 1,p (Ω) with .p given by (4.19). By the trace theorem in fractional Sobolev spaces (see (7.9)), this implies that .γ (u) ∈ 1− 1 ,p

W p (Ω), and it is clear that .χ = γ (u) a.e. on .|. Then, .u ∈ U. Moreover, by ' (4.20), we also have .η ∈ (Lp (Ω))N , where .η(x) ∈ β(t, x, ∇u(x)) a.e. .x ∈ Ω. Since .ϕ is convex and continuous, it is also weakly l.s.c. on .L2 (Ω) and so .lim inf ϕ(un ) ≥ ϕ(u). n→∞

Passing to the limit in (4.41), as .n → ∞, we get that .ϕ(u) = d, so that .ϕ reaches its minimum on .U. Now, we make the connection between this solution and the solution to (4.34). Let λ ∞ .λ > 0, and define the variation .u = u + λψ, for all .ψ ∈ C (Q). We have .ϕ(u) ≤ λ ϕ(u ), for any .λ > 0. Replacing the expression of .ϕ, dividing by .λ, and letting .λ → 0,

4.2 The Strongly Coercive Case

93

we get {

{ (uψ + hη · ∇ψ)dx +

.

Ω

uψdσ − b(ψ) ≤ 0, |

for all .ψ ∈ C ∞ (Q). By density, this extends to all .ψ ∈ U . Changing .ψ to .−ψ and making the same calculus, we obtain the converse inequality, so that we find that the solution to (4.38) satisfies (4.37). Having this result, we can deduce in an iterative way that system (4.23)–(4.24) has a unique weak solution. It can be rewritten as {

{

.

Ω

h yi+1 ψdx + h

{ Ω

h ηi+1 · ∇ψdx +

|

h yi+1 ψdσ = bi+1 (ψ), ∀ψ ∈ U,

(4.42)

h (x) ∈ β(t h ' for .i = 0, . . . n − 1, where .ηi+1 i+1 , x, ∇yi+1 (x)) a.e. .x ∈ Ω and .bi+1 ∈ U is given by

{ .bi+1 (ψ)

:= Ω

{ h + hfi+1 ψ)dx

(yih ψ

+ |

h (yih + hgi+1 )ψdσ, ∀ψ ∈ U.

(4.43)

We have ) (|| || || h || || 2 |bi+1 (ψ)| ≤ ||yih ||L2 (Ω) + h ||fi+1 ||ψ||L2 (Ω) L (Ω) ) (|| || . || h || h || || || || + yi L2 (|) + h gi+1 L2 (|) ||ψ||L2 (|) , for .i = 0, . . . , n − 1. The procedure begins with Eq. (4.42) for .i = 0, where .b1 (ψ) satisfies the previous relation for .i = 0. Setting .b = b1 in .ϕ, we get that the corresponding problem (4.38) has a unique weak solution .y1h that verifies (4.42) for .i = 0. Next, we set .b = bi+1 in .ϕ, and by recurrence, we obtain a sequence of solutions h h 1,p (Ω), .y i+1 ∈ U, which satisfy (4.42) for all .i = 1, . . . , n − 1. In particular, .yi ∈ W with .p given by (4.19). h in (4.26) and use (4.4) getting To obtain the first estimate (4.28), we set .ψ = yi+1 {

{ .

Ω

h (yi+1 )2 dx + h

{ ≤ Ω

Ω

{ h (j (ti+1 , x, ∇yi+1 ) − j (ti+1 , x, 0))dx +

{ =

Ω

{

h (yi+1 )2 dx + h

{ h yih yi+1 dx +

|

{ Ω

h h ηi+1 · ∇yi+1 dx +

h yih yi+1 dσ + h

{ Ω

|

|

h (yi+1 )2 dσ

h (yi+1 )2 dσ

h h fi+1 yi+1 dx + h

(4.44) { |

Then, we sum up (4.44) from .i = 0 to .i = m − 1 ≤ n − 1, use (4.13),

h h gi+1 yi+1 dσ.

94

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . . m−1 || || Σ{ 1 || 1 || || h ||2 || h ||2 h . ||ym || + h j (t , x, ∇y )dx + ||y || 2 i+1 m i+1 L2 (Ω) L (|) 2 2 Ω i=0

≤

m−1 m−1 m−1 || || || h Σ || h Σ || h Σ || || h ||2 || h ||2 || h ||2 ||yi+1 || 2 + ||fi+1 || 2 + ||yi+1 || 2 L (Ω) L (Ω) L (|) 2 2 2 i=0

+

i=0

(4.45)

i=0

m−1 || || || h Σ || 1 || 1 || || h ||2 || ||2 || ||2 ||gi+1 || 2 + ||y0h || 2 + ||y0h || 2 , L (|) L (Ω) L (|) 2 2 2 i=0

and obtain (since j is continuous on .Q) || ||

||2

h ||

. ||ym ||

L2 (Ω)

|| ||2 || h || + ||ym || 2

+ 2C1 h

L (|)

≤ C0 + h

m−1 Σ ||

|| || h ||p ||∇yi+1 || p

m (|| || Σ || h ||2 ||yi || 2

L (Ω)

i=1

(4.46)

L (Ω)

i=0

|| ||2 || || + ||yih || 2

)

L (|)

,

where { .C0

| | | | (||f (t)||2L2 (Ω) + ||g(t)||2L2 (|) )dt + ||y0 ||2L2 (Ω) + ||y0 ||2L2 (|) + 2T |C10 | meas(Ω).

T

=h 0

Using a variant of the discrete Gronwall’s lemma (in a form proved, e.g., in [34]), we get || ||

||2

h ||

. ||ym ||

h

m || || Σ || h ||2 ||yi || 2

L (Ω)

i=1

L2 (Ω)

+h

|| ||2 || h || + ||ym || 2

L (|)

m || || Σ || h ||2 ||yi || 2 i=1

L (|)

|| ||2 || || ≤ 2eT (||y0h || 2

L (Ω)

|| ||2 || || + ||y0h || 2

L (|)

|| ||2 || || ≤ eT (||y0 ||2L2 (Ω) + ||y0h || 2

L (|)

+ C0 ), .

+ C0 ).

(4.47) (4.48)

By the last two relations (4.46) and (4.45), we obtain (4.28)–(4.30) and (4.33). For calculating the estimate (4.31), we set .ψ =

.

|| || || y h − y h ||2 || i+1 i || || || || 2 || h

L (Ω)

{ = Ω

h fi+1

|| || || y h − y h ||2 || i || + || i+1 || || 2 || h

{ + Ω

L (|)

h − yh yi+1 i

h

{ dx + |

h gi+1

h −y h yi+1 i h

h ηi+1 ·∇

h − yh yi+1 i

h

|| || h h ||2 || 1 || 1 || || yi+1 − yi || || h ||2 ≤ ||fi+1 || 2 + || || || 2 L (Ω) 2 2 || h

L (Ω)

in (4.26) and write

h − yh yi+1 i

h

dx

dσ

|| || h h ||2 || 1 || 1 || || yi+1 − yi || || h ||2 + ||gi+1 || 2 + || || || 2 L (|) 2 2 || h

.

L (|)

4.2 The Strongly Coercive Case

95

We use again (4.4) and sum up from .i = 0 to .i = m − 1 ≤ n − 1, getting ||

.h

m−1 h Σ || || yi+1

|| ||

i=0

||2 − yih || || || || 2 h

||

+h

i=0

L (Ω)

+h

m−1 Σ ||

|| || h ||2 ||gi+1 || 2

L (|)

i=0

m−1 h Σ || || yi+1

+2

|| ||

≤h

L (|)

m−1 Σ{ i=0

||2 − yih || || || || 2 h

Ω

m−1 Σ ||

|| || h ||2 ||fi+1 || 2

i=0

L (Ω)

(4.49)

h (j (ti+1 , x, ∇yih ) − j (ti+1 , x, ∇yi+1 ))dx.

By (4.12), we have .j (t, x, r)

≤

j (s, x, r) , for t, s ∈ [0, T ], |t − s| < 1/L. 1 − L |t − s|

Then, we compute m−1 Σ{ i=0

Ω

.(j (ti+1 , x, ∇yi

h

=

{

m−1 Σ{ Ω

i=0

≤ Lh

(j (ti+1 , x, ∇yih ) − j (ti , x, ∇yih ))dx +

m−1 Σ{ i=0

≤

h ) − j (ti+1 , x, ∇yi+1 ))dx

Ω

j (0, x, ∇y0 )dx Ω

{ j (ti+1 , x, ∇yih )dx +

j (0, x, ∇y0 )dx Ω

{ m−1 { Lh Σ j (ti , x, ∇yih )dx + j (0, x, ∇y0 )dx ≤ C, 1 − Lh Ω Ω i=0

for h sufficiently small, .h h.

The step function .y h defined by (4.50) is called an h-approximating solution to (4.1)–(4.3). Also, we set, for all .i = 1, . . . n, .f

h

(t) = fih , t ∈ [(i − 1)h, ih),

96

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . .

g h (t) = gih , t ∈ [(i − 1)h, ih), β(t, x, ∇y h (t)) = β(ti , x, ∇yih ), t ∈ [(i − 1)h, ih), ηh (t) = ηih , t ∈ [(i − 1)h, ih), j (t, x, ∇y h (t)) = j (ti , x, ∇yih ), t ∈ [(i − 1)h, ih).

We see that .ηh (t, x) ∈ β(t, x, ∇y h (t)) a.e. on .Q. Then, we deduce from (4.28)–(4.33) the estimates || || || || || || || h || . ||y (t)|| + ||γ (y h (t))|| 2 ≤ C, for t ∈ [0, T ], . 2 L (Ω)

{

L (|)

T

dt ≤ C, .

(4.52)

|| || h || y (t + h) − y h (t) ||2 || || || 2 dt ≤ C, . || h L (Ω)

(4.53)

|| || || γ (y h (t + h)) − γ (y h (t)) ||2 || || || 2 dt ≤ C, . || h L (|)

(4.54)

L (Ω)

0

{ 0

{

T 0

|| || || h ||p ||∇y (t)|| p

(4.51)

T

{

T

{ j (t, x, ∇y h (t))dxdt ≤ C,

0

(4.55)

Ω

with C independent of .h. Also, (4.51) and (4.52) imply that { .

0

T

|| || || h ||p ||y (t)|| 1,p W

(Ω)

dt ≤ C,

(4.56)

where .p given by (4.19). Definition 4.3 We call a weak solution to problem (4.1)–(4.3) a function .y ∈ L2 (Q), with .∇y

∈ Lp (0, T ; (Lp (Ω))N ), γ (y) ∈ L2 (Σ), j (·, ·, ∇y) ∈ L1 (Q),

such that there exists .η

'

∈ (Lp (Q))N , η(t, x) ∈ β(t, x, ∇y(t, x)) a.e. (t, x) ∈ Q,

satisfying { .

{

−

{

yφt dxdt = Q

η · ∇φdxdt − Q

yφt dσ dt Σ

(4.57)

4.2 The Strongly Coercive Case

97

{ =

{ f φdxdt +

Q

{ gφdσ dt +

Σ

{ y0 φ(0)dx +

Ω

y0 φ(0)dσ, |

for all .φ ∈ W 1,2 (0, T ; L2 (Ω)), .∇φ ∈ (Lp (Q))N , .γ (φ) ∈ W 1,2 (0, T ; L2 (|)), .φ(T ) = 0.

Theorem 4.4 below is the main result of this section. Theorem 4.4 Let us assume (4.21). Then, under hypotheses .(H1 ), .(H2 ), .(H5 ), problem (4.1)–(4.3) has at least one weak solution .y, which satisfies .y

∈ W 1,2 (0, T ; L2 (Ω)), γ (y) ∈ W 1,2 (0, T ; L2 (|)),

(4.58)

∇ · η ∈ L2 (Q), η(t, x) ∈ β(t, x, ∇y(t, x)) a.e. (t, x) ∈ Q. Moreover, y is a strong solution to (4.1)–(4.3), that is, − ∇ · η = f, a.e. in Q, .

(4.59)

γ (η) · ν + yt = g, x a.e. on Σ, .

(4.60)

.yt

y(0) = y0 , in Ω.

(4.61)

The solution y is given by .y

= lim y h strongly in Lr (Q), h→0

(4.62)

with .y h defined by (4.50), for .r = 2 if .p ≥ 2 and for .r = p, if .p ∈ (1, 2). The solution is unique in the class of functions satisfying (4.58)–(4.61), and the map .(y0 , γ (y0 )) → (y(t), γ (y(t))) is Lipschitz from .L

2

(Ω) × L2 (|) to C([0, T ]; L2 (Ω)) × C([0, T ]; L2 (|)).

Proof The proof is done in three steps. We note that for .y0 ∈ U we have .j (0, ·, ∇y0 ) ∈ L1 (Ω). Step 1. Existence of a weak solution. By (4.51)–(4.55), it follows that one can select a subsequence such that as .h → 0, we have yh γ (y h ) ∇y h . h y (t + h) − y h (t) h γ (y h (t + h)) − γ (y h (t)) h

→y weak∗ in L∞ (0, T ; L2 (Ω)), → γ (y) weak∗ in L∞ (0, T ; L2 (|)), → ∇y weakly in (Lp (Q))N , →l

weakly in L2 (Q),

→ l1

weakly in L2 (Σ).

98

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . .

The last two assertions are proved by a direct calculus. For some .δ > 0, we take .φ ∈ Mδ , where .Mδ

= {φ ∈ C ∞ (Q); φ(t, x) = 0 on t ∈ [T − δ, T ]},

and calculate (without writing the argument x for the functions .y h and .φ) {

{

y h (s + h) − y h (s) φ(s)dxds h Ω { T −h { { T −h { 1 h 1 h y (s + h)φ(s)dxds − y (s)φ(s)dxds = 0 0 Ω h Ω h { T{ { T −h { 1 h 1 h = y (s)φ(s − h)dxds − y (s)φ(s)dxds h h h 0 Ω Ω { T −h { { T { 1 h 1 h = y (s)φ(s − h)dxds + y (s)φ(s − h)dxds h T −h Ω h Ω h { h{ { T −h { 1 h 1 h − y (s)φ(s)dxds − y (s)φ(s)dxds 0 h Ω h Ω h { h{ { T −h { φ(s) − φ(s − h) h 1 h y (s)dxds − y (s)φ(s)dxds =− h h h 0 Ω Ω { T { 1 h + y (s)φ(s − h)dxds. T −h Ω h

T .

0

In this calculus, .φ(t, x) = 0 for .t ∈ [T − h, T ] since we can take .δ > h. Next, { .

−

h{

1 h y (s)φ(s)dxds h 0 Ω { { { { 1 h 1 h y h (s)(φ(s) − φ(0))dxds − y h (s)φ(0)dxds =− h 0 Ω h 0 Ω { { { || 1 h h 1 h || || h || φ(0) y (s)dsdx ≤ ||y (s)|| 2 ||φ(s) − φ(0)||L2 (Ω) ds − L (Ω) h 0 h 0 Ω { { { 1 h ||φ(s) − φ(0)||L2 (Ω) ds − φ(0)y0 dx + e(h) → − φ(0)y0 dx, ≤C h 0 Ω Ω

where .e(h) → 0 as .h → 0. Proceeding in the same way for the last term, we get { .

T

{

T −h Ω

{ +

T

1 h y (s)φ(s − h)dxds = h {

T −h Ω

{

T

{

T −h Ω

1 h 1 y (s)φ(T − h)dxds ≤ C h h

1 h y (s)(φ(s − h) − φ(T − h))dxds h {

T T −h

||φ(s − h) − φ(T − h)||L2 (Ω) ds,

4.2 The Strongly Coercive Case

99

as .φ(T − h) = 0. Again by the continuity of .φ, we obtain that { .

T

{

T −h Ω

1 h y (s)φ(s − h)dxds → 0, as h → 0. h

All these yield { T{ .

y h (t + h) − y h (t) φ(t)dxdt = − h

lim

h→0 0

Ω

{ T{ y(t) 0

Ω

dφ (t)dxdt − dt

{ φ(0)y0 dx, Ω

for any .φ ∈ Mδ . Therefore, we get, in the sense of distributions, that { T{ .

lim

h→0 0

Ω

y h (t + h) − y h (t) dy φ(t)dxdt = (φ), for any φ ∈ C0∞ (Q), h dt

(4.63)

' and so, .l = dy dt in .D (Q) (the space of Schwartz distributions on .Q). Moreover, by (4.53), we still have

| | | dy | | ≤ C ||φ|| 2 , for φ ∈ C ∞ (Q) ∩ Mδ , (φ) L (Q) 0 | dt |

.|

dy 2 with C independent of .δ, and so . dy dt ∈ L (Q) and .l = dt a.e. on .Q. Thus, .y ∈ 1,2 2 W (0, T − δ; L (Ω)), and since .δ is arbitrary, we infer that .y ∈ W 1,2 (0, T ; L2 (Ω)). Proceeding in the same way for the time differences on .|, we get that

{ lim

.

h→0

{

γ (y h (t + h)) − γ (y h (t)) φ(t)dσ dt h 0 {| T { { dφ γ (y(t)) (t)dσ dt − φ(0)y0 dσ, for φ ∈ Mδ . =− dt 0 | | T

(4.64)

Therefore, we obtain that .l1

=

dγ (y) dγ (y) a.e. on Σ, ∈ L2 (0, T − δ; L2 (|)) dt dt

and so, finally .γ (y) ∈ W 1,2 (0, T ; L2 (|)). We deduce that .ξ = ∇y a.e. on .Q, by the same argument used in Proposition 4.2. By passing to the limit in (4.55) and using the weak lower semicontinuity of the convex integrand, we get .j (t, ·, ∇y) ∈ L1 (Q). The next step is to prove (4.62). A simple way to show it is to use a compactness argument in the space of vectorial functions with bounded variations on .[0, T ]. We have that .y h ∈ BV ([0, T ]; L2 (Ω)), the space of functions with bounded variation from .[0, T ] to .L2 (Ω), see Definition 7.12 in the Appendix. Then, the total variation of h .y is

100

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . .

.V (y

h

np || || Σ || || h ; [0, T ]) = sup ≤ C, ||y (si ) − y h (si−1 )|| 2 L (Ω) P ∈P i=1

(4.65)

where C is a constant and .P = {P = (s0 , . . . , snp ); P is a partition of .[0, T ]} is the set of all partitions of .[0, T ]. Indeed, if we consider an equidistant partition (e.g., with .si = ti ), we have by (4.31) that (n−1 || Σ || || || h ||y (ti+1 ) − y h (ti )|| .

i=0

)2 L2 (Ω)

≤n

n−1 || ||2 Σ || h || ||yi+1 − yih || 2 i=0

= nh · h

L (Ω)

||

n || h Σ || yi+1 i=1

|| ||

||2 − yih || || || || 2 h

(4.66) ≤ T C.

L (Ω)

At this point, we discuss separately the cases .p ≥ 2 and .p ∈ (1, 2). Let .p ≥ 2. By (4.66), we also have .y h ∈ BV ([0, T ]; (H 1 (Ω))' ). On the basis of this relation, (4.51), and since .L2 (Ω) is compact in .(H 1 (Ω))' , we can apply the strong version of Helly theorem for the infinite dimensional case (see Theorem 7.13, see also [21], Remark 1.127, p. 48). We deduce that .y

h

(t) → y(t) strongly in (H 1 (Ω))' for t ∈ [0, T ].

(4.67)

Next, applying Lemma 5.1 in [56], p. 58, we have that for any .ε > 0 there exists a constant .Cε such that . ||w||L2 (Ω)

≤ ε ||w||H 1 (Ω) + Cε ||w||(H 1 (Ω))' , ∀w ∈ H 1 (Ω).

(4.68)

This lemma applied for .w = y h (t) − y(t) yields .

||2 || ||2 ||2 || 1 || || || || || || || h ≤ ε ||y h (t) − y(t)|| 1 + Cε ||y h (t) − y(t)|| 1 ' . ||y (t) − y(t)|| 2 L (Ω) H (Ω) (H (Ω)) 2

Integrating with respect to t on .(0, T ), we obtain that .

1 2

{

T 0

||2 || || || h ||y (t) − y(t)|| 2

L (Ω)

{ ≤ε 0

T

dt

|| ||2 || h || ||y (t) − y(t)||

H 1 (Ω)

{ dt + Cε 0

T

|| ||2 || h || ||y (t) − y(t)||

(H 1 (Ω))'

dt.

The last term on the right-hand side tends to 0 as .h → 0, by (4.67), and the coefficient of .ε is bounded, by (4.56). Therefore, { . lim sup

h→0

0

T

||2 || || || h ||y (t) − y(t)|| 2

L (Ω)

dt ≤ Cε for any ε > 0,

4.2 The Strongly Coercive Case

101

and since .ε is arbitrary, we get (4.62), with .r = 2. Let .p ∈ (1, 2). We shall prove that .y

h

→ y strongly in Lp (Q), as h → 0.

(4.69)

From (4.66), it follows that .y h ∈ BV ([0, T ]; Lp (Ω)). We assert that there exists a Banach space X such that .Lp (Ω) ⊂ X, with compact injection. For example, .X = ' rN (W 1,r (Ω))' , i.e., if .W 1,r (Ω) ⊂ Lp (Ω), with .p ' < N−r if .N > r, and .p ' = r if .N ≤ r. || h || || Applying again the strong Helly theorem and since . y (t)||Lp (Ω) ≤ C, we get .y

h

(t) → y(t) strongly in X, uniformly with t ∈ [0, T ].

(4.70)

Then, we apply the argument before for the triplet .W 1,p (Ω) ⊂ Lp (Ω) ⊂ X and use (4.70) and (4.56). We get .

1 2

{

||p || || || h ||y (t) − y(t)|| p dt L (Ω) 0 { T || ||p || h || ≤ε ||y (t) − y(t)|| 1,p T

W

0

{ (Ω)

T

dt + Cε 0

|| ||p || h || ||y (t) − y(t)|| dt → 0, X

as .ε → 0, whence (4.69) follows. Let us fix .t ∈ [0, T ]. By (4.51), on a subsequence, we have .y

h

(t) → ϑ(t) weakly in L2 (Ω), as h → 0, for each fixed t ∈ [0, T ],

and since we have either (4.67) or (4.70), we get by the limit uniqueness that .ϑ(t) = y(t), a.e. on .Ω. In particular, it follows that (T ) → y(T ) weakly in L2 (Ω), as h → 0.

.y

h

h

→ f strongly in L2 (0, T ; L2 (Ω)), as h → 0.

(4.71)

g h → g strongly in L2 (0, T ; L2 (|)), as h → 0.

(4.72)

We have .f

Relation (4.20) yields | |

. |η

h

| | |p−1 | | | (t, x)| ≤ C3 |∇y h (t, x)| + C30 ,

for .ηh (t, x) ∈ β(t, x, ∇y h (t, x)) a.e. on .Q, and so we conclude that .(ηh )h is bounded ' in .(Lp (Q))N . Therefore, on a subsequence, we have .η

h

'

→ η weakly in (Lp (Q))N , as h → 0,

102

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . .

and it remains to prove that .η(t, x) ∈ β(t, x, ∇y(t, x)), a.e. .(t, x) ∈ Q. Summing up (4.44) from .i = 0 to .n − 1, we get n−1 {

|| Σ 1 || || h ||2 ||yn || 2 + h L (Ω) 2

.

i=0

≤h

n−1 { Σ Ω

i=0

Ω

h h ηi+1 · ∇yi+1 dx +

h h fi+1 yi+1 dx + h

n−1 { Σ i=0

|

|| 1 || || h ||2 ||yn || 2 L (|) 2

h h gi+1 yi+1 dσ +

1 1 ||y0 ||2L2 (Ω) + ||y0 ||2L2 (|) , 2 2

whence, replacing the definition of .y h (t), we obtain { T{ ||2 ||2 1 || 1 || || h || || || (T ) + h ηh · ∇y h dxdt + ||y h (T )|| 2 ||y || 2 L (Ω) L (|) 2 2 0 Ω { T{ { T{ 1 1 ≤h f h y h dxdt + h g h y h dσ dt + ||y0 ||2L2 (Ω) + ||y0 ||2L2 (|) . 2 2 0 0 Ω |

.

This yields, by passing to the limit as .h → 0, {

.

1 ηh · ∇y h dxdt ≤ − ||y(T )||2L2 (Ω) (4.73) 2 Q h→0 { { 1 1 1 + fydxdt + gydσ dt + ||y0 ||2L2 (Ω) + ||y0 ||2L2 (|) − ||y(T )||2L2 (|) . 2 2 2 Q Σ lim sup

We write (4.23)–(4.24) in the following form, after replacing the functions .yih by .y h , and integrating with respect to .t ∈ (0, T ) { .

Q

{

y h (t + h, x) − y h (t, x) φ(t, x)dxdt + h

ηh (t + h, x) · ∇φ(t, x)dxdt Q

{

γ (y h (t + h, x)) − γ (y h (t, x)) φ(t, x)dσ dt h Σ { { h = f (t + h, x)φ(t, x)dxdt + g h (t + h, x)φ(t, x)dσ dt +

Q

(4.74)

Σ

for any .φ ∈ C ∞ (Q), with .φ(T , x) = 0. We pass to the limit as .h → 0, using the convergences previously deduced, and get { .

{

−

yφt dxdt + Q

{ η · ∇φdxdt −

Q

{

=

{

f φdxdt + Q

{

gφdσ dt + Σ

(4.75)

yφt dσ dt Σ

{ y0 φ(0)dx +

Ω

y0 φ(0)dσ, |

4.2 The Strongly Coercive Case

103 '

where .η = lim ηh weakly in .(Lp (Ω))N . h→0

Taking into account that .y ∈ W 1,2 (0, T ; L2 (Ω)), we have {

{

{

yφt dxdt = −

.

yt φdxdt −

Q

y0 φ(0)dx,

Q

Ω

and so we obtain { { { { { . yt φdxdt + η · ∇φdxdt + yt φdσ dt = f φdxdt + gφdσ dt. Q

Q

Σ

Q

(4.76)

Σ

By density, this extends to all .φ ∈ W 1,2 (0, T ; L2 (Ω)) with .∇φ ∈ (Lp (Q))N , and in particular, for .φ = y. Finally, we have { .

1 1 η · ∇ydxdt = − ||y(T )||2L2 (Ω) + ||y0 ||2L2 (Ω) 2 2 Q { { 1 1 + fydxdt + gφdσ dt − ||y(T )||2L2 (|) + ||y0 ||2L2 (|) . 2 2 Q Σ

(4.77)

Comparing with (4.73), we deduce that {

T

{

{ η · ∇y dxdt ≤ h

. lim sup

h→0

0

η · ∇ydxdt

h

Ω

Q

and since the operator .z → β(t, x, z) is maximal monotone in the dual pair .(Lp (Q))N — p' N .L (Q)) , we get .η(t, x) ∈ β(t, x, ∇y(t, x)), a.e. on Q (see e.g., [13], p. 41, Corollary 2.4). Hence, y is a weak solution to (4.1)–(4.3). Step 2. Existence of a strong solution. By (4.76), we see that {

{

{

yt φdxdt +

.

Q

η · ∇φdxdt = Q

Q

f φdxdt, ∀φ ∈ C0∞ (Q).

Since we have { .(−∇

· η)(φ) = Q

η · ∇φdxdt, ∀φ ∈ C0∞ (Q), '

and recalling that .yt , .f ∈ L2 (Q), .η ∈ (Lp (Q))N , we get (4.59) in .D' (Q), as claimed. Because .∇ · η ∈ L2 (Q), it follows that .γ (η)

· ν ∈ L2 (0, T ; H −1/2 (|))

is well-defined (see e.g., [2], or Theorem 1.2 in [77]), and the following formula holds

104

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . .

{

{

{

φ(t)∇ · η(t)dx = −

.

η(t) · ∇φ(t)dx +

Ω

Ω

φ(t)(γ (η(t)) · ν)dσ, |

(4.78)

a.e. t ∈ (0, T ).

Now, we multiply (4.59) by .φ ∈ W 1,2 (0, T ; L2 (Ω)) with .∇φ ∈ (Lp (Q))N , .γ (φ) ∈ = 0, and by (4.78), we get that

W 1,2 (0, T ; L2 (|)), .φ(T ) { .

{

−

{

yφt dxdt − Q

{

y0 φ(0)dx − Ω

{

φη · νdσ dt + Σ

η · ∇φdxdt = Q

f φdxdt. Q

(4.79) After replacing (4.79) in (4.57), we obtain that {

{ γ (y)φt dσ dt +

.

Σ

{ y0 φ(0)dσ −

|

{ φη · νdσ dt +

Σ

gφdσ dt = 0, Σ

whence we deduce (4.60) in the sense of distributions and also a.e. on .Σ, since, as seen d γ (y) ∈ L2 (Σ). earlier, . dt Step 3. Continuous dependence on data. Let us consider two solutions y and .y to (4.1)– (4.3), corresponding to the data .(y0 , f, g) and .(y0 , f , g), respectively, in the class of functions satisfying (4.58)–(4.61). We multiply the difference of the two Eqs. (4.59) corresponding to these data by .(y − y)(t) and integrate on .Ω. We get .

1 d ||y(t) − y(t)||2L2 (Ω) + 2 dt

{ (η(t) − η(t)) · (∇y(t) − ∇y(t))dx Ω

1 d ||γ (y(t)) − γ (y(t))||2L2 (|) 2 dt { = (f (t) − f (t))(y(t) − y(t))dt,

+

Ω

where .η(t, x) ∈ β(t, x, ∇y(t, x)), .η(t, x) ∈ β(t, x, ∇y(t, x)) a.e. .(t, x) ∈ Q. Using the monotonicity of .β and integrating with respect to t, we obtain that .

||y(t) − y(t)||2L2 (Ω) + ||γ (y(t)) − γ (y(t))||2L2 (|) ⎛

⎞ ||y0 − y0 ||2L2 (Ω) + ||γ (y0 ) − γ (y0 )||2L2 (|) { T || ⎜ ⎟ || ⎜ + ⎟ ||f (t) − f (t)||2 2 dt ⎟ , ∀t ∈ [0, T ], ≤C⎜ (Ω) L ⎜ ⎟ 0 { T ⎝ ⎠ 2 ||g(t) − g(t)||L2 (|) dt + 0

as claimed.

u

4.3 The Weakly Coercive Case

105

4.3 The Weakly Coercive Case We assume that hypotheses .(H1 ), .(H3 )−(H5 ) characterize weakly coercive functions j and .j ∗ . As mentioned earlier, without loss of generality, we may assume (4.13). A standard example in this case is .β(t, x, r)

= a(t, x) log(|r|N + 1)sign(r) + a(t, x)

|r|N , |r|N + 1

where .a ∈ C 1 (Q), .a > 0 and .sign(r) = r/ |r|N for .r /= 0, .sign(0) = {r; .|r|N ≤ 1} is the multivalued function signum. On the other hand, monotone functions .r → β(t, x, r) with exponential growth and symmetric at .±∞, in the sense of (4.11), are accepted by the current hypotheses. First, we note down the following simple lemma, for later use. Lemma 4.5 Let .u

∈ (L1 (Ω))N , w ∈ (L1 (Ω))N , j (·, ·, u) ∈ L1 (Ω), j ∗ (·, ·, w) ∈ L1 (Ω).

Then, under assumption (4.11), we have .u

· w ∈ L1 (Ω).

(4.80)

Proof By the Legendre–Fenchel relations + j ∗ (t, x, ω) ≥ ω · r, ∀r, ω ∈ RN , ∀(t, x) ∈ Q,

(4.81)

+ j ∗ (t, x, ω) = ω · r iff ω ∈ ∂j (t, x, r), ∀(t, x) ∈ Q,

(4.82)

.j (t, x, r)

.j (t, x, r)

we have .j (t, x, u(x))

+ j ∗ (t, x, w(x)) ≥ u(x) · w(x), ∀(t, x) ∈ Q,

and this yields { u(x) · w(x)dx < ∞.

.

Ω

We write (4.81) for .(−u∗ ) .j (t, x, −u

∗

(x)) + j ∗ (t, x, w(x)) ≥ −u∗ (x) · w(x), ∀(t, x) ∈ Q,

and use (4.11) to obtain

106

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . .

{

{

{ (−u · w) dx ≤ γ1

.

j (t, x, u)dx + γ2 meas(Ω) +

Ω

Ω

j ∗ (t, x, w)dx < ∞.

Ω

u

Therefore, we get (4.80), as claimed.

We define the space .U1

= {z ∈ L2 (Ω); z ∈ W 1,1 (Ω), γ (z) ∈ L2 (|)}.

(4.83)

For t fixed in .[0, T ], .h > 0, and .w1 ∈ L2 (Ω), .w2 ∈ L2 (|), let us consider the problem .

.u

u − h∇ · β(t, x, ∇u) e w1 in Ω, u + hβ(t, x, ∇u) · ν e w2 on |.

(4.84)

As in the previous case, we call a weak solution to problem (4.84) a function ∈ U1 , such that .j (t, ·, ∇u) ∈ L1 (Ω), and there exists .η

∈ (L1 (Ω))N , η(x) ∈ β(t, x, ∇u(x)) a.e. x ∈ Ω, j ∗ (t, ·, η) ∈ L1 (Ω),

(4.85)

satisfying {

{ (uψ + hη · ∇ψ)dx +

.

Ω

uψdσ = b(ψ), ∀ψ ∈ C 1 (Ω),

(4.86)

|

with b given by (4.35) for all .u ∈ U1 . Problem (4.84) has a unique solution given by the unique minimizer of the functional .ϕ : L2 (Ω) → R { { ⎧ { 1 ⎨1 u2 dx + h j (t, x, ∇u)dx + u2 dσ − b(u), if u ∈ U1 , ϕ(u) . = 2 Ω (4.87) 2 | Ω ⎩ +∞, otherwise. Next, we prove the equivalence between (4.84) and the minimization problem Minimize {ϕ(u); u ∈ U1 } .

.

(4.88)

Proposition 4.6 Problem (4.84) has a unique solution that is the minimizer of .ϕ. Proof Let .λ > 0, and consider the approximating regularized problem .u

− h∇ · (βλ (t, x, ∇u) + λ∇u) = w1 in Ω,

u + h(βλ (t, x, ∇u) + λ∇u) · ν = w2 on |,

(4.89)

4.3 The Weakly Coercive Case

107

where .βλ is the Yosida approximation of .β (see (7.22) =

.βλ (t, x, r)

1 (1 − (1 + λβ(t, x, ·))−1 )r, ∀r ∈ RN . λ

(4.90)

Its potential (i.e., the Moreau regularization of .j, (7.35).) is given by { .jλ (t, x, r)

= inf N s∈R

} |r − s|2N + j (t, x, s) 2λ

(4.91)

|2 1 || | |(1 + λβ(t, x, ·))−1 r − r | + j (t, x, (1 + λβ(t, x, ·))−1 r), N 2λ

=

and the function .jλ has the following properties: .

jλ (t, x, r) ≤ j (t, x, r) for all r ∈ RN , (t, x) ∈ Q, λ > 0, limλ→0 jλ (t, x, r) = j (t, x, r), for all r ∈ RN , (t, x) ∈ Q.

(4.92)

We deduce that the solution to (4.89) is provided by the unique minimizer of the problem .Minimize

{ } ϕλ (u); u ∈ L2 (Ω) ,

(4.93)

where .ϕλ : L2 (Ω) → R,

.ϕλ (u)

=

{ { ⎧1{ 1 2 ⎪ u dx + h j (t, x, ∇u)dx + u2 dσ ⎪ λ ⎪ ⎨2 Ω 2 | { Ω ⎪ ⎪ ⎪ ⎩

+λ +∞,

Ω

(4.94)

|∇u|2N dx − b(u), if u ∈ W 1,2 (Ω), otherwise.

Indeed, by applying Proposition 4.2, we obtain a weak solution .uλ ∈ W 1,2 (Ω) that satisfies { { { { . uλ ψdx = h βλ (t, x, ∇uλ ) · ∇ψdx + λ ∇uλ · ∇ψdx + uλ ψdσ Ω

Ω

Ω

{

|

{

=

w1 ψdx + Ω

w2 ψdσ, ∀ψ ∈ W 1,2 (Ω).

(4.95)

|

In particular for .ψ = uλ , this yields .

1 2

{

{ Ω

u2λ dx +

{ βλ (t, x, ∇uλ ) · ∇uλ dx + λh Ω

Ω

{ |∇uλ |2N dx +

|

u2λ dσ

108

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . .

{

{

=

w1 uλ dx + Ω

(4.96)

w2 uλ dσ, |

whence we obtain the estimate { { { { |∇uλ |2N dx + u2λ dσ ≤ C. . u2λ dx + 2h jλ (t, x, ∇uλ )dx + 2λh Ω

Ω

Ω

(4.97)

|

By C, we denote a positive constant independent of .λ. Replacing the definition (4.91) (second line) of .jλ , we get {

{ .

Ω

u2λ dx

+h Ω

{

|2 1 || | |(1 + λβ(t, x, ·))−1 ∇uλ − ∇uλ | dx N λ j (t, x, (1 + λβ(t, x, ·))−1 ∇uλ )dx

+2h {

Ω

+2λ Ω

(4.98)

{ |∇uλ |2N dx +

|

u2λ dσ ≤ C.

Consequently, each term on the left-hand side in (4.98) is bounded independently of .λ, and, in particular, {

j (t, x, (1 + λβ(t, x, ·))−1 ∇uλ )dx ≤ C, ∀λ > 0.

.

(4.99)

Ω

By (4.99), (4.7), and the Dunford–Pettis theorem (see Theorem 7.3), we can deduce the N weakly compactness of the sequence .((1 + λβ(t, x, ·))−1 ∇uλ )λ>0 in .(L1 (Ω)) { . Indeed, −1 denoting .zλ = (1 + λβ(t, x, ·)) ∇uλ , we have to show that the integrals . S |zλ |N dx, with .S ⊂ Ω, .ε > 0 there exists { are equi-absolutely continuous, meaning that for every2C |z | .δ such that . dxdt < ε whenever meas .(S) < δ. Let .Mε > λ N S ε , where C is the λ) |z | ≥ M for . > RM , by (4.7). If constant in (4.99), and let .RM be such that . j (t,x,z ε λ N |zλ |N ε .δ < , then 2RM {

{ |zλ | dxdt ≤

.

S

{

{x;|zλ (x)|N ≥RM }

≤ Mε−1

{

|zλ | dx +

{x;|zλ (x)|N 0.

(4.108)

This yields (by the definition (4.91) for .jλ∗ ) { .

Ω

|2 1 || | |(1 + λβ −1 (t, x, ·))−1 βλ (t, x, ∇uλ ) − βλ (t, x, ∇uλ )| dx N 2λ { j ∗ (t, x, (1 + λβ −1 (t, x, ·))−1 βλ (t, x, ∇uλ ))dx + Ω

{ =

Ω

jλ∗ (t, x, βλ (t, x, ∇uλ ))dx ≤ C.

(4.109)

110

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . .

Arguing as above, on the basis of (4.8) and the Dunford–Pettis theorem, we deduce that ) ( the sequence . (1 + λβ −1 (t, x, ·))−1 βλ (t, x, ∇uλ ) λ>0 is weakly compact in .(L1 (Ω))N , and so, on a subsequence, as .λ → 0, we get .(1

.(1

+ λβ −1 (t, x, ·))−1 βλ (t, x, ∇uλ ) → η weakly in (L1 (Ω))N ,

(4.110)

+ λβ −1 (t, x, ·))−1 βλ (t, x, ∇uλ ) − βλ (t, x, ∇uλ ) → 0 strongly in (L2 (Ω))N ,

which implies .βλ (t, x, ∇uλ )

→ η weakly in (L1 (Ω))N .

(4.111)

Then, by (4.108) and the weak lower semicontinuity of the convex integrand, we infer that .j

∗

(t, ·, η) ∈ L1 (Ω).

(4.112)

Now, we pass to the limit in (4.106), taking into account (4.98) and (4.110), and we get { .

βλ (t, x, ∇uλ ) · ∇uλ dx

lim sup λ→0

Ω

≤

{{

1 h

w1 u∗ dx +

(4.113)

{

Ω

w2 u∗ dσ −

{

|

(u∗ )2 dx −

} (u∗ )2 dσ .

{

Ω

|

Then, letting .λ → 0 in (4.95) and recalling (4.80), we obtain { .

u∗ ψdx + h

Ω

{

{ η · ∇ψdx + Ω

{

u∗ ψdσ =

{ w1 ψdx +

|

Ω

w2 ψdσ, |

∀ψ ∈ C 1 (Ω). This is extended by density for all .ψ ∈ L2 (Ω) ∩ W 1,1 (Ω), .γ (ψ) ∈ L2 (|), and in particular for .ψ = u∗ . We obtain {

η · ∇u∗ dx =

.

Ω

1 h

{{ Ω

w1 u∗ dx +

{

w2 u∗ dσ − |

{

(u∗ )2 dx − Ω

} (u∗ )2 dσ .

{ |

(4.114)

By (4.113) and (4.114), we finally obtain that {

{ βλ (t, x, ∇uλ ) · ∇uλ dx ≤

. lim sup

λ→0

Ω

η · ∇u∗ dx.

(4.115)

Ω

Since .∇uλ → ∇u∗ weakly in .(L1 (Ω))N , .βλ (t, x, ∇uλ ) → η weakly in .(L1 (Ω))N , and using (4.113), we deduce by [13], p. 38, Lemma 2.3, that

4.3 The Weakly Coercive Case

111

.η(x)

∈ β(t, x, ∇u∗ (x)) a.e. x ∈ Ω.

In order to get the latter relation, we note that by (4.115) we have {

{

η · (∇u∗ − θ )dx, ∀θ ∈ (L1 (Ω))N ,

(jλ (t, x, ∇uλ ) − jλ (t, x, θ ))dx ≤

.

Ω

Ω

and letting .λ → 0, we get by (4.91), (4.105), and (4.92) { .

(j (t, x, ∇u∗ ) − j (t, x, θ ))dx ≤

Ω

{

η · (∇u∗ − θ )dx, ∀θ ∈ (L1 (Ω))N .

Ω

Since .θ is arbitrary, we get that .η(x) ∈ ∂j (t, x, ∇u∗ (x)), as desired. Passing to the limit in (4.95), we get { .

(u∗ ψ + hη · ∇ψ)dx +

Ω

{

u∗ ψdσ = b(ψ), ∀ψ ∈ W 1,2 (Ω),

|

where b is defined by (4.35) for all .u ∈ U1 . By density, this extends to all .ψ ∈ L2 (Ω) ∩ W 1,1 (Ω), with .γ (ψ) ∈ L2 (|). Hence, .u∗ is the weak solution to (4.84). Uniqueness of ∗ 2 .u , as weak solution, is immediate. Moreover, it also follows that .∇ · η ∈ L (Ω). u Definition 4.7 Let .y0

∈ U1 , f ∈ L2 (Q), g ∈ L2 (Σ).

(4.116)

We call a weak solution to problem (4.1)–(4.3) a pair .(y, η), .y

∈ L2 (Q) ∩ L1 (0, T ; W 1,1 (Ω)), γ (y) ∈ L2 (Σ), j (·, ·, ∇y) ∈ L1 (Q),

(4.117)

η ∈ (L1 (Q))N , η(t, x) ∈ β(t, x, ∇y(t, x)), a.e. on Q, j ∗ (·, ·, η) ∈ L1 (Q), which satisfies { { { − yφt dxdt + η · ∇φdxdt − yφt dσ dt Q Q Σ { { { { . = f φdxdt + y0 φ(0)dx + y0 φ(0)dσ + gφdσ dt, Q

Ω

|

(4.118)

Σ

for all .φ ∈ W 1,2 (0, T ; L2 (Ω)) ∩ L1 (0, T ; W 1,1 (Ω)), .γ (φ) ∈ W 1,2 (0, T ; L2 (|)), with .φ(T ) = 0.

By Lemma 4.5, it is clear that the second term on the left-hand side in (4.118) makes sense. Theorem 4.8 Let us assume (4.116) and .j (0, ·, ∇y0 ) ∈ L1 (Ω). Then, under hypotheses .(H1 ), .(H3 )–.(H5 ), problem (4.1)–(4.3) has at least one weak solution. Moreover, y is a

112

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . .

strong solution to (4.1)–(4.3), that is, it satisfies (4.59)–(4.61). Finally, y is given by .y

= lim y h strongly in L1 (Q),

(4.119)

h→0

where .y h is defined by (4.50). The solution is unique in the class of functions satisfying (4.117), (4.59)–(4.61). Proof Let us consider the time-discretized system (4.23)–(4.25) whose weak solution is defined as in Definition 4.1, by replacing U by .U1 . We claim that system (4.23)–(4.25) has a unique weak solution that satisfies || ||

||

h || L2 (Ω)

. ||ym ||

|| || || h || + ||γ (ym )||

+h

L2 (|)

||

.

h

i=0

|| ||

m−1 Σ{ i=0

m−1 Σ ||

|| || h || ||∇yi+1 ||

||2 − yih || || || || 2 h

+h

L (Ω)

Ω

h j (ti+1 , x, ∇yi+1 )dx

|| || h h ||2 || γ (yi+1 ) − γ (yi ) || || || || || 2 h

m−1 Σ || i=0

(4.120)

≤ C, m = 1, . . . , n,

L1 (Ω)

i=0 m−1 h Σ || || yi+1

+h

≤ C,

L (|)

(4.121)

m = 1, . . . , n, where C is a positive constant, independent of h. The proof can be lead as in Proposition 4.2 (see (4.44)), using the hypothesis corresponding to the weakly coercive case. Next, we define .y h by (4.50), and on the basis of (4.120) and (4.121), we write

.

|| || || h || ||y (t)||

L2 (Ω)

|| || || || + ||γ (y h (t))||

{ L2 (|)

+

j (t, x, ∇y h (t))dxdt ≤ C, Q

(4.122)

for t ∈ [0, T ], { .

0

T

|| || h || y (t + h) − y h (t) ||2 || || || 2 dt || h L (Ω) || { T || || γ (y h (t + h)) − γ (y h (t)) ||2 || || + || || 2 dt ≤ C. h 0 L (|)

(4.123)

Using these estimates and the Dunford–Pettis compactness theorem in .L1 (Q), we can select a subsequence such that, as .h → 0,

4.4 The Semigroup Approach

yh γ (y h ) ∇y h . h y (t + h) − y h (t) h γ (y h (t + h)) − γ (y h (t)) h

113

→y weak* in L∞ (0, T ; L2 (Ω)), → γ (y) weak* in L∞ (0, T ; L2 (|)), → ∇y weakly in (L1 (Q))N , →

dy dt

weakly in L2 (Q),

→

dy dt

weakly in L2 (Σ).

We denote .X = W 1,r (Ω) with .r > N. Then, .W 1,r (Ω) is compact in .L∞ (Ω), and .L1 (Ω) is compact in .X ' = (W 1,r (Ω))' . We have that .y h ∈ BV ([0, T ]; L2 (Ω)), which implies that .y h ∈ BV ([0, T ]; X ' ). By the Helly theorem, it follows that .y

h

(t) → y(t) strongly in X ' , uniformly with t ∈ [0, T ].

Using again the Aubin–Lions–Simon Lemma 7.9 in the Appendix and taking into account that .W 1,1 (Ω) ⊂ L1 (Ω) ⊂ X ' , we deduce that .y

h

→ y strongly in L1 (Q), as h → 0.

The next part of the proof follows as in Theorem 4.4, and so it will be skipped.

(4.124) u

Remark 4.9 The singular case .β(t, x, r) = ρsign(r) (which is relevant in the study of diffusion systems with singular energy) is ruled out by the present approach, but, as seen later, the corresponding problem (4.1)–(4.3) is well posed, however, in the space of functions with bounded variation on .Ω.

4.4 The Semigroup Approach We discuss both strongly coercive and weakly coercive cases, so that we shall assume that either hypotheses .(H1 ), .(H2 ), .(H5 ) or .(H1 ), .(H3 )−(H5 ) are satisfied. Here, we assume that .β is independent of .t, that is .β ≡ β(x, r). It shall turn out that in this time-invariant case Theorems 4.4 and 4.8 can be proved by the nonlinear contraction semigroup theory that leads to sharper regularity results for the solution .y. On the space .X = L2 (Ω) × L2 (|), endowed with the standard Hilbertian structure, we consider the operator .A : D(A) ⊂ X → X, defined by ( ) ( ) ( ) u −∇ · β(x, ∇u) u .A = , ∀ ∈ D(A), z β(x, ∇u) · ν z

(4.125)

114

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . .

⎧( ) ⎫ ⎪ ⎪ ⎨ u ∈ X; u ∈ U -, z = γ (u), ∃η(x) ∈ β(x, ∇u(x)) a.e. x ∈ Ω,⎬ D(A) . = . z ⎪ ⎪ ⎩ ⎭ 1 N 2 2 η ∈ (L (Ω)) , ∇ · η ∈ L (Ω), η · ν ∈ L (|)

(4.126) - = U in the strongly coercive case, that is under hypothesis .(H2 ), and .U -= Here, .U U1 in the weakly coercive case under the hypotheses .(H3 ) − (H4 ). In (4.125), by .β(x, ∇u), we mean, as usually, any measurable section .η of .β(x, ∇u) satisfying (4.126).. Then, the system − ∇ · β(x, ∇y) e f in Q, .

(4.127)

β(x, ∇y) · ν + yt e g on Σ, .

(4.128)

.yt

y(0) = y0 in Ω,

can be written as ( ) ( ) ( ) d y(t) y(t) f (t) +A e , a.e. t ∈ (0, T ), dt z(t) z(t) g(t) ( ) ( ) . y y0 (0) = . z z0

(4.129)

(4.130)

Lemma 4.10 The operator .A is maximal monotone in .X. Proof It is easily seen that .A is monotone, that is, ( .

( ) ( ) ( ) ( ) ( )) u u u u u−u ≥ 0, ∀ , A −A , ∈ D(A). z z−z z z z X

In fact, this follows by the Green formula { .

−

{ v∇ · ηdx = −

Ω

{ γ (v)η · νdσ +

|

η · ∇vdx, η(x) ∈ β(x, ∇u(x)), a.e. x ∈ Ω, Ω

for all .u, v ∈ U , in the strongly coercive case, or .u, v ∈ U1 in the weakly coercive case. On the other hand, the range .R(I + A) is all of .X, for all .λ > 0. Indeed, equation ( ) ( ) ( ) u w1 w1 .(I + A) = for all ∈X z w2 w2 reduces to Eq. (4.34) (or (4.84)), for which existence has been previously proved.

u

Then, by the standard existence theorem for the Cauchy problem associated with nonlinear maximal monotone operators (see, e.g., [13], p. 151), for

4.4 The Semigroup Approach

( .

115

) y0 z0

∈ D(A), f ∈ W 1,1 (0, T ; L2 (Ω)), g ∈ W 1,1 (0, T ; L2 (|)),

there is a unique function ( ) y . ∈ W 1,∞ (0, T ; L2 (Ω)) × W 1,∞ (0, T ; L2 (|)), z

which satisfies (4.130) a.e., and also in the following stronger sense, d+ . dt

(

) y(t) z(t)

(

(

+ A

) y(t) z(t)

( −

f (t) g(t)

))◦ = 0, ∀t ∈ [0, T ),

where, for each closed convex set C , .C ◦ stands for the minimal section of C . Moreover, if .f = 0, .g = 0, we have the exponential formula ( .

) y(t) z(t)

( = lim

n→∞

t I+ A n

)−n (

) y0 z0

in L2 (Ω) × L2 (|),

uniformly in t on compact intervals. We are led, therefore, to the following sharper versions of Theorems 4.4 and 4.8. Theorem 4.11 Let .y0 ∈ U .(respectively, .U1 in the weakly coercive case.) be such that .∇

· β(·, ∇y0 ) ∈ L2 (Ω), β(·, ∇y0 ) · ν ∈ L2 (|),

and let .f

∈ W 1,1 (0, T ; L2 (Ω)), g ∈ W 1,1 (0, T ; L2 (|)).

Then, there is a unique .y ∈ W 1,∞ (0, T ; L2 (Ω)) with .γ (y) ∈ W 1,∞ (0, T ; L2 (|)), which satisfies d+ y(t, x) − (∇ · β(x, ∇y(t, x)) − f (t, x))◦ = 0 in [0, T ) × Ω, dt+ . d y(t, x) + (β(x, ∇y(t, x)) · ν − g(t, x))◦ = 0 on [0, T ) × |, dt y(0, x) = y0 in Ω. Moreover, (4.62) holds.

The condition .∇ · β(·, ∇y0 ) ∈ L2 (Ω) means, of course, that there is .η0 measurable, such that .η0 (x) ∈ β(x, ∇y0 (x)), a.e. .x ∈ Ω, .∇ · η0 ∈ L2 (Ω). We also note that the operator .A is the subdifferential of the function .o : X → R ∪ {+∞},

116

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . .

⎧{ 2 ( ) ⎪ ⎪ ⎨ j (x, ∇u(x))dx, z = γ (u) ∈ L (|), u Ω .o = if u ∈ W 1,1 (Ω), j (·, ∇u) ∈ L1 (Ω) ⎪ z ⎪ ⎩ +∞, otherwise.

This is the energy functional associated with system (4.127)–(4.129). ( ) ( ) u u Indeed, for . ∈ D(A), ∈ D(o), we have z z ( ) ( ) { u u .o −o = (j (x, ∇u) − j (x, ∇u))dx z z Ω {

{ η · (∇u − ∇u)dx = −

≤ Ω

(u − u)∇ · ηdx Ω

{ +

(η · ν)(z − z)dσ, |

for .η ∈ (L1 (Ω))N , .η(x) ∈ β(x, ∇u(x)) a.e. .x ∈ Ω. Here we have used the Green formula { { { . v · ∇udx = − u∇ · vdx + γ (u)v · νdσ, Ω

Ω

|

which is valid for all .u ∈ W 1,1 (Ω) ∩ L2 (Ω) and .v ∈ (L1 (Ω))N such that .γ (u)(v

· ν) ∈ L1 (|) and u∇ · v ∈ L1 (Ω).

In virtue of Lemma 4.5, .v = η satisfies this condition. This implies that .A ⊂ ∂o, and since .A is maximal monotone, we infer that .A = ∂o, as claimed. Then, by Theorem 4.11 in [13], p. 158, it follows that for all ( .

) y0 z0

∈ D(o), f ∈ L2 (Q), g ∈ L2 (Σ),

( ) y problem (4.130) has a unique solution . ∈ W 1,2 (0, T ; X). Hence, under the z above assumptions, there is a unique solution .y ∈ W 1,2 (0, T ; L2 (Ω)) with .γ (y) ∈ W 1,2 (0, T ; L2 (|)) to (4.127)–(4.129).

By Theorem 4.5 in [13], p. 164, we also have in this case the following asymptotic result for the solution y to (4.127)–(4.129). Theorem 4.12 Let .y0 ∈ W 1,p (Ω), .2 ≤ p < ∞, and

4.4 The Semigroup Approach .f (t)

117

≡ f ∈ L2 (Ω), g(t) ≡ g ∈ L2 (|).

Assume that the set of equilibrium states for (4.1), .K

{ = y ∈ U ; ∇ · β(·, ∇y) ∈ L2 (Ω), β(·, ∇y) · ν ∈ L2 (|), ∇ · β(x, ∇y) = f in Ω, β(x, ∇y) · ν = g on |} ,

is nonempty. Then, for .t → ∞, we have .y(t)

→ y∞ weakly in L2 (Ω),

(4.131)

γ (y(t)) → γ (y∞ ) weakly in L (|), 2

where .y∞ ∈ K.

Taking into account that, as easily seen by (4.130), we have (

) y(t) z(t)

.o

)

( ≤o

y0 γ (y0 )

, for all t ≥ 0,

it follows by (4.131) and by the compactness of .W 1,p (Ω) in .Lq (Ω), that for .t → ∞, .y(t)

→ y∞ strongly in Lq (Ω),

where .1 ≤ q < NNp −p if .N > p , .q = p if .N ≤ p in the strongly coercive case, and .q = 1 in the weakly coercive case. In other words, the solution y is strongly convergent to an equilibrium solution .y∞ to system (4.127)–(4.129). One of the main advantages of the semigroup approach is its flexibility to incorporate other nonlinear terms in the basic equations (4.127)–(4.128). An example is the problem ∂y − ∇ · β(x, ∇y) + a1 (x, y) e f (t, x) in Q, . ∂t

(4.132)

∂y p−2 − ∇ · (|∇| y|N −1 ∇| y) + β(x, ∇y) · ν + a2 (x, y) e g(t, x) on Σ, . ∂t

(4.133)

.

y(0, x) = y0 in Ω,

(4.134)

where .β satisfies assumption (4.6), .ai : Ω × R → R, .i = 1, 2 are continuous and ≥ 2. Here, .∇| y is the Riemannian gradient of .y, that is, .∇| y = (∂τ1 y, . . . , ∂τN −1 y), where .∂τi y is the directional derivative of y along the tangential directions .τi at each point on .| and

.p

118

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . .

{ .

−

{ |∇| u|p−2 ∇| y · ∇| vdσ.

v∇ · (|∇| u|p−2 ∇| u)dσ = |

|

Problem (4.132)–(4.134) can be written as (4.130), where ( ) ( ) −∇ · β(x, ∇u) + a1 (x, u) u .A = , p−2 z −∇ · (|∇| u|N−1 ∇| u) + β(x, ∇u) · ν + a2 (x, u)

(4.135)

( ) u for all . ∈ D(A), where z ⎧( ) ⎫ ⎪ ⎪ u ⎪ ⎪ ⎪ ∈ X; u ∈ U, z = γ (u), ∃η(x) ∈ β(x, ∇u(x)) a.e. x ∈ Ω,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z ⎨ ⎬ . D(A) . = such that ⎪ ⎪ ⎪ ⎪ 1 N 2 p ⎪ ⎪ η ∈ (L (Ω)) , ∇ · η + a1 (·, u) ∈ L (Ω), |∇| u|N−1 ∈ L (|), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p−2 ⎩ ⎭ ∇ u) + η · ν + a (·, u) ∈ L2 (|) −∇ · (|∇ u| |

N−1

|

2

(4.136)

If .y → ai (x, y), .i = 1, 2, are monotone (or more generally quasi-monotone, that is, .λy + ai (x, y) are monotone for some .λ > 0) and .|ai (x, r)| ≤ C |r|q−1 , for .r ∈ R, where q is as before, then, arguing as above, it follows that the operator .A is maximal monotone in X (or quasi m-accretive if .ai are quasi monotone), and in fact it is a subdifferential operator. Then, we get for problem (4.132)–(4.134) the following existence result: Theorem 4.13 Let .y0

∈ U, such that ∇| y0 ∈ (Lp (|))N−1 ,

and let .f

∈ L2 (0, T ; L2 (Ω)), g ∈ L2 (0, T ; L2 (|)).

Then, there is a unique solution .y ∈ W 1,2 (0, T ; L2 (Ω)) to (4.132)–(4.134), such that .γ (y)

∈ W 1,2 (0, T ; L2 (|)) and ∇| y ∈ (Lp (Σ))N−1 .

We are going to give some particular examples.

4.4 The Semigroup Approach

119

4.4.1 The Obstacle Problem Consider the following free boundary problem associated with the Wentzell boundary condition, namely, yt − ∇ · β(x, ∇y) yt − ∇ · β(x, ∇y) . yt + β(x, ∇y) · ν y(0, x)

≥ e e =

f, f, g, y0 ,

y ≥ 0, in Q, in {(t, x) ∈ Q; y(t, x) > 0}, on Σ, in Ω.

(4.137)

This problem can be written as

.

d dt

(

) y(t) z(t)

( +A

) y(t) z(t)

(

) ( ) y(t) f (t) +B e , a.e. t ∈ (0, T ), z(t) g(t) ( ) ( ) y0 y (0) = , z γ (y0 )

(4.138)

where ( ) ( ) ( ) y a(y) y .B = , ∀ ∈ X = L2 (Ω) × L2 (|) z 0 z

and .a : R → R is the multivalued function .a(s) = 0 for .s > 0, .a(0) = (−∞, 0], = ∅ for .s < 0. ( ( ) ( )) ( ) y y y Taking into account that . A ≥ 0 for all . ,B ∈ D(A) ∩ D(B), z z z X it follows that .A + B is maximal monotone and therefore .A + B = ∂o1 , where ⎧ ( ) ( ) ⎪ ⎨ o y , for y ∈ L2 (Ω), y ≥ 0 a.e. in Ω, y .o1 = z ⎪ z ⎩ +∞, otherwise.

.a(s)

Then, applying the general existence theory, we infer the following result. Theorem 4.14 Let .y0

∈ W 1,1 (Ω), y0 ≥ 0 a.e. in Ω, j (·, ∇y0 ) ∈ L1 (Ω).

Then, problem (4.138) has a unique solution .y ∈ W 1,2 (0, T ; L2 (Ω)).

More generally, one might take instead of a a general maximal monotone graph in .R × R. This case is studied in [42].

120

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . .

4.4.2 The Total Variation Wentzell Flow Let us consider now the singular case .j (x, r) ≡ ρ |r|, .r ∈ RN , or, equivalently, ⎧ ⎨ρ r , if r /= 0, |r|N .β(x, r) = ρ sign(r) = ⎩ {r; |r| ≤ ρ}, if r = 0. N

Then, problem (4.127)–(4.129) reduces to − ρ∇ · sign(∇y) e f, in Q, .

(4.139)

ρ sign(∇y) · ν + yt e g, on Σ, .

(4.140)

y(0) = y0 , in Ω.

(4.141)

.yt

As mentioned earlier, this problem is not covered by the previous weakly coercive case, and as a matter of fact, it cannot be treated in the .W 1,1 (Ω) space, but in the space .BV (Ω) of functions with bounded variation on .Ω, that is, ⎧ ⎨ .BV (Ω)=

⎩

{{ u ∈ L (Ω); ||Du|| = 1

||ϕ||

sup ≤1 RN )

L∞ (Ω;

u∇ · ϕdx; ϕ ∈ Ω

C0∞ (Ω; RN )

⎫ }⎬ ⎭

< ∞.

We recall (see, e.g., [4]) that, for each .u ∈ BV (Ω), there is the trace .γ (u) ∈ L1 (|; dHN −1 ), where .dHN −1 is the Hausdorff measure on .|, defined by {

{ u∇ · ψdx = −

.

Ω

{ ψd(∇u) +

Ω

uψ · νdHN −1 , ∀ψ ∈ C 1 (RN , RN ).

|

Here, .∇u (the gradient of u in the sense of distributions.) is a Radon measure on .Ω. Let us define the energy functional .o : L2 (Ω) × L2 (|) → (−∞, +∞], ( ) { u ρ ||Du|| , if u ∈ BV (Ω) ∩ L2 (Ω), z = γ (u) ∈ L2 (|), .o = z +∞, otherwise.

It is easily seen that .o is convex and l.s.c on .X = L2 (Ω) × L2 (|). Let .∂o : X → X be its subdifferential. Then, for each ( .

the problem

) y0 z0

∈ D(o), f ∈ L2 (0, T ; L2 (Ω)), g ∈ L2 (0, T ; L2 (|)),

4.4 The Semigroup Approach

d . dt

121

( ) ( ) ( ) y ξ f (t) + (t) = (t), a.e. t ∈ (0, T ), . z η g (

) ξ(t) η(t)

( ∈ ∂o

(4.142)

) y(t) z(t)

, a.e. t ∈ (0, T ), .

( ) ( ) y y0 (0) = , in Ω, z z0

(4.143)

(4.144)

( ) ( ) y ξ 1,2 has a unique solution . ∈ W (0, T ; X), . ∈ L2 (0, T ; X). z η Taking into account (see, e.g., [2]) that for all .u, v ∈ BV (Ω) ∩ L2 (Ω) and .ζ ∈ ∞ (L (Ω))N , { . ||Du||

≤ ||Dv|| −

{

(ζ · ν)(u − v)dHN −1 ,

(u − v)∇ · ζ dx − Ω

|

where .||ζ ||(L∞ (Ω))N ≤ 1, .∇ · ζ ∈ L2 (Ω), we may interpret .t → y(t) as a solution to system (4.139)–(4.141). This is the total variation Wentzell flow. The operator .∂o (and implicitly system (4.142)–(4.144)) is, however, hardly to be described in explicit terms, so that a better insight into problem (4.139)–(4.141) can be gained by taking into account that the solution to (4.142)–(4.144) is the limit of the finite difference scheme provided by the iteration process ( .

) yi+1 zi+1

( + h∂o

) yi+1 zi+1

( e

) yi zi

, i = 0, 1,

or, equivalently, .yi+1

{ } { { 1 1 |u − yi |2 dx + |γ (u) − γ (yi )|2 dσ . = arg min ρh ||Du|| + u 2 Ω 2 |

Remark 4.15 The previous results naturally extend to the case of nonlinear functions .β : Ω × R N → R N , which are not of gradient type with respect to .r ∈ R N . Namely, it suffices to assume that .β ≡ β(x, r) is continuous on .Ω × R N , monotone with respect to .r, that is .(β(x, r)

− β(x, r)) · (r − r) ≥ 0, ∀r, r ∈ RN ,

(4.145)

and that it satisfies .β(x, r) . |β(x, r)|N

p

· r ≥ α1 |r|N , ∀r ∈ RN , p−1

≤ α2 |r|N

+ α3 , ∀r ∈ RN ,

(4.146) (4.147)

122

4 Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary. . .

where .2 ≤ p < ∞, .α1 , α2 > 0, .α3 ∈ R. Let us consider .U

=

{( ) } y ∈ U × L2 (|); γ (u) = z z

endowed with the natural norm and denote by .U' the dual space, in the duality induced by - : U → U' , defined by the pivot space .X. Then, the operator .A / .

A

( ) ( ) ( )\ { ϕ1 u ϕ1 β(x, ∇u) · ∇ϕ1 dx, ∀ = , ∈ U, z ϕ2 U ' , U ϕ2 Ω

is, by the Browder theory (see, e.g., [13], p. 81), maximal monotone in .U × U' , and so its restriction ( ) ( ) u - u ∩X .AX =A z z to X is maximal monotone in .X×X. Then, Theorem 4.11 remains true in the present situation.

Chapter 5

A Nonlinear Control Problem in Image Denoising

In this chapter, we present an optimal control problem related to image denoising and restoration techniques, formulated with the purpose of denoising an initial blurred image. Image denoising consists in the removal of noise from a noisy image, that is, the restoration of the true image. From a mathematical perspective, image denoising is an inverse problem and its solution is not unique. In the last decades, models based on partial differential equations (PDEs) have been widely used in digital signal and image processing, because they are able to realize an equilibrium between the noise reduction, undesired effect removing, and image detail preservation. Since noise, edge, and texture are high-frequency components, it is difficult to distinguish them in the process of denoising, and the denoised images could inevitably lose some details. Feature-preserving image enhancement still represents a challenging open task for the image processing researchers. The classic image filters are seriously affected by unintended effects, such as the blurring, and cannot preserve essential image details, especially the boundaries. The secondorder, as well as the fourth-order nonlinear PDE restoration approaches overcome these problems, preserving the features such as edges, corners, and sharp structures. These techniques encourage the diffusion only within the regions, by performing the denoising along but not across the edges. These aspects have been reported in the literature (see, e.g., [32, 33, 50]). Nonlinear PDE restoration approaches, expressed as second-order anisotropic diffusion schemes or variational models, have been developed since P. Perona and J. Malik proposed their PDE denoising model in 1987 (see [67]). Rigorous mathematical works on this topic are, e.g., [16, 18– 20, 23]. Here we give an example of the use of Legendre–Fenchel relations in a nonlinear PDE restoration approach, expressed by a second-order anisotropic diffusion. This goes back to solve a nonconvex optimal control problem .(P ) with the state and controller connected on a manifold described by a nonlinear elliptic equation. The diffusion term .β, which now is a vector, is provided by a potential .j. The problem is solved for a weakly coercive potential (as introduced in Chap. 2), while the case © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Marinoschi, Dual Variational Approach to Nonlinear Diffusion Equations, PNLDE Subseries in Control 102, https://doi.org/10.1007/978-3-031-24583-1_5

123

124

5 A Nonlinear Control Problem in Image Denoising

with a potential with a polynomial growth is briefly discussed at the end. The problem is solved in a series of steps, beginning with the proof of the existence -), equivalent of an optimal control in .(P ) in Theorem 5.4. Then, a new problem .(P to .(P ), is introduced in order to create the advantage of working with a linear PDE instead of the nonlinear one. This is achieved by defining a new state connected by an appropriate equation with the state of the primal problem. In the case of an unfriendly cost functional (e.g., not differentiable), a technique often used in control theory is to study an approximating control problem involving a regular cost -ε ) for functional. This suggests introducing an approximating control problem .(P a cost functional written by means of Legendre–Fenchel relations. This problem -) in the sense that the is appropriately set such that to approximate problem .(P -). sequence of approximating optimal pairs in .(Pε ) tends to an optimal pair of .(P This and a desired strong convergence for the state proved in a certain space are provided in Theorem 5.7. Thus, the determination of the approximating optimality conditions in Proposition 5.8 concludes the discussion due to the proof of their convergence to an optimal pair for .(P ). Moreover, the approximating optimality conditions can be used for writing algorithms if numerical simulations are intended to be achieved.

5.1 Problem Presentation and Preliminaries We assume that .o is an open bounded subset of .RN , having the boundary .| = ∂o of class .C 1 , .ν is the outer normal to the boundary, and denote by .|·|N the Euclidean norm in the space .RN and by .| · | the norm in .R. Let .q ≥ 2, and let .λ be a positive real value. As explained before, in image restoration, the purpose is to reconstruct an image y from an observed blurred image .y obs lying in the domain .o. For the gray-level images, .o is a two-dimensional domain, while for color images, its dimension is three. In the first case, the real value y represents the intensity of the gray color. The denoising process is performed here by the action of a control u by the minimization problem Minimize

.

(u,y)∈U

{ ) } { ( |u(x)|q λ J (u, y) = + (y(x) − y obs (x))2 dx , q 2 o

(P )

where { U = (u, y); u ∈ Lq (o), y ∈ L2 (o) ∩ W 1,1 (o), . (u, y) satisfies (5.2), (5.3) below}

(5.1)

−∇ · β(∇y) = u in o, β(∇y) · ν = 0 on |,

(5.2)

.

5.1 Problem Presentation and Preliminaries

125

{ u(x)dx = 0.

.

(5.3)

o

We specify that (5.2) is written for the moment in the sense of distributions. The unusual restriction (5.3) for the control u is included to ensure the existence of a solution to (5.2), which otherwise may not exist for each .u. Its role will be clearly seen later. We also note that the admissible set U is not empty because, e.g., if we take y a constant and .u = 0, the functional .J (u, y) is finite and .(u, y) ∈ U. We shall study this problem, under the following hypotheses: (i) .j : RN → R is a convex differentiable function such that β(r) = ∇j (r) for all r ∈ RN .

.

(5.4)

(ii) j and its conjugate .j ∗ are weakly coercive .

j (r) j ∗ (r) → ∞, → ∞, as |r| → ∞. |r| |r|

(5.5)

(iii) Symmetry at infinity: There exist .γ1 > 0, .γ2 ≥ 0 such that j (r) ≤ γ1 j (−r) + γ2 , γ1 > 0, γ2 ≥ 0.

(5.6)

j (r) ≥ 0 for all r ∈ R, j (0) = 0.

(5.7)

.

(iv) Positiveness .

(v) .j ∗ is differentiable. We recall that .j ∗ : RN → R is the conjugate of j defined by j ∗ (ω) = sup (r · ω − j (r)) N r∈R

.

(5.8)

and that it is convex and continuous. The hypothesis .j (0) = 0 may be assumed without loss of generality by redefining .j (r) as .j (r) − j (0), and this implies that j ∗ (ω) ≥ 0 for all ω ∈ R.

.

By (5.4), it follows that .β : RN → RN is continuous and monotone, that is, (β(r) − β(r)) · (r − r) ≥ 0, for any r, r ∈ RN .

.

(5.9)

126

5 A Nonlinear Control Problem in Image Denoising

We recall that the functions j and .j ∗ satisfy the Legendre–Fenchel relations .

j (r) + j ∗ (ω) − r · ω ≥ 0 for all r, ω ∈ RN , .

(5.10)

j (r) + j ∗ (ω) − r · ω = 0 if and only if ω = β(r).

(5.11)

Moreover, we recall that (5.5) is equivalent to the following relations (see Proposition 7.35 in the Appendix), .

{ } sup |r|N ; r = β −1 (ω), |ω|N ≤ M ≤ WM , .

(5.12)

{ } sup |ω|N ; ω = β(r), |r|N ≤ M ≤ YM ,

(5.13)

respectively, where .M, .WM , .YM are positive constants. Below there are two standard examples of functions j having the previous properties: α+1 j (r) = |r|N , β(r) = |r|αN sign(r), α > 0,

(5.14)

.

and j (r) = log(|r|N + 1) |r|N , β(r) = log(|r|N + 1)sign(r) +

.

r |r|N + 1

,

(5.15)

where sign is the multivalued signum function { sign(r) =

.

, if r = / 0 N {r ∈ R ; |r|N ≤ 1}, if r = 0. r |r|N

Functions j and .j ∗ with the property (5.5) can be appropriately used in image denoising processes because this property is essential for edges preserving in the denoising process. Other types of potentials j leading to more regular solutions have the effect of shadows removal. We recall that by the Rellich–Kondrachov embedding inequalities (see Theorem 7.6 in the Appendix), we have '

W 1,1 (o) ⊂ Lq (o), with compact injections,

.

where q ' ∈ [1, p∗ ), p∗ =

(5.16)

N if N > 1 and p∗ = ∞ if N = 1. N −1

Actually, the exponent q in .(P ) is chosen to be exactly the conjugate of .q ' previously specified in (5.16)

5.2 Existence for the Minimization Problem

⎧( ) 1 −1 ⎨ 1− ' > N, if N ≥ 2, .q = q ⎩ 2, if N = 1.

127

(5.17)

We remark that if .N = 1, .q ' ≥ 1, but in this case a natural choice is .q ' = q = 2. The proof of the well-posedness of the state system (5.2) is not necessary for obtaining the further results. Essential is the proof of the existence of an optimal -), control in .(P ) done in Theorem 5.4. Then, a modified control problem .(P equivalent to .(P ), is introduced in order to allow the calculation of optimality conditions. However, the optimality conditions are computed first for an approxi-ε ) in Proposition 5.8, after the proof of the convergence of the mating problem .(P -ε ) to .(P ) in Theorem 5.7. An algorithm constructed for approximating problem .(P solving numerically .(Pε ) is briefly described for the interested readers.

5.2 Existence for the Minimization Problem We recall the following result proved relying on relations (5.6), (5.10) in Chap. 2, Lemma 2.6, the first part. Lemma 5.1 If .v, .w ∈ (L1 (o))N , such that .j (v) ∈ L1 (o), .j ∗ (w) ∈ L1 (o), then v · w ∈ L1 (o).

.

Definition 5.2 Let .u ∈ Lq (o). We call a solution to (5.2) a function .y ∈ L2 (o), such that .

∇y ∈ (L1 (o))N , β(∇y) ∈ (L1 (o))N , j (∇y) ∈ L1 (o), j ∗ (β(∇y)) ∈ L1 (o),

which satisfies the weak form { { . β(∇y) · ∇φdx = uφdx, for all φ ∈ C 1 (o). o

(5.18)

(5.19)

o

The next result proves that relation (5.19) can be written for a less regular test function, and so the definition of a solution to (5.2) can be extended for such functions. Lemma 5.3 Let y be a solution to (5.2). Then, {

{ β(∇y) · ∇φdx =

.

o

uφdx, o

for all φ ∈ W 1,1 (o) ∩ L2 (o) with j (∇φ) ∈ L1 (o).

(5.20)

128

5 A Nonlinear Control Problem in Image Denoising

Proof We take .φ ∈ W 1,1 (o) ∩ L2 (o) with .j (∇φ) ∈ L1 (o) and perform an approximation using a mollifier. Let .ρ be a function with the following properties: ρ ∈ C ∞ (RN ), ρ(x) ≥ 0, ρ(x) = ρ(−x),

{

.

RN

ρ(x)dx = 1.

A mollifier .ρε is defined as ρε (x) =

.

1 (x ) , ε > 0. ρ εN ε

By approximating .φ by its convolution with a mollifier, namely by { φε (x) = (φ ∗ ρε )(x) =

.

RN

ρε (x − ξ )φ(ξ )dξ,

we obtain .φε ∈ C 1 (o), and φε → φ strongly in W 1,1 (o) ∩ L2 (o), as ε → 0.

.

(5.21)

Consequently, φε → φ a.e. in o, as ε → 0.

.

Moreover, by definition, it follows that .|φε (x)| + |∇φε (x)| < ∞ for all .x ∈ o. By Egorov theorem (see Theorem 7.4 in the Appendix), it follows that for any .δ > 0, there exists .oδ ⊂ o, with meas(.o\oδ ) ≤ δ, such that φε → φ, ∇φε → ∇φ uniformly in oδ , as ε → 0.

.

Recall that .β is continuous. It follows that β(∇y) · ∇φε → β(∇y) · ∇φ uniformly in oδ , and also in L1 (oδ ).

.

On the other hand, the convexity of j implies ∇φε · β(∇y) ≤ j (∇φε ) + j ∗ (β(∇y)) ≤ j (∇(φ ∗ ρε )) + j ∗ (β(∇y))

.

≤ j (∇φ) ∗ ρε + j ∗ (β(∇y)), and this yields { .

∇φε · β(∇y)dx

lim sup ε→0

o\oδ

5.2 Existence for the Minimization Problem

{

( ) j (∇φ) ∗ ρε + j ∗ (β(∇y)) dx

≤ lim sup {

129

(5.22)

o\oδ

ε→0

( ) j (∇φ) + j ∗ (β(∇y)) dx.

= o\oδ

Since y is a solution to (5.2), using (5.19) with the regularization .φε , we have {

{

{

uφε dx =

.

o

β(∇y) · ∇φε dx + oδ

{

{

( ) j (∇φ) ∗ ρε + j ∗ (β(∇y)) dx.

β(∇y) · ∇φε dx +

≤ oδ

β(∇y) · ∇φε dx o\oδ

o\oδ

Therefore, by (5.21) and (5.22) letting .ε → 0, we obtain that {

{

{

uφdx ≤

.

o

( ) j (∇φ) + j ∗ (β(∇y)) dx.

β(∇y) · ∇φdx + oδ

(5.23)

o\oδ

But .∇φ, .∇y ∈ (L1 (o))N , .β(∇y) ∈ (L1 (o))N , .j (∇φ), .j ∗ (β(∇y)) ∈ L1 (o), and thus, Lemma 5.1 entails that .β(∇y) · ∇φ ∈ L1 (o), allowing us to deduce that { .

lim

δ→0 o\oδ

{ β(∇y) · ∇φdx = 0, lim

δ→0 o\oδ

( ) j (∇φ) + j ∗ (β(∇y)) dx = 0.

By letting .δ go to 0 in (5.23), we deduce that {

{ uφdx ≤

.

o

β(∇y) · ∇φdx. o

Then, by changing .φ by .−φ, the reverse inequality is obtained, and so (5.20) follows, as claimed. u n Now, we are able to show the existence of a solution to problem .(P ). Theorem 5.4 Let .y obs ∈ L2 (o), q = 2 if N = 1 and q > N if N ≥ 2.

.

(5.24)

Then, problem .(P ) has at least a solution .(u∗ , y ∗ ). Proof We see that .J (u, y) ≥ 0 implying that the infimum d exists and it is positive. We take a minimizing sequence, .(un , yn )n ∈ U, that is, .un ∈ Lq (o), .yn ∈ L2 (o) ∩ W 1,1 (o), such that .

−∇ · β(∇yn ) = un , in o, β(∇yn ) · ν = 0, on |

(5.25)

130

5 A Nonlinear Control Problem in Image Denoising

{ un (x)dx = 0

.

(5.26)

o

and { ( d ≤ J (un , yn ) =

.

o

) |un (x)|q 1 λ + (yn (x) − y obs (x))2 dx ≤ d + . q 2 n

(5.27)

It follows that .(un )n≥1 and .(yn )n≥1 lie in bounded subsets of .Lq (o) and .L2 (o), respectively. Since .yn is a solution to (5.25), it follows by Definition 5.2 that yn ∈ L2 (o), ∇yn ∈ (L1 (o))N , β(∇yn ) ∈ (L1 (o))N ,

.

j (∇yn ) ∈ L1 (o), j ∗ (β(∇yn )) ∈ L1 (o) and {

{ β(∇yn ) · ∇ψdx =

.

o

un ψdx, for all ψ ∈ C 1 (o).

(5.28)

o

By Lemma 5.1, it follows that .β(∇yn ) · ∇yn ∈ L1 (o), and by Lemma 5.3, we can set .ψ = yn in (5.28), getting {

{ β(∇yn ) · ∇yn dx =

.

o

o

un yn dx ≤ ||un ||L2 (o) ||yn ||L2 (o) ≤ C,

(5.29)

independently of .n. We stress that since .q ≥ 2, the last integral in the previous relation makes sense. Still by C we further denote several constants independent of .n. Also, by relation (5.11), we can write { { ∗ . (j (∇yn ) + j (β(∇yn ))dx = β(∇(yn )) · ∇yn dx ≤ C (5.30) o

o

and prove that the sequences .(∇yn )n and .(β(∇yn ))n are weakly compact in (L1 (o))N , using the Dunford–Pettis theorem (see Theorem 7.3). Let us apply it first for .(∇yn ). We have to show that this{ sequence is equicontinuous, meaning that for any .ε > 0, there exists .δ such that . A |∇yn | dx ≤ ε, if meas.(A) < δ with A measurable in .o. Indeed, let .ε be positive, and take .Mε > 2C ε with C the constant in (5.30). By the property (5.5), it follows that

.

.

Let .δ
Mε as |∇yn |N > RMε . |∇yn |N

, and calculate using (5.31)

(5.31)

5.2 Existence for the Minimization Problem

131

{

{ |∇yn | dx =

.

A

{ ≤ ≤

{ {x∈A;|∇yn |N >RMε }

{x∈A;|∇yn |N >RMε }

1 Mε

{

A

|∇yn |N dx +

{x∈A;|∇yn |N ≤RMε }

|∇yn |N dx

j (∇yn ) dx + RMε meas(A) Mε

j (∇yn )dx + RMε δ ≤

ε C ε + RMε = ε, Mε 2RMε

{

because . A j (∇yn )dx ≤ C by (5.30). This implies by the aforementioned theorem that .(∇yn )n is weakly compact in .(L1 (o))N . Similarly, the weakly compactness property is shown for .(β(∇yn ))n . Taking into account all previous results, we can select a subsequence of .(un , yn )n , denoted still by the subscript .n, and get as .n → ∞ that un yn . ∇yn β(∇yn ) −∇ · β(∇yn )

→ u∗ → y∗ → ∇y ∗ →ζ →ξ

weakly in Lq (o), weakly in L2 (o), weakly in (L1 (o))N , weakly in (L1 (o))N , weakly in Lq (o).

Since .yn ∈ W 1,1 (o), we get by the compactness (5.16) that '

yn → y ∗ strongly in Lq (o),

.

and so un yn → u∗ y ∗ weakly in L1 (o).

.

Also, we see that {

{

u∗ dx = 0.

un dx →

.

o

(5.32)

o

By passing to the limit in (5.30) as .n → ∞, we obtain on the basis of the convexity, weakly lower semicontinuity and positiveness of j and .j ∗ that j (∇y) ∈ L1 (o), j ∗ (ζ ) ∈ L1 (o).

.

Next, by passing to the limit in (5.28), we get that {

{ ζ · ∇φdx =

.

o

o

u∗ φdx, for all φ ∈ C 1 (o).

132

5 A Nonlinear Control Problem in Image Denoising

According to Lemma 5.3, this is true, in particular, also for .φ = y ∗ . This means that ∗ .y is a solution in the sense of distributions to the problem .

− ∇ · ζ = u∗ in o, ζ · ν = 0 on |,

and these equations are satisfied also a.e. on .o, because .ζ ∈ (L1 (o))N . Now, we go back to (5.29) and observe that {

{ .

u∗ y ∗ dx =

β(∇yn ) · ∇yn dx =

lim sup n→∞

o

{

o

ζ · ∇y ∗ dx.

o

It follows (see [13], p. 38) that .ζ = β(∇y ∗ ) a.e. on .o, and so .ξ = ∇ · β(∇y ∗ ) ∈ Lq (o). These show that .(u∗ , y ∗ ) is a solution to (5.2)–(5.3) and together with (5.32) infer that .(u∗ , y ∗ ) ∈ U. Now, we can pass to the limit in (5.27) and obtain that ∗ ∗ .J (u , y ) = d, which proves that .(P ) has at least a solution. u n -), and the motivation of this At this point, we introduce a modified problem .(P artifice will be seen a little later. As a matter of fact, a new state is introduced with an appropriate connection with the state y, and the expression of functional J remains unchanged, but the admissible set is modified by including the relation between the two states. For approaching this problem, we shall work with the space { V = {z ∈ H (o);

zdx = 0}.

1

.

(5.33)

o

We recall that with the norm ({ .

)2 )1/2

({

|||z||| =

|∇z| dx + 2

o

zdx

(5.34)

o

is equivalent to the norm .||z||H 1 (o) (see [28], p. 286), so that .|||z||| = ||∇z||L2 (o) . We are ready to write the new minimization problem ({ (

) ) |u(x)|q λ + (y(x) − y obs (x))2 dx , q 2

-) (P

- = {(u, y, z); (u, y) ∈ U, z ∈ V , β(∇y) = ∇z a.e. on o}. U

-) (U

.

min

(u,y,z)∈U

o

where .

Taking into account that .(u, y) ∈ U verifies (5.2), it follows that .z verifies the boundary value problem .

− ∆z = u in o,

(5.35)

5.3 Approximating Problem

133

∇z · ν = 0 on |, { where . o udx = 0. It is known that (5.35) has a solution that moreover is unique { due to the state restriction . o zdx = 0. It satisfies the estimate C ||z||H 1 (o) ≤ ||z||V ≤ ||u||L2 (o) ,

.

(5.36)

with C a constant. The restriction (5.3) allows the existence of a solution to (5.35), -) has the advantage that it and the justification of its choice is clear. Problem .(P involves a linear PDE system (5.35) instead of the nonlinear one (5.2). -) Corollary 5.5 Under the hypotheses of Theorem 5.4, the minimization problem .(P has at least a solution .(u∗ , y ∗ , z∗ ). Proof The proof is based on all arguments developed in Theorem 5.4. The minimizing sequence should satisfy in addition .β(∇yn ) = ∇zn , where .zn turns out to be the unique solution to .

− ∆zn = un in o,

(5.37)

∇zn · ν = 0 on |, satisfying .c ||zn ||H 1 (o) ≤ ||un ||L2 (o) . In Theorem 5.4, in particular, .un → weakly in .L2 (o). Then, we get that .zn → z∗ weakly in .H 1 (o), {and by weak form of (5.37), we obtain that .(u∗ , z∗ ) satisfy (5.35). Moreover, . o zn dx { ∗ ∗ ∗ ∗ o z dx; hence, (.u , y , z ) ∈ U .

u∗ the → u n

5.3 Approximating Problem The calculus of the first-order conditions of optimality directly in problem .(P ), and consequently, a gradient-type algorithm for the numerical computation of the optimality conditions, would formally involve the directional derivative of .β. A possibility to avoid this inconvenient is to work instead with the gradients of j and .j ∗ . This suggests introducing an approximating control problem .(Pε ) involving these functions connected by the duality Legendre–Fenchel relations, and this will entail a simpler computation of the optimality conditions. Thus, we introduce Minimize Jε (u, y, z),

.

(u,y,z)∈U

-ε ) (P

where { ( Jε (u, y, z) =

.

o

) |u|q λ obs 2 + (y − y ) dx q 2

(5.38)

134

5 A Nonlinear Control Problem in Image Denoising

+

1 ε

{

(j (∇y) + j ∗ (∇z) − ∇y · ∇z)dx o

and .

{ -ε = (u, y, z); u ∈ Lq (o), y ∈ W 1,1 (o) ∩ L2 (o), z ∈ H 1 (o), U

(5.39)

j (∇y) ∈ L1 (o), j ∗ (∇z) ∈ L1 (o), ∇y · ∇z ∈ L1 (o), { { } zdx = 0, udx = 0 . − ∆z = u on o, ∇z · ν = 0 on |, o

o

-ε was used in order to The admissible set does not depend on .ε, but the notation .U -ε ) approximates .(P ) indicate that it corresponds to problem .(Pε ). We show that .(P in some sense. Another advantage of this problem is that it involves a linear state -, we have by system instead of the nonlinear one in problem .(P ). If .(u, y, z) ∈ U the Green formula { { { . ∇y · ∇zdx = − y∆zdx = uydx. (5.40) o

o

o

Thus, .Jε can be equivalently written { (

) |u|q λ obs 2 .Jε (u, y, z) = + (y − y ) dx q 2 o { 1 + (j (∇y) + j ∗ (∇z) − uy)dx. ε o

(5.41)

In the next proofs, both expressions (5.38) and (5.41) will be many times exploited. -ε ), the state z and the control u are connected by We see that in problem .(P relation (5.35) that has a unique solution as specified before. Theorem 5.6 Let .y obs ∈ L2 (o), and let q and .q ' be as in (5.16) and (5.24). Then, -ε ) has at least a solution .(u∗ε , yε∗ , zε∗ ). problem .(P Proof The functional .Jε (u, y, z) is nonnegative because the second integral in (5.38) is nonnegative by the second Legendre–Fenchel relation. Then, taking a minimizing sequence .{un , yn , zn }, we have dε ≤ Jε (un , yn , zn ) ≤ dε +

.

where

1 , for n ≥ 1, n

(5.42)

5.3 Approximating Problem

135

{ un ∈ Lq (o), yn ∈ W 1,1 (o) ∩ L2 (o), zn ∈ H 1 (o), .

{o

j (∇yn ) ∈ L1 (o), j (∇zn ) ∈ L1 (o), ∇yn · ∇zn ∈ L1 (o),

zn dx = 0, un dx = 0,

o

and .zn is the solution to .

−∆zn = un in o, ∇zn · ν = 0 on |.

(5.43)

Since the second term in (5.38) is nonnegative, we have { ( .

o

) |un |q λ obs 2 + (yn − y ) dx ≤ Jε (un , yn , zn ) ≤ dε + 1. q 2

Thus, .(un )n and .(yn )n are bounded in .Lq (o) and .L2 (o), respectively, independently of .n, with .q ≥ 2. Also, since the first integral in .(Pε ) is nonnegative, we can write { { 1 1 . un yn dx + dε + 1 (5.44) (j (∇yn ) + j ∗ (∇zn ))dx ≤ ε o ε o and so { .

o

(j (∇yn ) + j ∗ (∇zn ))dx ≤ ||un ||L2 (o) ||yn ||L2 (o) + Cε

with .Cε independent of .n. In the same way as in Theorem 5.4, we prove that the sequences .(∇yn )n and .(∇zn )n are weakly compact in .(L1 (o))N , and so, we can select a subsequence .n → ∞, such that un yn ∇y n . y u n n { un dx o

→ u∗ε → yε∗ → ∇yε∗ ∗ ∗ →u {ε yε

weakly in Lq (o), ' weakly in L2 (o) ∩ W 1,1 (o) and strongly in Lq (o), weakly in (L1 (o))N , weakly in L1 (o),

u∗ε dx = 0.

→ o

'

'

The strong convergence in .Lq (o) is due to the compactness .W 1,1 (o) in .Lq (o). Moreover, by (5.43) and (5.36), it follows that .(zn )n is bounded in V and so zn →

.

By (5.43), we still have

zε∗

{ weakly in V and o

zε∗ dx = 0.

136

5 A Nonlinear Control Problem in Image Denoising

{

{ zn · ∇φdx =

.

o

un φdx for all φ ∈ C 1 (o),

(5.45)

o

and we get at limit { .

o

zε∗

{ · ∇φdx = o

u∗ε φdx for all φ ∈ C 1 (o).

This is the weak form of the boundary value problem .

−∆zε∗ = u∗ε in o, ∇zε∗ · ν = 0 on |.

(5.46)

By passing to the limit, as .n → ∞, in (5.44) and (5.42), we obtain that .j (∇yε∗ ) and ∗ 1 ∗ ∗ ∗ .j (zε ) are in .L (o) and that .Jε (uε , yε , zε ) = dε , respectively.. In conclusion, we ∗ ∗ ∗ -ε ). have proved that .(uε , yε , zε ) ∈ U and that it is a solution for .(P u n Theorem 5.7 Let q and .q ' be as in (5.16), (5.24), and let .(u∗ε , yε∗ , zε∗ ) be an optimal -ε ). Then, pair for problem .(P u∗ε → u weakly in Lq (o), ' ∗ .y → y weakly in W 1,1 (o) and strongly in Lq (o), ε z weakly in V . zε∗ → -). Moreover, .(u, y ,z) is optimal in .(P -ε ), using either the expression (5.41) or (5.38), Proof If .(u∗ε , yε∗ , zε∗ ) is optimal in .(P we can write ) { ( || ∗ ||q { ( ) uε λ ∗ 1 obs 2 + (yε − y ) dx + j (∇yε∗ + j ∗ (∇zε∗ ) − u∗ε yε∗ )dx . q 2 ε o o ) { ( q |u| λ (5.47) + (y − y obs )2 dx ≤ q 2 o { 1 + (j (∇y) + j ∗ (∇z) − ∇y · ∇z)dx, ε o -ε . for all .(u, y, z) ∈ U -), that is .u∗ ∈ Lq (o), .y ∗ ∈ L2 (o) ∩ Let .(u∗ , y ∗ , z∗ ) be an optimal pair in .(P W 1,1 (o), and .(u∗ , y ∗ ) satisfies (5.2). Also, .∇z∗ = β(∇y ∗ ) a.e. on .o, and so .z∗ ∈ V and .z∗ is the unique solution to (5.35) corresponding to .u∗ . Moreover, according to Definition 5.2, we have β(∇y ∗ ) ∈ (L1 (o))N , j (∇y ∗ ), j ∗ (β(∇y ∗ )) ∈ L1 (o).

.

5.3 Approximating Problem

137

-ε . Let us set in (5.47) .u = u∗ , .y = y ∗ , and .z = z∗ . This triplet belongs to .U It follows that the last integral on the right-hand side in (5.47) vanishes, and the remainder is bounded by a constant independent of .ε. This allows us to use similar arguments as in Theorem 5.6 to select a subsequence .(ε → 0) such that u∗ε yε∗ zε∗ . ∗ ∗ { uε yε u∗ε dx

→u →y →z →uy { →

weakly in Lq (o), ' weakly in W 1,1 (o) ∩ L2 (o) and strongly in Lq (o), weakly in V , weakly in{ L1 (o), { udx = 0,

o

o

o

zε∗ dx →

zdx = 0, o

and .(u,z) satisfies (5.35). Moreover, by (5.47), the sequence ηε =

.

1 ε

{ o

( ) j (∇yε∗ + j ∗ (∇zε∗ ) − u∗ε yε∗ )dx

is bounded by a constant; hence, { 0≤

.

o

( ) j (∇yε∗ + j ∗ (∇zε∗ ) − ∇yε∗ · ∇zε∗ )dx = εηε .

By passing to the limit as .ε → 0 in the previous relation, we get { 0≤

.

z) − uy )dx y ) + j ∗ (∇(j (∇-

o

{

≤ lim

ε→0 o

{

= lim

ε→0 o

( ) j (∇yε∗ + j ∗ (∇zε∗ ) − u∗ε yε∗ )dx ( ) j (∇yε∗ + j ∗ (∇zε∗ ) − ∇yε∗ · ∇zε∗ )dx = 0,

which yields {

z) − uy )dx = y ) + j ∗ (∇(j (∇-

.

o

{

z) − ∇y · ∇z)dx = 0. y ) + j ∗ (∇(j (∇-

o

By the Legendre–Fenchel relations, the integrand in the last integral is nonnegative, and so it follows that j (∇y ) + j ∗ (∇z) − ∇y · ∇z = 0, a.e. on o,

.

which leads to the conclusion that

138

5 A Nonlinear Control Problem in Image Denoising

∇z = β(∇y ), a.e. on o.

.

Moreover, by passing to the limit in (5.35) corresponding to .(zε∗ , u∗ε ), it follows that the limit .z is the solution to .

− ∆z =u in o, ∇z · ν = 0 on |.

-ε . Recalling (5.47), It turns out that .(u, y ) satisfies (5.2) and .(u, y ,z) ∈ U ) { ( || ∗ ||q uε λ ∗ obs 2 + (yε − y ) dx ≤ Jε (u∗ε , yε∗ , zε∗ ) . q 2 o ) { ( ∗q |u | λ + (y ∗ − y obs )2 dx, ≤ q 2 o and passing to the limit as .ε → 0, we obtain { ( .

o

) ) { ( ∗q ||u | u|q λ λ + (y − y obs )2 dx ≤ + (y ∗ − y obs )2 dx, q 2 q 2 o

-). which proves that .(u, y ,z) is an optimal pair in .(P

u n

5.3.1 Optimality Conditions for the Approximating Problem - be an optimal pair in .(P -ε ), and let .λ be a real value. We take v Let .(u∗ε , yε∗ , zε∗ ) ∈ U and Y such that { q .v ∈ L (o), vdx = 0, Y ∈ L2 (o) ∩ W 1,1 (o), o

and define uσε = u∗ε + σ v, yεσ = yε∗ + σ Y.

.

Then, problem .

− ∆zεσ = uσε in o, ∇zεσ · ν = 0 on |

has a unique solution, and we deduce that Z is the unique solution to

5.3 Approximating Problem

139

.

−∆Z = v in o, ∇Z · ν = 0 on |,

(5.48)

where zεσ − zε∗ weakly in H 1 (o). σ λ →0

Z = lim '

.

-ε ), such Proposition 5.8 Let .q ≥ 2. If .(u∗ε , yε∗ , zε∗ ) is an optimal pair for problem .(P that .β −1 (∇zε∗ ) ∈ (L2 (o))N , the conditions of optimality read .

| | |u∗ |q−2 u∗ = 1 y ∗ − pε + C ∗ , ε ε ε ε ε

(5.49)

with .Cε∗ a constant, where the adjoint variables .pε , .yε∗ , and .zε∗ are the solutions to 1 ∇ · β −1 (∇zε∗ ) in o, ε . 1 ∇pε · ν = β −1 (∇zε∗ ) · ν on |, ε

(5.50)

−∇ · β(∇yε∗ ) + λε(yε∗ − y obs ) = u∗ε in o, β(∇yε∗ ) · ν = 0 on |,

(5.51)

−∆zε∗ = u∗ε in o, ∇zε∗ · ν = 0 on |.

(5.52)

∆pε =

.

.

-ε ). Then, Proof Assume that .(u∗ε , yε∗ , zε∗ ) is optimal in .(P -ε . Jε (u∗ε , yε∗ , zε∗ ) ≤ Jε (u, y, z) for all (u, v, z) ∈ U

.

In particular for .(u, v, z) = (uσε , yεσ , zεσ ), we have Jε (u∗ε , yε∗ , zε∗ ) ≤ Jε (uσε , yεσ , zεσ ).

.

By subtracting the left-hand side term from the right-hand one, dividing by .σ , and passing to the limit as .σ → 0, we obtain .

{{ ( ) | ∗ |q−2 ∗ |u | u v + λ(y ∗ − y obs )Y dx o

ε

1 + ε

ε

{ ( o

(5.53)

ε

β(∇yε∗ ) · ∇Y

+β

−1

(∇zε∗ ) · ∇Z

− u∗ε Y

− vyε∗

)

} dx ≥ 0.

140

5 A Nonlinear Control Problem in Image Denoising

We introduce the adjoint problem (5.50) and note it has a unique solution .pε ∈ V (defined in (5.33)) that satisfies the weak form { { 1 . ∇pε · ∇ψdx = β −1 (∇zε∗ ) · ∇ψdσ for all ψ ∈ H 1 (o). (5.54) ε o o Multiplying (5.48) by .pε , we have {

{ ∇pε · ∇Zdx =

.

o

(5.55)

vpε dx, o

which compared with (5.54) gives 1 . ε

{ β

−1

o

(∇zε∗ ) · ∇Zdx

{

{

=

∇pε · ∇Zdx = o

vpε dx.

(5.56)

o

By (5.53), (5.56), and (5.55), we obtain { ( .

o

) 1 1 λ(yε∗ − y obs ) − ∇ · β(∇yε∗ ) − u∗ε Y dx ε ε ( ) { | ∗ |q−2 ∗ 1 ∗ | | v uε uε − yε + pε dx ≥ 0. + ε o

Replacing now .σ by .−σ , we obtain the inverse inequality, so that the final relation is ) { ( 1 1 ∗ ∗ obs ∗ . (5.57) λ(yε − y ) − ∇ · β(∇yε ) − uε Y dx ε ε o ( ) { | |q−2 ∗ 1 ∗ + v |u∗ε | uε − yε + pε dx = 0, ε o which is true for any Y and v. In particular, for .v = 0, we get { ( .

o

λ(yε∗

−y

obs

) 1 1 ∗ ∗ ) − ∇ · β(∇yε ) − uε Y dx = 0, for all Y ∈ L2 (o), ε ε

which means that .yε∗ is a solution in the sense of distributions to (5.51). Next, .Y = 0 leads to ( ) { { | |q−2 ∗ 1 ∗ . v |u∗ε | uε − yε + pε dx = 0, for all v ∈ Lq (o), with vdx = 0. ε o o | |q−2 ∗ This implies that .|u∗ε | uε{ − 1ε yε∗ + pε = Cε∗ , with .Cε∗ a constant that may be deduced from the condition . o u∗ε dx = 0. n u

5.3 Approximating Problem

141

For example, we observe that constant .Cε∗ in (5.49) can be easily determined in { the case .N = 1, when .q = 2, from the condition . o u∗ε dx = 0, that is, ∗ .Cε

1 = meas(o)

) { ( 1 ∗ pε − yε dx. ε o

(5.58)

In the case .N ≥ 2, when .q > N, the determination of .Cε∗ could be done numerically. Remark 5.9 We observe that the optimality conditions are obtained in Proposition 5.8 for .q ≥ 2. However, in Theorem 5.6, we proved the existence of an optimal triplet .(u∗ε , yε∗ , zε∗ ) in the case .N ≥ 2 only if .q > 2. We bring into question if it is possible to have a result in the limit case as .q → 2 when for .N ≥ 2. To give an answer, we rely on the following argument. Let .δ be positive, arbitrary small, and -) and .(P -ε ) with .q = 2 + δ and .q ' = 2+δ . We call consider the control problems .(P 1+δ -δ ) and .(P -ε,δ ). Both problems have solutions .(u∗ , y ∗ ) and .(u∗ , y ∗ , z∗ ), them .(P δ δ ε,δ ε,δ ε,δ respectively. We check what happens at limit as .δ → 0 in the approximating ∗ , z∗ ) be an optimal pair for the approximating problem with .ε fixed. Let .(u∗ε,δ , yε,δ ε,δ -ε,δ ) and proceed as in the proof of Theorem 5.7. Then problem .(P { .

⎛| ⎞ | | ∗ |q λ ∗ ⎜ |uε,δ | ⎟ + (yε,δ − y obs )2 ⎠ dx ⎝ q 2 o 1 + ε

{ o

( ) ∗ ∗ ∗ j (∇yε,δ + j ∗ (∇zε,δ ) − u∗ε,δ yε,δ )dx

{ ( ≤ o

(5.59)

) { |u|q 1 λ obs 2 + (y − y ) dx + (j (∇y) + j ∗ (∇z) − uy)dx, q 2 ε o

-ε . In particular, let us take .u = z = 0, and so, the left-hand side for all .(u, y, z) ∈ U is bounded by a constant independent of .δ. Then, on a subsequence .(δ → 0), we have u∗ε,δ → u^ε weakly in Lq (o), .

(5.60)

.

'

∗ yε,δ → y^ε weakly in W 1,1 (o) ∩ L2 (o) and strongly in Lq (o), . { { ∗ u∗ε,δ yε,δ → u^ε y^ε weakly in L1 (o), u∗ε,δ dx → u^ε dx = 0, o

∗ zε,δ → z^ε weakly in V ,

and .z^ε satisfies equation

(5.61)

o

(5.62)

142

5 A Nonlinear Control Problem in Image Denoising .

− ∆^ zε = u^ε in o,

(5.63)

∇ z^ε · ν = 0 on | in the sense of distributions. Therefore, problem .(Pε,δ ) with .q = 2+δ converges to a -ε , as .δ → 0. This discussion was done for giving an argument limit .(u^ε , y^ε , z^ε ) ∈ U for the consideration of the exponent .q = 2 (which makes easier the determination of .Cε∗ ) in the further algorithm, to ensure that the scheme is stable.

5.3.2 Numerical Algorithm A numerical algorithm can be written on the basis of the optimality conditions -ε ) can be computed previously deduced. The optimal state .(u∗ε , .yε∗ , .zε∗ ) in .(P from Eqs. (5.49)–(5.52), by an iterative steepest descent gradient algorithm. For simplicity, we shall indicate neither the subscript .ε in all functions related to -ε ) nor the superscript .∗ for the optimal pair. problem .(P For the reader’s convenience, we specify that Eq. (5.50) is equivalent to the following minimization problem: { ( Minimize o(p) =

.

u∈U1

o

) 1 1 −1 2 |∇p| − β (∇z) · ∇p dx, 2 ε

(5.64)

where .β −1 (∇z) ∈ (L2 (o))N and { } { 1 .U1 = p ∈ H (o); pdx = 0 .

(5.65)

o

Therefore, according to the steepest descent formula (see [3]), we have iteratively pn+1 = pn − ρ n ∇o(pn ),

.

(5.66)

where .n ∈ {0, . . . , N } is the number of iterations. A possible choice for the step .ρ n is given by { } o(pn + ρ n w n ) = min o(pn + ρw n ); ρ ≥ 0 .

.

(5.67)

Similarly, Eq. (5.51) is equivalent to the following minimization problem (for .u ∈ Lq (o)) Minimize u(y) =

.

y∈U2

( ) ) { ( 1 2 j (β(∇y)) + λε y − y obs y − uy dx, 2 o

(5.68)

5.3 Approximating Problem

143

{ } U2 = y ∈ L2 (o) ∩ W 1,1 (o); β(∇y) ∈ (L1 (o))N , j (β(∇y)) ∈ L1 (o) (5.69) and so .

y n+1 = y n − ρ n ∇u(y n ).

.

(5.70)

The proof of the existence of a unique solution to (5.68) is very technical and can be led in a similar way as in Proposition 4.6 in Chap. 4. Since 1 ∇o(p) = −∆p + ∇ · β −1 (∇z) ε

.

and ∇u(y) = −∇ · β(∇y) + ελ(y − y obs ) − u,

.

the formulas (5.70) and (5.66) yield (y n+1 − y n )(x1 , x2 ) = ρ n (∇ · β(∇y n ) − ελ(y n − y obs ) + un )(x1 , x2 ),

.

(p

.

n+1

( ) 1 −1 n n − p )(x1 , x2 ) = ρ ∆p − β (∇z ) (x1 , x2 ). ε n

n

(5.71) (5.72)

We complete these systems by the iterations corresponding to (5.49) and (5.52) 1 n y (x1 , x2 ) − pn (x1 , x2 ) + C n , ε

(5.73)

(zn+1 − zn )(x1 , x2 ) = ρ n (∆zn + un )(x1 , x2 ),

(5.74)

un (x1 , x2 ) =

.

.

where .(x1 , x2 ) is a pixel in .2D. Having computed these expressions, we present a sketch of the iterative algorithm for .q = q ' = 2 : Algorithm Step 0. Set .n = 0, and fix .un , .y n , .C n , .zn . Step 1. Compute .y n+1 by (5.71). Step 2. Compute .pn+1 by (5.72). Step 3. Compute .C n+1 by (5.58). C n+1 =

.

1 meas(o)

Step 4. Compute .zn+1 by (5.74). Step 5. Compute .un+1 by

{ o

( ) 1 pn+1 − y n+1 dx. ε

(5.75)

144

5 A Nonlinear Control Problem in Image Denoising

n+1

u

.

(

= u +λ n

) 1 n+1 n+1 n+1 , y −p +C ε

(5.76)

with .λ ∈ R. Step 6. Stop criterion: if { .

o

| |2 | n+1 | − y n | dx ≤ εstop |y

with .εstop small, fixed, then .un+1 is the optimal value. If not, set .n := n + 1 and go to Step 1. The system (5.71)–(5.74) discretized with respect to the space variables and the corresponding algorithm were presented in [20].

5.4 A Potential with a Polynomial Growth We put down a few lines referring to the case of a potential with a polynomial growth. In this case, we introduce the minimization problem Minimize

.

(u,y)∈U1

{ { J (u, y) =

( o

) } |u(x)|q σ obs 2 dx , + (x)) (y(x) − y q' 2

(P1 )

where { U1 = (u, y); u ∈ Lq (o), y ∈ L2 (o), ∇y ∈ (Lp (o))N ,

.

(5.77)

(u, y) satisfies problem (5.78)–(5.79)} ,

.

− ∇ · β(∇y) e u in o,

(5.78)

β(∇y) · ν e 0 on ∂o, { u(x)dx = 0.

.

(5.79)

o

Here, the state equation (5.78) is written in the sense of distributions (this being rigorously explained in Definition 5.10 later), and .y obs is given in .L2 (o). In image denoising, the less is p, the better denoising result is obtained because a smaller value of p serves to the aim of a better image edges preserving. This justifies the interest in problem .(P ) for .p > 1, but close to 1.

5.4 A Potential with a Polynomial Growth

145

The following hypotheses are assumed: (i1 ) (i2 )

p > 1 and .q ≥ 2. j : RN → R is a proper convex function such that

.

.

.

.

β(r) = ∂j (r) for all r ∈ RN .

(5.80)

p

(5.81)

.

(i3 )

.

j has the property p

C1 |r|N + C10 ≤ j (r) ≤ C2 |r|N + C20 ,

.

with .C1 , C2 positive. By (5.80), it follows that .β : RN → RN is maximal monotone (and possibly multivalued), that is, N (η − . η) · (r − r) ≥ 0, for any r, r ∈ R ,

(5.82)

where .η ∈ β(r), η ∈ β(r), and R(I + β) = RN

.

(5.83)

where I is the identity operator and R is the range. It should be remarked that the state equation (5.78) may be viewed as an elliptic equation in divergence form with a discontinuous diffusion term .β, in which the jumps have been filled in, actually providing the multivalued function .β. This is the reason for which we may see the equation as a singular one. Definition 5.10 Let q be defined as in (5.17), and let .u ∈ Lq (o). We call a solution to (5.78) a function .y ∈ L2 (o), such that .∇y ∈ (Lp (o))N , which satisfies {

{ ξ(x) · ∇ψ(x)dx =

.

o

u(x)ψ(x)dx,

(5.84)

o

for all .ψ ∈ L2 (o) with ∇ψ ∈ (Lp (o))N , and some .ξ(x) ∈ β(∇y(x)), a.e. .x ∈ o. Consequently to hypothesis (5.81), the functions j and .j ∗ enjoy the properties (1.11)–(1.13) proved in Lemma 1.1, in Chap. 1. Let .y ∈ L2 (o), such that p N p' N .∇y ∈ (L (o)) . Then, by (1.11)–(1.13), it follows that .ξ ∈ (L (o)) , if .ξ(x) ∈ β(∇y(x)), a.e. .x ∈ o, and the left-hand side in (5.84) makes sense. In this case, Lemma 5.3 is no longer necessary. The results obtained in the current section are not directly deduced from those previously exposed in this chapter, and we refer to [62] for the proofs in this case.

Chapter 6

An Optimal Control Problem for a Phase Transition Model

In this chapter, we present an optimal control problem for a system governed by the Penrose–Fife phase transition model. The phase-field model considered here, due to Penrose and Fife in [65] and [66], is a thermodynamically consistent model for the description of the kinetics of phase transition and separation processes in binary materials in terms of the absolute temperature .θ and the order parameter .ϕ. Loosely speaking, the order parameter represents the local fraction of one of the two components. The Penrose–Fife model consists of a system coupling a singular heat equation for the absolute temperature .θ with a nonlinear equation that characterizes the evolution of the phase variable .ϕ. These equations are accompanied by initial data for .θ and .ϕ and by boundary conditions, considered here of Robin type for .θ and of homogeneous Neumann type for .ϕ, chosen according to physical considerations. A vast literature is devoted to the well-posedness of this model, the long-time behavior of solutions in terms of both attractors and convergence of single trajectories to stationary states and some associated control problems (see, e.g., [36, 37, 40, 55], and the references in [38]). The aim of the control problem discussed in this chapter is to force a sharp interface separation between the states of the system by means of a distributed heat source and a boundary heat source as controllers, while keeping its temperature at a certain average level .θf . This is an optimal control problem with state and control restrictions, and actually, it is another example of how to use the Legendre–Fenchel duality relations in solving optimal control problems.

6.1 Presentation of the Problem Let .Ω be an open bounded domain of .R3 , having the boundary .Γ sufficiently smooth. We assume here that the phase transition takes place in .Ω in the interval .(0, T ), with T finite. The Penrose–Fife system we are interested in reads © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Marinoschi, Dual Variational Approach to Nonlinear Diffusion Equations, PNLDE Subseries in Control 102, https://doi.org/10.1007/978-3-031-24583-1_6

147

148

6 An Optimal Control Problem for a Phase Transition Model

θt − Δβ(θ ) + ϕt = u, in Q := (0, T ) × Ω, .

(6.1)

1 1 − , in Q, . θc θ

(6.2)

.

ϕt − Δϕ + (ϕ 3 − ϕ) = −

∂β(θ ) = α(x)(β(θ ) − v), on Σ := (0, T ) × Γ, . ∂ν ∂ϕ = 0, on Σ, . ∂ν

(6.3) (6.4)

θ |t=0 = θ0 , in Ω, .

(6.5)

ϕ|t=0 = ϕ0 , in Ω,

(6.6)

where .ν is the unit outward normal to .Γ, θc is the transition temperature, u is the distributed heat source, and v is the boundary heat source. The system is studied under the following hypotheses: β ∈ C 1 (0, ∞). .β(r) behaves like .−c1 /r closed to 0 and like .c2 r in a neighborhood of .+∞, for some constants .c1 and .c2 . .(iii) The parameter .α has the property (i) .(ii) .

.

α ∈ H 1 (Γ ) ∩ L∞ (Γ ), 0 < αm ≤ α(x) ≤ αM a.e. x ∈ Γ ,

.

(6.7)

with .αm , αM constants. For the sake of simplicity, we can assume that 1 β(r) = − + r r

.

(6.8)

because this is frequently chosen in several types of phase transition and phase separation models both in liquids and in crystalline solids (according to, e.g., [36] where similar growth conditions are postulated). We shall use the sign function (denoted here by .S) translated by .θc ⎧ r < θc ⎨ −1, .S(r) = [−1, 1], r = θc ⎩ 1, r > θc

(6.9)

to set the third control variable in our problem. Let us introduce the optimal control problem Minimize J (u, v, η) for all (u, v, η) ∈ K1 × K2 × K3 ,

.

subject to (6.1)–(6.6), where

(P )

6.1 Presentation of the Problem

J (u, v, η) =

λ1 2

149

( (θ − θf )2 dxdt +

λ2 2

( (ϕ − η)2 dxdt,

(6.10)

.

K1 = {u ∈ L∞ (Q); um ≤ u(t, x) ≤ uM a.e. (t, x) ∈ Q}, .

(6.11)

K2 = {v ∈ L∞ (Σ); vm ≤ v(t, x) ≤ vM a.e. (t, x) ∈ Σ}, .

(6.12)

K3 = {η ∈ L∞ (Q); η(t, x) ∈ S(θ (t, x)) a.e. (t, x) ∈ Q},

(6.13)

.

Q

Q

and .um , uM , vm , vM are fixed real values. The positive constants .λ1 , λ2 are weights that may ensure more importance to one term or the other in .(P ). Generally, we take θf ∈ L2 (Q).

.

(6.14)

Problem .(P ) is formulated such that to enforce the formation of a sharp interface between the two phases by the constraint .η ∈ K3 . If by the control problem one intends to preserve the system separated into two phases by the ||sharp interface, it || should be added that .θf must belong to a neighborhood of .θc , i.e., .||θf − θc ||L2 (Q) ≤ δ, with .δ rather small. As in the previous chapter, the approach of the control problem assumes the proof of the following results: the existence for the state system, with new supplementary regularity of the state in Theorem 6.2, and the existence of at least one solution to problem .(P ), represented by an optimal triplet of controllers .(u, v, η) and the corresponding pair of states .(θ, ϕ) in Theorem 6.3. Due to the singularity induced by the graph representing the sharp interface, the conditions of optimality cannot be deduced directly for .(P ). In order to avoid working with the graph .S(θ ), we introduce an approximating problem .(Pε ) in which the constraint .η ∈ S(θ ) is replaced by an equivalent relation based on the Legendre–Fenchel relations between a certain proper convex lower semicontinuous function j and its conjugate .j ∗ . In this case, j is the potential of .S. This approximating problem has at least one solution that is an appropriate approximation of a solution to .(P ). This last assertion relies on the convergence result of .(Pε ) to .(P ) given in Theorem 6.5. Then, the question concerning the computation of the optimality conditions is rigorously examined. A second approximation is represented by a penalized minimization problem .(Pε,σ ) in which j is replaced by its Moreau–Yosida regularization. The optimality conditions for .(Pε,σ ) are provided by explicit expressions in Proposition 6.10. Some estimates and the proof of the strong convergence (as .σ → 0) of the sequence of controllers, performed in Theorem 6.11, allow the passage to the limit as .σ → 0 in order to recover the form of the controllers in problem .(Pε ). Recalling Theorem 6.5, the optimal controller in .(P ) is obtained as the limit of a sequence of optimal controllers in .(Pε ), on the basis of the convergence of .(Pε ) to .(P ).

150

6 An Optimal Control Problem for a Phase Transition Model

6.2 Existence in the State System and Control Problem Let V be the Sobolev space .H 1 (Ω) endowed with the standard scalar product (( .

||ψ||V =

)1/2

( |∇ψ(x)|2 dx +

|ψ(x)|2 dx

Ω

(6.15)

,

Ω

and identify .L2 (Ω) with its dual space, such that .V ⊂ L2 (Ω) ⊂ V ' with dense and compact embeddings. We recall that if .α satisfies (6.7), then the norm (

( |∇ψ(x)|2 dx +

|||ψ||| =

.

Ω

α(x) |ψ(x)|2 ds

(6.16)

Γ

is equivalent to .||ψ||V , and we have the inequality (( 2 . ||ψ||V

)

( |∇ψ(x)| dx +

≤ CP

|ψ(x)| ds , ∀ψ ∈ V

2

Ω

2

(6.17)

Γ

(see [64, p. 20]), with .CP depending on .Ω. The positive constants .C, Ci , i = 0, 1, 2 . . . , used in the next statements and proofs may differ from line to line. Definition 6.1 Let θ0 ∈ L2 (Ω), θ0 > 0 a.e. in Ω, ln θ0 ∈ L1 (Ω), ϕ0 ∈ H 1 (Ω),

.

u ∈ L2 (Q), v ∈ L2 (Σ), α satisfies (6.7).

(6.18)

We call a solution to (6.1)–(6.6) a pair .(θ, ϕ) such that θ ∈ L2 (0, T ; V ) ∩ C([0, T ]; L2 (Ω)) ∩ W 1,2 (0, T ; V ' ), β(θ ), .

ϕ ∈ L2 (0, T ; H 2 (Ω)) ∩ C([0, T ]; V ) ∩ W 1,2 (0, T ; L2 (Ω)),

1 ∈ L2 (0, T ; V ), θ (6.19)

which satisfies (6.1)–(6.4) in the form ( .

0

T

/

\ ( ( dθ dϕ (t), ψ1 (t) ψ1 dxdt dt + ∇β(θ ) · ∇ψ1 dxdt + dt Q ( ( Q dt ( V ' ,V αβ(θ )ψ1 dσ dt = uψ1 dxdt + αvψ1 dσ dt, + Σ

Q

Σ

(6.20)

6.2 Existence in the State System and Control Problem

( .

Q

dϕ ψ2 dxdt + dt

(

( ∇ϕ · ∇ψ2 dxdt + (ϕ 3 − ϕ)ψ2 dxdt Q ) ( (Q 1 1 ψ2 dxdt, − = θ Q θc

151

(6.21)

for any .ψ1 , ψ2 ∈ L2 (0, T ; V ), and the initial conditions (6.5)–(6.6). The next statement collects a series of properties of solutions to (6.1)–(6.6). The proof is particularly technical, and we indicate some arguments and provide detailed calculations only for some estimates. Theorem 6.2 Let assumptions (6.18) hold. Then (6.1)–(6.6) have a unique solution, with the properties θ > 0 a.e. on Q, ln θ ∈ L∞ (0, T ; L1 (Ω)),

.

and it satisfies the estimates || || || 1 || .||θ ||L2 (0,T ;V ) + ||θ ||L∞ (0,T ;L2 (Ω)) + ||θ ||W 1,2 (0,T ;V ' ) + || || ≤ C, . || θ || 2 L (0,T ;V ) (6.22) ||ϕ||L2 (0,T ;H 2 (Ω)) + ||ϕ||L∞ (0,T ;V ) + ||ϕ||W 1,2 (0,T ;L2 (Ω)) ≤ C. (6.23) Let us set .θ¯ := θ1 − θ1 , .ϕ¯ := ϕ1 − ϕ2 , .u¯ := u1 − u2 , .v¯ := v1 − v2 , where .(θ1 , ϕ1 ) and .(θ2 , ϕ2 ) are the solutions of (6.1)–(6.6) corresponding, respectively, to the data .u1 , v1 and .u2 , v2 , with the same initial data .θ0 , .ϕ0 and the same coefficient .α. Then, the following continuous dependence estimate of the solution with respect to the data ||θ¯ ||2L2 (Q) + ||ϕ|| ¯ 2C([0,T ];L2 (Ω)) + ||ϕ|| ¯ 2L2 (0,T ;V )

.

(6.24)

) ( ≤ C ||u|| ¯ L2 (Q) + ||v|| ¯ 2L2 (Σ) holds with the positive constant C independent of .ui , vi , .1 = 1, 2. If, in addition to (6.18), we suppose that ϕ0 ∈ H 2 (Ω),

.

∂ϕ0 = 0 on Γ, ∂ν

(6.25)

then we have ϕ ∈ L∞ (Q) ∩ L∞ (0, T ; H 2 (Ω)) ∩ W 1,2 (0, T ; V )

.

(6.26)

152

6 An Optimal Control Problem for a Phase Transition Model

and .

||ϕ||L∞ (Q) + ||ϕ||L∞ (0,T ;H 2 (Ω)) + ||ϕ||W 1,2 (0,T ;V ) ≤ C.

(6.27)

Further, if, in addition to (6.18), we still assume 1 ∈ L∞ (Ω), . θ0

θ0 ,

.

u ∈ L2 (0, T ; L6 (Ω)), v ∈ L∞ (Σ), v ≤vM a.e. on Σ,

(6.28) (6.29)

then we have 1 ∈ L∞ (Q) θ

θ,

.

(6.30)

and .

|| || || 1 || || ||θ ||L∞ (Q) + || ≤ C. || θ || ∞ L (Q)

(6.31)

Proof The existence of solutions to (6.1)–(6.6) can be proved by adapting the proof given in [36] to the case of .α non-constant in (6.3). The uniqueness of solutions has been proved in [37], and it has been then generalized to the case of less regular data (satisfying assumptions (6.18)) in [68], where also a continuous dependence result of the solution with respect to the data has been shown. We also refer to the above-mentioned papers for the proof of estimates (6.22)–(6.23). n Proof of Estimate (6.24) Following the lines of Theorem 1 in [36] and Theorem 3.5 in [68], we write (6.1) for .(θ1 , ϕ1 ) and for .(θ2 , ϕ2 ), and these being two solutions to (6.1)–(6.6) corresponding, respectively, to the data .u1 , v1 and .u2 , v2 , with the same initial data .θ0 , .ϕ0 , and with the same coefficient .α. We set .θ¯ := θ1 − θ1 , .ϕ ¯ := ϕ1 − ϕ2 , .u¯ := u1 − u2 , .v¯ := v1 − v2 and denote by .∗ the standard time convolution operator ( (a ∗ b)(t) =

.

t

a(t − τ )b(τ )dτ, t ∈ (0, T ].

0

Taking the difference of Eqs. (6.1) for .θ1 and .θ2 and integrating with respect to time, we obtain ( .

0

t

< > θ¯ (τ ), ψ1 (τ ) V ' ,V dτ +

( t( 0

1 ∗ ∇ (β(θ1 ) − β(θ2 )) · ∇ψ1 dxdτ Ω

6.2 Existence in the State System and Control Problem

( t(

+

0

ϕψ ¯ 1 dxdτ +

0

Ω

( t(

=

( t(

0

α (1 ∗ (β(θ1 ) − β(θ2 ))) ψ1 dσ dτ

153

(6.32)

Γ

¯ ψ1 dxdτ + (1 ∗ u)

( t( 0

Ω

α (1 ∗ v) ¯ ψ1 dσ dτ , Γ

for any .ψ1 ∈ L2 (0, T ; V ). Then, we set as test function .ψ1 = β(θ1 ) − β(θ2 ), and using the monotonicity properties of the function .θ |→ −1/θ, we find out that ( t(

¯ 2 dxdτ + |θ|

.

0

Ω

+

(

1 2

|∇(1 ∗ (β(θ1 ) − β(θ2 )) (t))|2 dx

( t( 0

+ ≤

ϕ¯ (β(θ1 ) − β(θ2 )) dxdτ

Ω

(

1 2

α |1 ∗ (β(θ1 ) − β(θ2 )) (t)|2 dσ Γ

( t( 0

+

(6.33)

Ω

(1 ∗ u) ¯ (β(θ1 ) − β(θ2 )) dxdτ

Ω

( t( 0

α (1 ∗ v) ¯ (β(θ1 ) − β(θ2 )) dσ dτ .

Γ

Then, taking the differences of (6.2), testing by .ψ2 = ϕ, ¯ and exploiting the monotonicity of .ϕ |→ ϕ 3 , we have that 1 2

.

( |ϕ(t)| ¯ 2 dx +

( t( 0

Ω

≤

( t( 0

|∇ ϕ| ¯ 2 dxdτ

(6.34)

Ω

|ϕ| ¯ 2 dxdτ + Ω

) ( t( ( 1 1 − + ϕdxdτ ¯ . θ1 θ2 0 Ω

Now, summing up (6.33) and (6.34), we take advantage of a cancelation of one term due to the special form (6.8) of .β. Then, in view of assumption (6.7) on .α, we get ( t( .

0

Ω

|θ |2 dxdτ + ||1 ∗ (β(θ1 ) − β(θ2 )) (t)||2V

2 +||ϕ(t)|| ¯ L2 (Ω)

+

( t( 0

|∇ ϕ| ¯ 2 dxdτ Ω

(6.35)

154

6 An Optimal Control Problem for a Phase Transition Model

(( t (

≤ C1

0

+

( t( 0

( t(

|ϕ¯ θ¯ |dxdτ +

0

Ω

|(1 ∗ u) ¯ (β(θ1 ) − β(θ2 )) |dxdτ Ω

|α (1 ∗ v) ¯ (β(θ1 ) − β(θ2 )) |dσ dτ +

) |ϕ| ¯ 2 dxdτ .

( t( 0

Γ

Ω

We estimate the integrals on the right-hand side of (6.35) as follows. Using Young’s inequality (see (7.5) in the Appendix), we deduce that ( t(

¯ |ϕ¯ θ|dxdτ ≤ δ1

.

0

( t( 0

Ω

Ω

|θ¯ |2 dxdτ + Cδ1

( t( 0

|ϕ| ¯ 2 dxdτ ,

Ω

for some .δ1 > 0 to be chosen later. Integrating by parts in time and using again Young’s inequality, we obtain ( t(

|(1 ∗ u) ¯ (β(θ1 ) − β(θ2 )) |dxdτ

.

0

Ω

( ≤

|1 ∗ u|(t)|1 ¯ ∗ (β(θ1 ) − β(θ2 )) |(t)dx Ω

+

( t( 0

|u¯ (1 ∗ (β(θ1 ) − β(θ2 ))) |dxdτ

(6.36)

Ω

2 ≤ δ2 ||1 ∗ (β(θ1 ) − β(θ2 )) (t)||2V + Cδ2 ||1 ∗ u(t)|| ¯ L2 (Ω)

+

( t( 0

(

t

|u| ¯ 2 dxdτ + 0

Ω

||1 ∗ (β(θ1 ) − β(θ2 )) (τ )||2L2 (Ω) dτ,

and, due to (6.7), ( t( 0

.

Γ

|α (1 ∗ v) ¯ (β(θ1 ) − β(θ2 )) |dσ dτ ( |α||1 ∗ v|(t)|1 ¯ ∗ (β(θ1 ) − β(θ2 )) |(t)dσ ≤ ( Γt ( + |α v¯ (1 ∗ (β(θ1 ) − β(θ2 ))) |dσ dτ 0

Γ

0

Γ

(6.37)

2 ≤ δ3 ||1 ∗ (β(θ1 ) − β(θ2 )) (t)||2V + Cδ3 ||1 ∗ v(t)|| ¯ L2 (Γ ) ( t( ( t 2 + |v| ¯ dσ dτ + ||1 ∗ (β(θ1 ) − β(θ2 )) (τ )||2L2 (Γ ) dτ . 0

n

6.2 Existence in the State System and Control Problem

155

Relying on the estimates (6.35)–(6.37) and choosing the constants .δi , i = 1, 2, 3, such that .C1 (δ1 + δ2 + δ3 ) < 1, we infer that

.

( t( .

0

Ω

|θ |2 dxdτ + ||1 ∗ (β(θ1 ) − β(θ2 )) (t)||2V ( t( 2 + ||ϕ(t)|| ¯ + |∇ ϕ| ¯ 2 dxdτ L2 (Ω) 0

((

t

≤ C2 0

Ω

||ϕ(τ ¯ )||2L2 (Ω) dτ

( + 0

t

) ||1 ∗ (β(θ1 ) − β(θ2 )) (τ )||2V dτ

+ ||u|| ¯ 2L2 ((0,t)×Ω) + ||v|| ¯ 2L2 ((0,t)×Γ ) , whence, using a standard Gronwall’s lemma, we deduce the desired (6.24). Proof of Estimate (6.27) In order to prove the regularity (6.26) and estimate (6.27), we can proceed formally by testing (6.2) by .−Δϕt and integrating by parts using (6.4). This choice should be made rigorous in the framework of a regularized scheme, e.g., of Faedo–Galerkin type, but in order to not complicate the presentation we proceed formally here. Performing this computation and integrating the resulting equation over .(0, t), .t ∈ (0, T ], we get .

( t 1 1 ||∇ϕτ (τ )||2L2 (Ω) dτ (6.38) ||Δϕ(t)||2L2 (Ω) − ||Δϕ0 ||2L2 (Ω) + 2 2 0 || ( t || ( t( || 1 || || || ≤ ∇(ϕ 3 − ϕ)∇ϕt dxdτ . || θ (τ )|| ||∇ϕτ (τ )||L2 (Ω) dτ + 0 0 Ω V

For the first integral on the right-hand side, we use the Young inequality together with estimate (6.22). The last integral is treated as follows: ( t( 0 .

∇(ϕ 3 − ϕ)∇ϕt dxdτ ( t( ) ||ϕ(τ )||2L6 (Ω) + 1 ||∇ϕ(τ )||L6 (Ω) ||∇ϕτ (τ )||L2 (Ω) dτ ≤C (0 ( t 1 t 2 ||∇ϕτ (τ )||L2 (Ω) dτ + C (1 + ||Δϕ(τ )||2L2 (Ω) )dτ, ≤ 4 0 0 Ω

(6.39)

where the Hölder and Young inequalities have been used together with the Gagliardo–Nirenberg inequality ( [28], p. 241) and the previous estimate (6.23). By rearranging in (6.38) and using once more (6.23) for the boundedness of .ϕ in 2 2 .L (0, T ; H (Ω)), we obtain the estimate ||Δϕ||L∞ (0,T ;L2 (Ω)) + ||∇ϕt ||L2 (Q) ≤ C,

.

156

6 An Optimal Control Problem for a Phase Transition Model

which, together with (6.22)–(6.23), the standard elliptic regularity results, and the continuous embedding of .L∞ (0, T ; H 2 (Ω)) into .L∞ (Q) in 3D, gives the desired (6.27). n Proof of Estimate (6.31) We use a Moser-type technique to prove the .L∞ -bound for p .θ . The procedure consists in testing (6.1) by .(p + 1)θ , .p ∈ (1, ∞). This estimate is formally computed. In order to perform it rigorously, we would need to introduce a regularized system and then pass to the limit. However, since the procedure is quite standard, we prefer to perform only the formal estimate here. Testing (6.1) by p .(p + 1)θ , we obtain (

d . dt

θ

p+1

Ω

( | ( | |2 |2 4p(p + 1) | p+1 | | p−1 | 2 2 | dx |∇θ | dx + |∇θ 2 (p − 1) Ω Ω ( + (p + 1) αθ p+1 dσ

4p dx + p+1

Γ

(

(

= (p + 1)

αθ p−1 dσ + (p + 1) Γ

αvθ p dσ Γ

( (u − ϕt )θ p dx .

+ (p + 1) Ω

Using now assumptions (6.18) and (6.28), we get d . dt

( θ Ω

p+1

4p dx + p+1

( | ( | |2 |2 | p+1 | | p+1 | 2 |∇θ | dx + (p + 1)αm |θ 2 | dσ Ω

(6.40)

Γ

(

(

≤ (p + 1)αM

θ

p−1

dσ + (p + 1)αM vM

Γ

θ p dσ Γ

( (u − ϕt )θ p dx .

+ (p + 1) Ω

By the Young inequality used in the form '

1 bq 1 1 aq + q ' /q ' , + ' = 1, .a · b ≤ e q q q q e we estimate the two boundary terms as follows: ( α. M Γ

αm (p−1) θ p−1 dσ ≤ 4 p+1

(

( θ p+1 dσ + Γ

4 αm

) (p−1) 2

p+1

αM2 meas(Γ ), . (p+1) (6.41)

6.2 Existence in the State System and Control Problem

(

(

(

αm p θ dσ ≤ 4 (p+1) p

αM vM Γ

157

θ

p+1

dσ +

Γ

4 αm

)p

(αM vM )p+1 meas(Γ ). (p+1) (6.42)

By estimates (6.41)–(6.42), (6.40) becomes d . dt

( θ

p+1

Ω

(( | ) ( | |2 | p+1 |2 | p+1 | | dx + δ |∇θ 2 | dx + |θ 2 | dσ Ω

Γ

( ≤ CR p+1 + (p + 1)

|u − ϕt |θ p dx, Ω

where .δ := min{ 32 αm , 83 } > 0 and .C, R are independent of p. Using then the continuous embedding of .V = H 1 (Ω) into .L6 (Ω) in 3D, we obtain d . dt

( θ

p+1

dx + δ

'

)1/3

(( θ

Ω

3(p+1)

( ≤ CR

dx

p+1

+ (p + 1)

|u − ϕt |θ p dx,

Ω

Ω

(6.43)

some .δ '

for > 0 always independent of p. Then, as (6.27) ensures the boundedness of .ϕt in .L2 (0, T ; L6 (Ω)), we can estimate the last integral ( .(p

+ 1)

|u − ϕt |θ p dx

(6.44)

Ω

≤ (p + 1)||(u − ϕt )(t)||L6 (Ω) δ' ≤ 2 δ' ≤ 2

θ

θ

3(p+1)

dx

θ

3 (p−1) 2 2

Ω

+ Cδ ' (p + 1)

dx

Ω

3(p+1)

Ω

)1/3

(( ((

)1/6 ((

((

2

θ

dx

dx

((

||(u − ϕt )(t)||2L6 (Ω)

)1/3 3(p+1)

)2/3

θ ((

+ C (p + 1) δ'

Ω

2

||(u − ϕt )(t)||2L6 (Ω)

3 4 (p−1)

Ω

)4/3 dx )

θ

p−1

dx

,

Ω

p−1

p−1

where we have also used the inequality .||θ 2 (t)||2L3/2 (Ω) ≤ C(Ω)||θ 2 (t)||2L2 (Ω) . Choosing now .p = 3 in (6.43)–(6.44) and integrating with respect to time, with the help of (6.22) and (6.28), we obtain ( 4 .||θ || ∞ L (0,T ;L4 (Ω))

(

≤C 1+ 0

T

) ||(u − ϕt )(t)||2L6 (Ω) dt

(6.45)

.

In general, integrating (6.43) from 0 to t , .t ∈ (0, T ], and using (6.44) and (6.28), we infer that ( ) ) (( ( 3

θ p+1 (t)dx ≤ C R p+1 + (p + 1)2 sup

.

Ω

[0,T ]

θ 4 (p−1) dx Ω

4/3

,

(6.46)

158

6 An Optimal Control Problem for a Phase Transition Model

where C depends on the data but not on p. At this point, we can introduce the sequence .p0

= 4, pk+1 =

4 pk + 2, 3

k ∈ N,

and take .p = pk+1 − 1 in (6.46), getting (

( θ

.

pk+1

)4/3 )

((

(t)dx ≤ C R

pk+1

+ (pk+1 ) sup 2

[0,T ]

Ω

pk

θ dx

.

Ω

We apply now Lemma A.1 in [54] with the choices .a = 4/3, .b = c = 2, .δ0 = 4, = pk . Thus, we deduce that

.δk

.

sup ||θ ||Lpk (Ω) ≤ C,

[0,T ]

where C is independent of k . Hence, letting k tend to .∞, we get .||θ ||L∞ (Q)

≤C.

(6.47)

Finally, we prove the .L∞ (Q)-bound for .1/θ . Hence, let us call .h = 1/θ and rewrite formally (6.2)–(6.3) as follows: ( ht − h2 Δ h − ( . ∂ − h− ∂ν

) 1 = −h2 (u − ϕt ) in Q, h) ( ) 1 1 = α h − + v on Σ . h h

(6.48)

We note that, due to the estimate (6.47), there exists a positive constant .C¯ (depending on the data) such that .h(t, x)

¯ a.e. (t, x) ∈ Q ¯. ≥ C,

Test now (6.48) by .php−1 , .p ∈ (1, ∞), getting .

d dt

( hp dx + Ω

+

4p(p + 1) (p + 2)2 4(p + 1) p

( | |2 | p+2 | |∇h 2 | dx Ω

( ( | | | p2 |2 dx + p αhp+2 dσ |∇h | Ω

( =p

Γ

( αhp dσ + p

Γ

( αvhp+1 dσ − p

Γ

(u − ϕt )hp+1 dx . Ω

6.2 Existence in the State System and Control Problem

159

Using now the Young inequality and the assumption (6.28), we obtain .

d dt

( hp dx + δ Ω

(( | ( | |2 |2 ) | p+2 | | p+2 | |∇h 2 | dx + |h 2 | dσ Ω

Γ

( ≤ CR p+2 + p

|u − ϕt |hp+1 dx, Ω

where .δ and R are positive constants independent of p. By recalling the continuous embedding of .H 1 (Ω) into .L6 (Ω) in 3D along with Hölder’s inequality, we have that .

d dt

(

hp dx + δ '

)1/3

(( |h|3(p+2) dx

Ω

Ω

( ≤ CR p+2 + p

|u − ϕt |hp+1 dx Ω

≤ CR p+2 + p||(u − ϕt )(t)||L6 (Ω) ||h ≤ CR p+2 +

p+2 2

p

(t)||L6 (Ω) ||h 2 (t)||L3/2 (Ω)

p δ ' p+2 p2 ||h 2 (t)||2L6 (Ω) + ' ||(u − ϕt )(t)||2L6 (Ω) ||h 2 (t)||2L3/2 (Ω) . 2 2δ

By integrating over .(0, t), .t ∈ (0, T ], and using the continuous embedding of .Lp (Ω) into .L3p/4 (Ω) as well as assumption (6.28), we infer that ( .

Ω

hp (t)dx + δ '

|h|3(p+2) dx Ω

( ≤C R

)1/3

((

p+2

+p

2

p+2

+p

2

( ≤C R

(

t

(

t

)

p ||(u − ϕt )(τ )||2L6 (Ω) ||h(τ )|| 3p dτ 0 L 4 (Ω)

0

)

p ||(u − ϕt )(τ )||2L6 (Ω) ||h(τ )||Lp (Ω) dτ

,

where C depends on the data, but not on p. Choosing now .p = 6 and applying the Gronwall’s lemma, we obtain the starting point for an iterating procedure that is completely analogous to the one in [55], p. 269. Hence, we obtain .||h||L∞ (Q)

=||1/θ ||L∞ (Q) ≤ C,

and this concludes the proof of Theorem 6.2. The next result proves the existence of a solution to problem .(P ).

(6.49) n

160

6 An Optimal Control Problem for a Phase Transition Model

Theorem 6.3 Assume that θ0 ∈ L2 (Ω), θ0 > 0 a.e. in Ω, ln θ0 ∈ L1 (Ω), ϕ0 ∈ H 1 (Ω),

.

(6.50)

hold. Then .(P ) has at least one solution. Proof We note that the admissible set U is not empty because by Theorem 6.2, the solution .(θ, ϕ) yields .J (u, v, η) < ∞, for whatever .(u, v, η) ∈ U. Since .J (u, v, η) ≥ 0, it follows that J has an infimum d and this infimum is nonnegative. Let .(un , vn , ηn )n≥1 be a minimizing sequence for .J. This means that .un ∈ K1 , vn ∈ K2 , ηn ∈ K3 , .(θn , ϕn ) is the solution to (6.1)–(6.6) corresponding to .un , vn , ηn , and the following inequalities take place ( d ≤ λ1

( (θn − θf )2 dxdt + λ2

.

Q

(ϕn − ηn )2 dxdt ≤ d + Q

1 , n ≥ 1. n

(6.51)

By selecting subsequences (denoted still by the subscript n), we deduce that un → u weak* in L∞ (Q), vn → v weak* in L∞ (Σ),

.

ηn → η weak* in L∞ (Q), as n → ∞, and .u ∈ K1 , v ∈ K2 , η ∈ K3 , since a convex set is closed if and only if it is weakly closed. By (6.22)–(6.23), we have θn → θ weakly in L2 (0, T ; V ) ∩ W 1,2 (0, T ; V ' ), as n → ∞, 1 → l weakly in L2 (0, T ; V ), as n → ∞, . θn ϕn → ϕ weakly in L2 (0, T ; H 2 (Ω)) ∩ W 1,2 (0, T ; L2 (Ω)) and weak* in L∞ (0, T ; V ), as n → ∞. By the Aubin–Lions–Simon lemma (see Lemma 7.9 in the Appendix), we have θn → θ strongly in L2 (0, T ; L2 (Ω)), as n → ∞,

.

ϕn → ϕ strongly in L2 (0, T ; V ), as n → ∞. Therefore, on a subsequence, it follows that θn → θ a.e. in Q, as n → ∞,

.

whence .

1 1 a.e. in Q, as n → ∞, → θn θ

6.2 Existence in the State System and Control Problem

161

entailing that .l = 1/θ a.e. in Q. Using the Egorov theorem (see Theorem 7.4 in the Appendix), we can conclude that .

1 1 strongly in Lp (Q), for all 1 ≤ p < 2, as n → ∞. → θn θ

On the other hand, in view of (6.8), the above convergences yield β(θn ) → β(θ ) weakly in L2 (0, T ; V ) and a.e. in Q, as n → ∞.

.

Since .(ϕn )n is bounded in .L∞ (0, T ; V ), we deduce that .(ϕn3 )n is bounded in ∞ 2 .L (0, T ; L (Ω)), and consequently, ϕn3 → l1 weak* in L∞ (0, T ; L2 (Ω)), as n → ∞.

.

We also note there exists a subsequence such that .ϕn → ϕ a.e. in Q. This implies that .ϕn3 → ϕ 3 a.e. in Q, and we infer that .l1 = ϕ 3 a.e. in .Q. Now, we recall that .ηn ∈ S(θn ) a.e. in Q, .θn → θ strongly in .L2 (Q) and .ηn → η weak* in .L∞ (Q). Since S is maximal monotone, it is strongly–weakly closed, and we deduce that .η ∈ S(θ ) a.e. in Q (Proposition 7.16 in the Appendix). Since the trace operator is continuous from V to .L2 (Γ ) (see Theorem 7.5 in the Appendix), we have that .||β(θn )||L2 (0,T ;L2 (Γ )) ≤ C and so .

β(θn )|Γ → β(θ )|Γ weakly in L2 (0, T ; L2 (Γ )), as n → ∞.

Passing to the limit as .n → ∞ in the weak forms (6.20)–(6.21) (

T

0

.

( Q

/

\ ( dθn (t), ψ1 (t) dt + ∇β(θn ) · ∇ψ1 dxdt dt V ' ,V ( (Q dϕn ψ1 dxdt + + αβ(θn )ψ1 dσ dt Σ ( Q dt ( =

un ψ1 dxdt + αvn ψ1 dσ dt, Σ ( ( dϕn ψ2 dxdt + ∇ϕn · ∇ψ2 dxdt + (ϕn3 − ϕn )ψ2 dxdt dt ( ( Q ) Q 1 1 ψ2 dxdt, − = θn Q θc Q

for all .ψ1 , ψ2 ∈ L2 (0, T ; V ), we obtain by the previous convergences that .(θ, ϕ) satisfies (6.20)–(6.21), which means that it is the solution to (6.1)–(6.6) corresponding to u and .v. At the end, we pass to the limit in (6.51) using the weakly lower semicontinuity property of the terms in J and get .J (u, v, η) = d. This implies that .(u, v, η) and the corresponding states .(θ, ϕ) are optimal in .(P ), and this ends the proof. n

162

6 An Optimal Control Problem for a Phase Transition Model

6.3 Approximating Control Problem (Pε ) We note that the cost functional J is not differentiable, and so, it is convenient to introduce an approximating problem .(Pε ) for an appropriate differentiable cost functional and to show its convergence in a suitable sense to .(P ). This is to be done by means of the Legendre–Fenchel relations. To this end, we introduce the convex function j : R → R, j (r) = |r − θc | ,

(6.52)

.

whose subdifferential is the graph S defined in (6.9), that is, ∂j (r) = S(r − θc ).

(6.53)

.

The conjugate of j is j ∗ (ω) = sup (ωr − j (r)), r∈R

.

and it has the expression j ∗ (ω) = ωθc + I[−1,1] (ω).

(6.54)

.

We recall the definition of the indicator function .IK of a closed convex set, .IK (r) = 0 if .r ∈ K and .IK (r) = +∞ otherwise, and the Legendre–Fenchel relations (see (7.38)–(7.39) in the Appendix) j (r) + j ∗ (ω) ≥ rω for all r, ω ∈ R;

j (r) + j ∗ (ω) = rω iff ω ∈ ∂j (r), (6.55) where .∂j denotes the subdifferential of j . Then, (6.55) reduces to .

j (r) + ωθc − ωr ≥ 0 for all r ∈ R, ω ∈ [−1, 1];

.

j (r) + ωθc − ωr = 0 iff ω ∈ S(r).

(6.56)

Let .ε > 0, and state the approximating problem as follows: Minimize Jε (u, v, η) for all (u, v, η) ∈ K1 × K2 × K[−1,1] ,

.

(Pε )

subject to (6.1)–(6.6), where Jε (u, v, η) =

.

λ1 2

( (θ − θf )2 dxdt + Q

λ2 2

( (ϕ − η)2 dxdt Q

(6.57)

6.3 Approximating Control Problem (Pε )

+

1 ε

163

( (j (θ ) + ηθc − ηθ )dxdt, Q

and K[−1,1] = {η ∈ L∞ (Q); |η(t, x)| ≤ 1 a.e. (t, x) ∈ Q}.

.

(6.58)

Proposition 6.4 Let the assumptions (6.50) hold. Then .(Pε ) has at least one solution. Proof By (6.56), we have that .Jε (u, v, η) ≥ 0, and we set dε = inf Jε (u, v, η) ≥ 0.

.

(u,v,η)

Let .(unε , vεn , ηεn )n be a minimizing sequence for .Jε , that is, dε ≤ Jε (unε , vεn , ηεn ) ≤ dε +

.

1 . n

(6.59)

Proceeding as in Theorem 6.3, we deduce that unε → u∗ε weak* in L∞ (Q),

.

vεn → vε∗ weak* in L∞ (Σ),

ηεn → ηε∗ weak* in L∞ (Q), as n → ∞, and .u∗ε ∈ K1 , vε∗ ∈ K2 , ηε∗ ∈ K[−1,1] . For the corresponding state, we infer that θεn → θε∗ weakly in L2 (0, T ; V ) ∩ W 1,2 (0, T ; V ' ) and strongly in L2 (Q), n ∗ 2 2 1,2 (0, T ; L2 (Ω)), .ϕ → ϕ ε ε weakly in L (0, T ; H (Ω)) ∩ W weak* in L∞ (0, T ; V ), and strongly in L2 (0, T ; V ), and ηεn θεn → ηε∗ θε∗ weakly in L2 (Q), as n → ∞.

.

Following the same arguments as in Theorem 6.3, we deduce that 1 θεn (ϕεn )3 . β(θ|εn ) β(θεn )|Γ

1 θε∗ → (ϕε∗ )3 → β(θε∗ ) | → β(θε∗ )|Γ →

weakly in L2 (0, T ; V ), as n → ∞, weakly in L2 (Q), as n → ∞, weakly in L2 (0, T ; V ), as n → ∞, weakly in L2 (0, T ; L2 (Γ )), as n → ∞,

164

6 An Optimal Control Problem for a Phase Transition Model

and .(θε∗ , ϕε∗ ) is a solution to (6.1)–(6.6) corresponding to (.u∗ε , vε∗ ). Since j is Lipschitz continuous and .θεn converges strongly to .θε∗ in .L2 (Q), we have j (θεn ) → j (θε∗ ) strongly in L2 (Q), as n → ∞.

.

We pass to the limit in{ (6.59) and exploit in the third term of .Jε the weak lower semicontinuity of . Q j (θεn )dx. We get .Jε (u∗ε , vε∗ , ηε∗ ) = dε . In conclusion, ∗ ∗ ∗ ∗ ∗ .(uε , vε , ηε ) and the corresponding state .(θε , ϕε ) are optimal in .(Pε ). n The next theorem proves the convergence of .(Pε ) to .(P ) as .ε → 0, showing thus that .(Pε ) is an appropriate approximation of .(P ). Theorem 6.5 Let hypotheses of Theorem 6.3 hold and .{(u∗ε , vε∗ , ηε∗ ), (θε∗ , ϕε∗ )} be optimal in .(Pε ). Then, we have that u∗ε → u∗ weak* in L∞ (Q), as ε → 0, .

(6.60)

vε∗ → v ∗ weak* in L∞ (Σ), as ε → 0, .

(6.61)

ηε∗ → η∗ weak* in L∞ (Q), as ε → 0,

(6.62)

.

θε∗ → θ ∗

.

weakly in L2 (0, T ; V ) ∩ W 1,2 (0, T ; V ' )

.

(6.63) and strongly in L2 (Q), ϕε∗ → ϕ ∗

weakly in L2 (0, T ; H 2 (Ω)) ∩ W 1,2 (0, T ; L2 (Ω)), (6.64) weak* in L∞ (0, T ; V ), and strongly in L2 (0, T ; V ),

where .(θ ∗ , ϕ ∗ ) is the solution to (6.1)–(6.6) corresponding to .(u∗ , v ∗ , η∗ ) and the pair .{(u∗ , v ∗ , η∗ ), (θ ∗ , ϕ ∗ )} is optimal in .(P ). Proof Let .{(u∗ε , vε∗ , ηε∗ ), (θε∗ , ϕε∗ )} be optimal in .(Pε ). Then, this pair satisfies Jε (u∗ε , vε∗ , ηε∗ ) ≤ Jε (u, v, η),

.

for any .(u, v, η) ∈ K1 × K2 × K[−1,1] . In particular, we set .u = u, v = v, η = η, θ, ϕ solving (6.1)– where .(u,v, η) is a solution to (.P ) with the corresponding state .(6.6). This entails that .η ∈ S(θ )= ∂j ( θ ) a.e. in .Q, η ∈ K3 . The previous inequality reads

6.3 Approximating Control Problem (Pε )

λ1 2 .

(

165

( λ2 (θε∗ − θf )2 dxdt + (ϕε∗ − ηε∗ )2 dxdt 2 Q Q ( 1 + (j (θε∗ ) + ηε∗ θc − ηε∗ θε∗ )dxdt ε Q( ( λ1 λ2 ≤ (θ − θf )2 dxdt + (ϕ −η)2 dxdt 2( Q 2 Q 1 + (j (θ) + ηθc − ηθ )dxdt. ε Q

(6.65)

By (6.56), we see that the last term on the right-hand side is actually zero, and so the right-hand side is bounded by a constant. In view of (6.11)–(6.12) and (6.58), by the boundedness of the optimal controllers .(u∗ε , vε∗ , ηε∗ ), we obtain (6.60)–(6.62). By (6.65), we obtain the boundedness of .θε∗ and .ϕε∗ in .L2 (Q). By Theorem 6.2, in particular using (6.22)–(6.23), we deduce (6.63)–(6.64). Then, writing the weak formulations (6.20)–(6.21) for the approximating state and passing to the limit as ∗ ∗ .ε → 0, we deduce that .(θ , ϕ ) is the solution to (6.1)–(6.6) corresponding to ∗ ∗ ∗ .(u , v , η ). It remains to show that .η∗ ∈ S(θ ∗ ) a.e. in .Q. We set ( .ζε = (j (θε∗ ) + ηε∗ θc − ηε∗ θε∗ )dxdt. Q

By (6.65), we observe that 0 ≤ γε :=

.

1 ζε ≤ C ε

for some constant C (independent of .ε). Thus, .ζε = εγε → 0, as .ε → 0. On the other hand, passing to the limit, we infer that (

(j (θ ∗ ) + η∗ θc − η∗ θ ∗ )dxdt ≤ lim ζε = 0.

0≤

.

ε→0

Q

We obtain that .j (θ ∗ (t, x))+η∗ (t, x)θc −η∗ (t, x)θ ∗ (t, x) = 0 a.e. .(t, x) ∈ Q, which by (6.56), implies that η∗ (t, x) ∈ ∂j (θ ∗ (t, x)) = S(θ ∗ (t, x)) a.e. (t, x) ∈ Q.

.

Finally, we pass to the limit in (6.65) as .ε → 0 and obtain by (6.13) J (u∗ , v ∗ , η∗ ) ≤ J (u,v, η),

.

for any .(u,v, η) ∈ K1 × K2 × K3 , with .(θ, ϕ ) solution to (6.1)–(6.6). This shows that .{(u∗ , v ∗ , η∗ ), (θ ∗ , ϕ ∗ )} is optimal in .(P ). n

166

6 An Optimal Control Problem for a Phase Transition Model

6.4 Optimality Conditions In this section, we compute the optimality conditions for the problem .(Pε ). We prove that any triplet of optimal controllers .(u∗ε , vε∗ , ηε∗ ) is represented by the expressions given in the further Theorem 6.11. However, this result is not immediately proved and needs some intermediate arguments because the functions j and .j ∗ are not regular. A second approximation introduced in order to regularize them is helpful for the determination of the optimality conditions. We recall some definitions and results beginning with (6.53) and its the Moreau regularization defined by } |r − s|2 + j (s) , for any r ∈ R, σ > 0. .jσ (r) = inf 2σ s∈R {

(6.66)

The function .jσ is convex, Lipschitz continuous along with its derivative, it can be still expressed as jσ (r) =

.

|2 1 || | |(I + σ S)−1 r − r | + j ((I + σ S)−1 r), 2σ

(6.67)

where I is the identity on .R, and has the properties 0 ≤ jσ (r) ≤ j (r) and lim jσ (r) = j (r), for any r ∈ R

.

σ →0

(6.68)

(Theorem 7.34 in the Appendix).

6.4.1 The Second Approximating Control Problem (Pε,σ ) Let .(u∗ε , vε∗ , ηε∗ ) be optimal in .(Pε ). Following the technique introduced in [12], we state the approximating penalized problem Minimize Jε,σ (u, v, η) for all (u, v, η) ∈ K1 × K2 × K[−1,1] ,

.

subject to (6.1)–(6.6), where Jε,σ (u, v, η) =

.

λ1 2

1 + ε +

1 2

(

( (θ − θf )2 dxdt + Q

λ2 2

( (ϕ − η)2 dxdt Q

1 (jσ (θ ) + ηθc − ηθ )dxdt + 2 Q

( Σ

(v − vε∗ )2 dσ dt +

1 2

( Q

( Q

(u − u∗ε )2 dxdt

(η − ηε∗ )2 dxdt.

(Pε,σ )

6.4 Optimality Conditions

167

One can easily see, by recalling Proposition 6.4, that problem .(Pε,σ ) has at least one solution. It remains to show that .(Pε,σ ) appropriately approximates .(Pε ), and we formally indicate this by .(Pε,σ ) → (Pε ) as .σ → 0. We shall see that due to the penalization technique represented by the inclusion of the last three terms in .Jε,σ , it is possible ∗ , η∗ ) to prove a strongly convergence of the optimal control sequence .(u∗ε,σ , vε,σ ε,σ ∗ ∗ ∗ precisely to a desired optimal control .(uε , vε , ηε ) in .(Pε ). Proposition 6.6 Under the hypotheses of Theorem 6.3, let .(u∗ε , vε∗ , ηε∗ ) be optimal ∗ , η∗ ) be optimal in .(P in .(Pε ) and .(u∗ε,σ , vε,σ ε,σ ). Then, we have that ε,σ u∗ε,σ → u∗ε weak* in L∞ (Q) and strongly in L2 (Q), as σ → 0, .

(6.69)

∗ vε,σ → vε∗ weak* in L∞ (Σ) and strongly in L2 (Σ), as σ → 0, .

(6.70)

∗ ηε,σ → ηε∗ weak* in L∞ (Q) and strongly in L2 (Q), as σ → 0,

(6.71)

.

∗ , ϕ ∗ ) converge to the optimal states .(θ ∗ , ϕ ∗ ) and the corresponding states .(θε,σ ε,σ ε ε ∗ ∗ ∗ that correspond to .(uε , vε , ηε ) in .(Pε ). ∗ , η∗ ) is optimal in .(P Proof We write that .(u∗ε,σ , vε,σ ε,σ ), that is, ε,σ ( ( λ1 λ 2 ∗ 2 . (θ ∗ − θf )2 dxdt + (ϕ ∗ − ηε,σ ) dxdt 2 Q ε,σ 2 Q ε,σ

+ + ≤

1 ε 1 2

( Q

( Q

1 + 2

∗ ∗ ∗ ∗ (jσ (θε,σ ) + ηε,σ θc − ηε,σ θε,σ )dxdt

(u∗ε,σ − u∗ε )2 dxdt +

(

λ1 2

(θ − θf )2 dxdt + Q

( Q

(6.72)

(u − u∗ε )2 dxdt

1 + 2

(

1 2

λ2 2

Σ

∗ (vε,σ − vε∗ )2 dσ dt +

( (ϕ − η)2 dxdt + Q

( (v Σ

− vε∗ )2 dσ dt

1 + 2

1 ε

1 2

( Q

∗ (ηε,σ − ηε∗ )2 dxdt

( (jσ (θ ) + ηθc − ηθ )dxdt Q

( Q

(η − ηε∗ )2 dxdt,

for all .u ∈ K1 , v ∈ K2 , η ∈ K[−1,1] , with .(θ, ϕ) denoting the corresponding solution to (6.1)–(6.6). In particular, we set .u = u∗ε , v = vε∗ , η = ηε∗ in (6.72), and so we can consider the corresponding solutions .θ = θε∗ , ϕ = ϕε∗ to (6.1)–(6.6) as well. Since the last three terms on the right-hand side of (6.72) vanish and .jσ (θε∗ ) ≤ j (θε∗ ) a.e. in Q, thanks to (6.68), it follows that the left-hand side is bounded independently of .σ. Consequently, by selecting subsequences (still denoted by the subscript .σ ), we get

168

6 An Optimal Control Problem for a Phase Transition Model

.

∗ → v weak* in L∞ (Σ), u∗ε,σ → uε weak* in L∞ (Q), vε,σ ε ∗ ∞ ηε,σ → ηε weak* in L (Q), as σ → 0,

(6.73)

and using the estimates (6.22)–(6.23) for the state system, we have, as .σ → 0, ∗ θε,σ → θε weakly in L2 (0, T ; V ) ∩ W 1,2 (0, T ; V ' ).

(6.74)

.

and strongly in L2 (Q), ∗ ϕε,σ → ϕε weakly in L2 (0, T ; H 2 (Ω)) ∩ W 1,2 (0, T ; L2 (Ω)),

(6.75)

weak* in L∞ (0, T ; V ), and strongly in L2 (0, T ; V ), where .(θε , ϕε ) is the solution to (6.1)–(6.6) corresponding to .(uε , vε , ηε ). Now we pass to the limit in (6.72) as .σ → 0. First, we assert that (

( j (θε )dxdt ≤ lim inf

.

σ →0

Q

Q

∗ jσ (θε,σ )dxdt,

(6.76)

∗ . Indeed, by (6.67), taking into account that j is where .θε is the limit of .θε,σ nonnegative, we have

1 . 2σ

( ( | |2 | −1 ∗ ∗ | ∗ jσ (θε,σ )dxdt ≤ constant, |(I + σ ∂j ) θε,σ − θε,σ | dxdt ≤ Q

Q

which implies that .

|| || || ∗ ∗ || lim ||(I + σ ∂j )−1 θε,σ − θε,σ ||

σ →0

L2 (Q)

= 0,

where I denotes the identity on .L2 (Q). Therefore, we deduce that ∗ (I + σ ∂j )−1 θε,σ → θε strongly in L2 (Q) as σ → 0.

.

(6.77)

Next, again by (6.67), we can infer that (

( j (θε )dxdt = lim

.

Q

σ →0 Q

∗ j ((I + σ ∂j )−1 θε,σ )dxdt

( ≤ lim inf σ →0

Q

∗ jσ (θε,σ )dxdt

by the Lipschitz continuity of j and (6.77). Then, passing to the limit in (6.72) as σ → 0, we get

.

6.4 Optimality Conditions

.

λ1 2

( (θε − θf )2 dxdt + Q

+

1 2

( Q

169

λ2 2

( (ϕε − ηε )2 dxdt + Q

(uε − u∗ε )2 dxdt +

1 2

( Σ

1 ε

( (j (θε ) + ηε θc − ηε θε )dxdt Q

(vε − vε∗ )2 dσ dt +

1 2

( Q

(ηε − ηε∗ )2 dxdt

( ( λ1 λ2 (θε∗ − θf )2 dxdt + (ϕ ∗ − ηε∗ )2 dxdt 2 Q 2 Q ε ( ( λ1 1 ∗ ∗ ∗ ∗ (j (θε ) + ηε θc − ηε θε )dxdt ≤ (θε − θf )2 dxdt + ε Q 2 Q ( ( 1 λ2 (ϕε − ηε )2 dxdt + (j (θε ) + ηε θc − ηε θε )dxdt. + 2 Q ε Q ≤

The second inequality can be written because .(u∗ε , vε∗ , ηε∗ ) is optimal in .(Pε ). Hence, we observe that uε = u∗ε , vε = vε∗ , ηε = ηε∗ a.e. in Q,

.

which implies .θε = θε∗ and .ϕε = ϕε∗ a.e. in .Q. Going back to (6.72), it follows that the convergences in (6.73) hold for the whole sequences, and moreover, u∗ε,σ → u∗ε strongly in L2 (Q),

.

∗ vε,σ → vε∗ strongly in L2 (Σ),

∗ ηε,σ → ηε∗ strongly in L2 (Q), as σ → 0,

which is the direct consequence of the penalization method used. The proof is ended. n

6.4.2 Optimality Conditions for (Pε,σ ) For a later use, we give an intermediate result with a quite technical proof, which will be applied for proving the existence of the solutions to the first-order variation system and of the dual system. We begin with the proof of the well-posedness of a generic problem Wt − a(t, x)ΔW + b(t, x)Φ = ω(t, x), in Q,

.

Φt − ΔΦ + c(t, x)Φ + d(t, x)Wt = g(t, x), in Q,

.

.

−

∂W = α(x)(W − γ (t, x)), on Σ, ∂ν

(6.78) (6.79) (6.80)

170

6 An Optimal Control Problem for a Phase Transition Model

∂Φ = 0, on Σ, ∂ν

(6.81)

W (0) = 0, Φ(0) = 0, in Ω.

(6.82)

.

.

Proposition 6.7 Let us assume the following conditions: .

a, b, c, d ∈ L∞ (Q), 0 < a0 ≤ a(t, x) ≤ ||a||L∞ (Q) =: ||a||∞ , a.e. (t, x) ∈ Q,

(6.83)

ω, g ∈ L2 (Q), γ ∈ W 1,2 (0, T , L2 (Γ )), α satisfies (6.7).

(6.84)

.

Then, the problem (6.78)–(6.82) has a unique solution .(W, Φ) with W ∈ L∞ (0, T ; V ) ∩ W 1,2 (0, T ; L2 (Ω)),

.

Φ ∈ L2 (0, T ; H 2 (Ω)) ∩ L∞ (0, T ; V ) ∩ W 1,2 (0, T ; L2 (Ω)). If .γ ≡ 0 in addition, we have that W ∈ L2 (0, T ; H 2 (Ω)).

.

(6.85)

Proof For proving the existence of the solution to (6.78)–(6.82), we split the proof into two parts. We assume first that all parameters are regular, define an appropriate operator and show that it satisfies the hypotheses of Lions’ theorem (see, e.g., [13], p. 177, Theorem 4.17), which ensures the existence of a unique solution, and determine some appropriate estimates. Next, we consider the system with regularized parameters and get an approximating solution owing to the previous results. The estimates and compactness properties allow us to pass to the limit and deduce that (6.78)–(6.82) has a solution, as claimed. Step 1. Assume first that all parameters .a, b, c, d ∈ C ∞ (Q). Let us replace .Wt into (6.79) from (6.78). We get Φt − ΔΦ + (c − bd)Φ + adΔW = g1 , in Q,

.

(6.86)

where .g1 = g − dω ∈ L2 (Q). Further, we focus on the system (6.78), (6.86), (6.80)–(6.82). We recall that .V = H 1 (Ω) with the norm (6.15), which is equivalent to the norm (6.16). We consider the product space .V = V ×V with the dual .V' = V ' ×V ' and define the pairing .

V' ,V = V ' ,V + δ V ' ,V ,

6.4 Optimality Conditions

171

for .(θ1 , θ2 ) ∈ V' , (ψ1 , ψ2 ) ∈ V, where the positive constant .δ will be later specified. We define the operator A(t) : V → V'

.

by / .

(

) ( W Φ

A(t)

)\ ψ1 ψ2

,

(6.87)

V' ,V

= V ' ,V + V ' ,V ( ( = (a∇W · ∇ψ1 + ψ1 ∇a · ∇W + bΦψ1 )dx + aα(W − γ )ψ1 dσ Ω

Γ

( +δ ( −δ

Ω

Γ

(∇Φ · ∇ψ2 + (c − bd)Φψ2 − ψ2 ∇W · ∇(ad) − ad∇W · ∇ψ2 )dx adαψ2 (W − γ )dσ

and introduce the Cauchy problem (

) Wt Φt

.

(

) ( ) W ω(t) (t) = , a.e. t ∈ (0, T ), Φ g1 (t) ( ) ( ) W 0 = . Φ 0

(t) + A(t)

(6.88)

We prove that .A is quasi-monotone. To this end, we denote .W = W1 − W2 , Φ = Φ1 − Φ2 and calculate / .E

(

: = A(t)

) ( W Φ

,

)\ W Φ

( ≥

Ω

+δ

V' ,V

(a |∇W |2 + W ∇a · ∇W + bΦW )dx + a0 αm ( Ω

( W 2 dσ Γ

(|∇Φ|2 + (c − bd)Φ 2 − Φ∇W · ∇(ad) − ad∇W · ∇Φ)dx

−δ ||a||∞ ||d||∞ αM

( ΦW dσ. Γ

By applying the Hölder’s and Young’s relations in an appropriate way, the trace theorem, and making some convenient arrangements of the terms, we obtain .E

(( ( ) ) ( 6 ||a||∞ ||d||∞ δ ||Φ||2V − C1 ≥ C ||W ||2V + δ 1 − W 2 dx − Φ 2 dx . a0 Ω Ω

172

6 An Optimal Control Problem for a Phase Transition Model

Now, for .δ chosen sufficiently small such that .1

−

6 ||a||∞ ||d||∞ δ ≥ δ0 > 0, a0

we obtain that / .

(

) ( W Φ

A(t)

,

)\ W Φ

'

V ,V

≥ C(W, Φ)2V − C2 (W, Φ)2L2 (Ω)×L2 (Ω) ,

which shows that .A is quasi-monotone from .V to .V' . Next, we show that .A is coercive, and for that, we set .(ψ1 , ψ2 ) = (W, Φ) in (6.87).. By similar operations as before, we obtain /

(

) (

)\ W A(t) , Φ . V' ,V ) (( ( ( ) 2 2 6 2 2 ≥ C ||W ||V + δ 1 − a0 ||a||∞ ||d||∞ δ ||Φ||V − C3 W dx − Φ dx − C4 . W Φ

Ω

Ω

Again, for .δ small enough, we obtain the coercivity of .A. In the above relations, i = 1, . . . , 5 may depend on the gradients of the parameters .a, b, c, d assumed to be in .C ∞ (Q), while C depends on .a0 and .αm . Finally, it is obvious that .A is continuous from .V to .V' . Indeed, if we take .(Wn , Φn )n such that .(Wn , Φn ) → (W, Φ) strongly in .V, we see by (6.87) that .Ci ,

( .A(t)

) Wn Φn

( →A

) W Φ

strongly in V' .

Consequently, we can apply Lions’ theorem and find that the Cauchy problem (6.88) has a unique solution .(W, Φ)

∈ L2 (0, T ; V) ∩ W 1,2 (0, T ; V' ).

(6.89)

Moreover, we assert that .

W ∈ W 1,2 (0, T ; L2 (Ω)) ∩ L∞ (0, T ; V ), ΔW ∈ L2 (0, T ; L2 (Ω)), Φ ∈ W 1,2 (0, T ; L2 (Ω)) ∩ L∞ (0, T ; V ) ∩ L2 (0, T ; H 2 (Ω)).

(6.90)

We prove this assertion first for .W, by considering the equation .

1 1 Wt − ΔW = (t, x) (ω(t, x) − b(t, x)Φ) , a(t, x) a

(6.91)

6.4 Optimality Conditions

173

with the Robin boundary condition (6.80) and .W (0) = 0. A rigorous proof is led by considering the system of finite differences of this equation (see, e.g., the proof of this part in [38]). Here, we proceed formally and multiply (6.91) by .Wt (t) and integrate over .Ω and then over .(0, t). We have 1 ||a||∞ .

(

( 1 αm ||Wτ (τ )||2L2 (Ω) dτ + ||∇W (t)||2L2 (Ω) + W 2 (t)dσ 2( ( 2 Γ 0 ( ( t t − αγ Wτ dσ dτ ≤ (ω − bΦ)Wτ dxdτ. t

0

0

Γ

Ω

By using the trace theorem (with the constant .cT r ) and the fact that .γ ∈ W 1,2 (0, T ; L2 (Γ )), we can write (

( t(

αγ Wτ dσ dτ =

.

0

Γ

≤

αm 4

αγ (t)W (t)dσ − Γ

( (

(

Γ t

Γ

α 2 γτ2 (t)dσ

Γ

||W (τ )||2H 1 (Ω) dσ

0

αγτ W dσ dτ 0

W 2 (t)dσ + C1

+cT2 r

( t(

(

+ C2 0

t

||αγτ (τ )||2L2 (Γ ) dτ.

Then, we have .

1 ||a|| 2 ∞

(

t

α1 ||W (t)||2V (6.92) 4 0 ( t ( t ||Φ||2L2 (Ω) dτ + cT2 r ||W (τ )||2H 1 (Ω) dσ + C2 ≤ C + C1 ||Wτ (τ )||2L2 (Ω) dτ +

0

0

for all .t ∈ [0, T ], where .α1 = min{1, αm }. By (6.91), it follows that .ΔW ∈ L2 (0, T ; L2 (Ω) and that (

t

.

0

||ΔW (τ )||2L2 (Ω) dτ

( ) ( t 2 ||Φ||L2 (Ω) dτ . ≤C 1+

(6.93)

0

Here, .C, C1 , C2 denote constants that may change from line to line, depending on the norms .||a||∞ , ||b||∞ , a0 , αm , ||ω||L2 (Q) , ||γ ||W 1,2 (0,T ;L2 (Γ )) . Now, we multiply (6.86) by .Φ(t) and integrate over .Ω and .(0, t). After a few calculations, using (6.93), we get .

( t 1 ||Φ(t)||2L2 (Ω) + ||∇Φ(τ )||2L2 (Ω) dτ 2 0 (( t ) ( t ||g1 (τ )||2L2 (Ω) dτ + ||ΔW (τ )||2L2 (Ω) dτ ≤C (

0

(

t

≤C 1+ 0

) ||Φ||2L2 (Ω) dτ .

0

174

6 An Optimal Control Problem for a Phase Transition Model

Applying Gronwall’s lemma, we obtain ( . ||Φ(t)|| 2 L (Ω)

2

+ 0

t

||∇Φ(τ )||2L2 (Ω) dτ ≤ C,

(6.94)

with C a constant depending on the .L∞ -norms of .a, b, c, d, a0 , αm , αM , .||ω||L2 (Q) , ||g||L2 (Q) , ||γ ||W 1,2 (0,T ;L2 (Γ )) . Proceeding by the same argument as for W , we still obtain that (

t

.

0

( ||ΔΦ(τ )||2L2 (Ω) dτ + ||Φ(t)||2V +

t

0

||Φτ (τ )||2L2 (Ω) dτ ≤ C,

(6.95)

for all .t ∈ [0, T ]. Then, .ΔΦ ∈ L2 (0, T ; L2 (Ω)) together with (6.81) implies that 2 2 .Φ ∈ L (0, T ; H (Ω)). By using (6.94) in (6.92) and (6.93), we deduce that ( .

0

t

( ||Wτ (τ )||2L2 (Ω) dτ + ||W (t)||2V +

t 0

||ΔW (τ )||2L2 (Ω) dτ ≤ C

(6.96)

for all .t ∈ [0, T ]. Step 2. Let us regularize the parameters .a, b, c, d by using a mollifier and defining for .ε > 0 .eε

= e ∗ ρε , e = a, b, c, d,

where .ρε was defined in Lemma 5.3 in Chap. 5. We consider system (6.78), (6.86), (6.80)–(6.82), equivalently problem (6.88) with the regularized parameters .aε , bε , cε , dε . Then, this regularized system has a solution .Wε

∈ L∞ (0, T ; V ) ∩ W 1,2 (0, T ; L2 (Ω)), ΔWε ∈ L2 (0, T ; L2 (Ω)), .

Φε ∈ L (0, T ; H (Ω)) ∩ W 2

2

1,2

∞

(0, T ; L (Ω)) ∩ L (0, T ; V ), 2

(6.97) (6.98)

satisfying estimates (6.94)–(6.96), in which the constants .C do not depend on .ε. Thus, we can select a subsequence such that Wε → W weakly in W 1,2 (0, T ; L2 (Ω)) and weak* in L∞ (0, T ; V ), . Φε → Φ weakly in W 1,2 (0, T ; L2 (Ω)) ∩ L2 (0, T ; H 2 (Ω)) and weak* in L∞ (0, T ; V ).

By Aubin–Lions–Simon lemma, we deduce that .

Wε → W strongly in L2 (0, T ; L2 (Ω)), Φε → W strongly in L2 (0, T ; V ),

6.4 Optimality Conditions

175

and so, by the trace theorem, we have

.

Wε |Γ → W |Γ weakly in L2 (0, T ); H 1/2 (Γ )), ∂Wε ∂W → weakly in L2 (0, T ); L2 (Γ )). ∂ν ∂ν

Moreover, .aε

→ a strongly in Lp (Q), for all p ≥ 1,

and we have similar convergences for the other parameters. Then, if a sequence is bounded in .L2 (0, T ; L2 (Ω)), we have on a subsequence

.(fε )ε

.aε fε

→ af weakly in L1 (Q), as ε → 0,

where f is the weak limit of .fε . As a matter of fact, it still follows that .aε fε

→ af weakly in Lq (Q), for 1 ≤ q < 2,

because by Hölder’s inequality we have q

. ||aε fε || q L (Q)

( =

|aε fε |q dxdt Q

)q/2 ((

(( |fε |2 dxdt

≤

2

) 2−q

|aε | 2−q dxdt

Q

2

Q

≤ C.

This applies to .aε ΔWε , bε Φε , cε Φε , dε (Wε )t . On the basis of the previous convergences, we can pass to the limit as .ε → 0 in (6.78), (6.86), (6.80)– (6.82) written for the approximating solution and obtain a weak limit in .L1 (Q). Consequently, it follows that the linear parabolic problem (6.78)–(6.82) has a unique solution, as stated in Proposition 6.7. The uniqueness follows by a standard technique. We write (6.78)–(6.82) for two solutions .(W 1 , Φ 1 ), (W 2 , Φ 2 ), multiply Eq. (6.78) of their difference divided by a by (.W 1 − W 2 )t , and integrate over .Q. After a few standard computations, we obtain ( .

0

t

|| || || ||2 ||(W 1 − W 2 )(τ )||2 2 dτ + ||∇(W 1 − W 2 )(t)||L2 (Ω) L (Ω) (

| | |(W 1 − W 2 )(t)|2 dσ ≤ C1

+ Γ

( 0

t

|| || ||(Φ 1 − Φ 2 )(τ )||2 2 dτ L (Ω)

for some constant .C1 depending only on the data in (6.83)–(6.84).

(6.99)

176

6 An Optimal Control Problem for a Phase Transition Model

Next, we multiply the difference of Eq. (6.79) written for those two solutions by − Φ2 ) and integrate over Q. Hence, it is a standard matter to infer that

.(Φ1

||(Φ1 − Φ2 )(t)||2L2 (Ω)

.

((

t

≤ C2

(6.100)

|| || ||(Φ 1 − Φ 2 )(τ )||2 2

L (Ω)

0

( dτ +

t

| | |(W 1 − W 2 )(τ )|2 dτ

)

0

for some positive constant .C2 . Then, combining (6.100) with (6.99), we obtain ( .

||(Φ1 − Φ2 )(t)||2L2 (Ω) ≤ C3

0

t

|| || ||(Φ 1 − Φ 2 )(τ )||2 2 dτ, L (Ω)

(6.101)

for all t ∈ [0, T ],

which leads to the uniqueness, by the Gronwall’s lemma.

n

Next, we assume the further regularity conditions (6.25) and (6.28), besides (6.18), in order we can take advantage of uniform .L∞ estimates for both components of a solution to (6.1)–(6.6). Let us go back to the computation of the optimality conditions for .(Pε,σ ). Let ∗ ∗ ∗ ∗ ∗ .(uε,σ , vε,σ , ηε,σ ) and .(θε,σ , ϕε,σ ) be optimal in .(Pε,σ ), and let .λ ∈ (0, 1). We introduce the variations uλε,σ = (1 − λ)u∗ε,σ + λu = u∗ε,σ + λu, u ∈ K1 ,

.

λ ∗ ∗ = (1 − λ)vε,σ + λv = vε,σ + λv , u ∈ K2 , vε,σ λ ∗ ∗ = (1 − λ)ηε,σ + λη = ηε,σ + λη, u ∈ K[−1,1] , ηε,σ

where ∗ ∗ u = u − u∗ε,σ , v = v − vε,σ , η = η − ηε,σ .

.

(6.102)

λ , ηλ ) has a Let us observe that system (6.1)–(6.6) corresponding to .(uλε,σ , vε,σ ε,σ λ , ϕ λ ), and unique solution .(θε,σ ε,σ λ ∗ λ ∗ θε,σ → θε,σ , ϕε,σ → ϕε,σ strongly in L2 (Q), as λ → 0.

.

(6.103)

This can be obtained by the estimate (6.24) combined with weak* compactness. We set -λ = Θ

.

λ − θ∗ ∗ θε,σ ϕ λ − ϕε,σ ε,σ -λ = ε,σ , Φ λ λ

(6.104)

6.4 Optimality Conditions

177

and claim that .

-λ → Φ weakly in L2 (Q), as λ → 0, -λ → Y and Φ Θ

(6.105)

where the limits Y and .Φ solve the system in variations ∗ Yt − Δ(β ' (θε.σ )Y ) + Φt = u, in Q, .

.

∗ 2 Φt − ΔΦ + (3(ϕε,σ ) − 1)Φ =

−

1 ∗ )2 (θε,σ

Y, in Q, .

∂ ' ∗ ∗ )Y ) − v ), on Σ, . (β (θε.σ )Y ) = α(x)(β ' (θε.σ ∂ν ∂Φ = 0, on Ω, . ∂ν Y (0) = 0, Φ(0) = 0, in Ω.

(6.106) (6.107) (6.108) (6.109) (6.110)

The proof of (6.105) is done in Proposition 6.8 below. First, we define a (very weak) solution to (6.106)–(6.110) as a pair of functions .Y ∈ L2 (Q), Φ ∈ L2 (0, T ; V ) that satisfies the system ( .

−

( Y ψt dxdt −

β

Q

Q

'

∗ (θε,σ )Y Δψdxdt

uψdxdt + Q

(6.111)

αv ψdσ dt, (

Φ(ψ1 )t dxdt +

∇Φ · ∇ψ1 dxdt +

Q

Φψt dxdt. Q

Σ

(

(

−

(

( =

−

(

Q

Q

∗ 2 3(ϕε,σ ) − 1)Φψ1 dxdt

(6.112) ( =

1 ∗ 2 Q (θε,σ )

Y ψ1 dxdt,

for all .ψ ∈ L2 (0, T ; H 2 (Ω)) ∩ W 1,2 (0, T ; L2 (Ω)), which solves the problem ψt + Δψ = −fQ , in Q,

.

∂ψ + αψ = 0, on Σ, ∂ν

(6.113)

ψ(T ) = 0, in Ω, for a generic .fQ ∈ L2 (Q), and for all .ψ1 ∈ L2 (0, T ; V ) ∩ W 1,2 (0, T ; L2 (Ω)) such that .ψ1 (T ) = 0.

178

6 An Optimal Control Problem for a Phase Transition Model

Proposition 6.8 Assume (6.50), (6.7), (6.25), and (6.28). Then the problem (6.106)–(6.110) has a unique solution .(Y, Φ) Y ∈ L2 (0, T ; L2 (Ω)),

(6.114)

.

Φ ∈ L2 (0, T ; H 2 (Ω)) ∩ L∞ (0, T ; V ) ∩ W 1,2 (0, T ; L2 (Ω)),

.

(6.115)

and the convergence properties in (6.105) hold. Proof First, we prove the existence and uniqueness of the solution to (6.106)– (6.110). Due to the hypotheses (6.28), we infer that the state system (6.1)–(6.6) ∗ ∗ , ϕ ∗ ) with both .θ for .u = u∗ε,σ and .v = vε,σ has the solution .(θε,σ ε,σ and ε,σ ∗ ∞ ∗ ∞ .1/θε,σ bounded in .L (Q), by (6.31). By (6.8), we see that .β(θε,σ ) ∈ L (Q). ∗ ∞ Also, relying on (6.25), we deduce by (6.27) the boundedness of .ϕε,σ in .L (Q) ∩ L∞ (0, T ; H 2 (Ω)). We integrate (6.106) and (6.108) with respect to .τ on .(0, t). We obtain ( t ( t ∗ Y (t, x) − Δ β ' (θε,σ (τ, x))Y (τ, x)dτ + Φ(t, x) = u(τ, x)dτ, 0 0 ( t ( t . ∂ ∗ ∗ (β ' (θε,σ (τ, x))Y (τ, x))dτ = α(x) (β ' (θε,σ (τ, x))Y − v (τ, x))dτ. − ∂ν 0 0 Then, we set (

t

W (t, x) :=

.

0

∗ (β ' (θε,σ (τ, x))Y (τ, x)dτ, (t, x) ∈ Q,

so that ∗ Wt (t, x) = β ' (θε,σ (t, x))Y (t, x), (t, x) ∈ Q.

.

(6.116)

Now, the system (6.106)–(6.110) can be replaced by ( .

) ( t 1 u(τ, x)dτ, in Q, . W − ΔW + Φ (t, x) = ∗ ) t β ' (θε,σ 0 ∗ Φt − ΔΦ + (3ϕε,σ − 1)Φ =

1

(6.117)

Wt , in Q, .

(6.118)

) ( ( t ∂W v (τ, x)dτ , on Σ, . (t, x) = α(x) W (t, x) − − ∂ν 0

(6.119)

∂Φ = 0, on Σ, . ∂ν

(6.120)

∗ )2 β ' (θ ∗ ) (θε,σ ε,σ

W (0) = 0, Φ(0) = 0, in Ω.

(6.121)

6.4 Optimality Conditions

179

At this point, we can apply Proposition 6.7 with the following choice: ∗ a(t, x) = b(t, x) = β ' (θε,σ (t, x)), ( t ∗ (t, x)) u(τ, x)dτ, ω(t, x) = β ' (θε,σ .

0

∗ c(t, x) = (3ϕε,σ − 1),

d(t, x) = −

1 , g(t, x) = 0, ∗ )2 β ' (θ ∗ ) (θε,σ ε,σ

( γ (t, x) =

t

v(τ, x)dτ, (t, x) ∈ Σ.

0

These coefficients satisfy (6.83)–(6.84); we conclude that (6.117)–(6.121) have a unique solution .(W, Ψ ) with W in the space .L∞ (0, T ; V ) ∩ W 1,2 (0, T ; L2 (Ω)). Then, owing to (6.116), it turns that (6.114) holds. Next, we have to show that if .(Y, Φ) fulfils the variational equalities in (6.111) and (6.112), then the pair .(W, Φ) with W specified by (6.116) solves the system (6.117)-(6.121). Indeed, taking an arbitrary .ζ such that ζ ∈ H 2 (Ω) such that

.

∂ζ + αζ = 0, on Γ, ∂ν

(6.122)

according to (6.113), we can choose .ψ(t, x) = (T − t)ζ (x), .(x, t) ∈ Q, in (6.111). Then, if .ζ also belongs to .D(Ω), integrating by parts in time, it is not difficult to recover the equality (6.117) in the sense of distributions in .Ω, for a.e. .t ∈ (0, T ). Once (6.117) is proved, we can compare the terms and find additional regularity for W (in particular, .ΔW ∈ L2 (0, T ; L2 (Ω)) in order to be able to get back to (6.111). This time we still use .ψ(t, x) = (T − t)ζ (x), but with an auxiliary function .ζ as in (6.122) to find the boundary condition (6.119) as well. A similar approach can be used in (6.112) by taking now .ψ1 (t, x) = (T −t)ζ1 (x), .(x, t) ∈ Q, with .ζ1 arbitrary first in .H01 (Ω), then in .H 1 (Ω) in order to arrive to an integrated version of (6.118) and (6.120). Then, it suffices to examine the regularity of .Φ and realize that (6.115), as well as (6.118) and (6.120), is satisfied. We prove now (6.105). As mentioned in Theorem 6.2, the solution to (6.1)–(6.6) is Lipschitz continuous with respect to the data. Relying on (6.24) and recalling Eq. (6.104), we can write .

|| λ ||2 ||Θ - || 2

L (Q)

|| λ ||2 || λ ||2 - || - || 2 + ||Φ + ||Φ ≤ ||u||2L2 (Q) + ||v ||2L2 (Σ) . C([0,T ];L2 (Ω)) L (0,T ;V )

(6.123) -λ and .Φ -λ are in the same spaces as It is obvious that, for each .λ, the functions .Θ ∗ ∗ .θε,σ and .ϕε,σ are., given by Theorem 6.2, and that they satisfy the system

180

6 An Optimal Control Problem for a Phase Transition Model

λ ) − β(θ ∗ ) β(θε,σ ε,σ -tλ = u, in Q, . +Φ λ -λ Θ λ 2 λ ∗ ∗ 2 -λ +((ϕε,σ -λ = -tλ −ΔΦ ) +ϕε,σ ϕε,σ +(ϕε,σ ) −1)Φ , in Q, . Φ λ θ∗ θε,σ ε,σ λ . t −Δ Θ

( ) λ ) − β(θ ∗ ) λ ) − β(θ ∗ ) β(θε,σ ∂ β(θε,σ ε,σ ε,σ = α(x) −v , on Σ, . − ∂ν λ λ -λ ∂Φ = 0, on Σ, . ∂ν -λ = 0, in Ω. -λ = 0, Φ Θ

(6.124) (6.125)

(6.126)

(6.127) (6.128)

By (6.123), we select a subsequence of .λ such that .

- weakly in L2 (Q), as λ → 0, -λ → Y Θ - weakly in L2 (0, T ; V ) and weak* in L∞ (0, T ; L2 (Ω)), as λ → 0. -λ → Φ Φ

Let us test (6.124) by .ψ given by (6.113) and (6.125) by .ψ1 ∈ W 1,2 (0, T ; L2 (Ω)) ∩ L2 (0, T ; V ) with .ψ1 (T ) = 0. Using the boundary conditions (6.126) and (6.127), we obtain ( .

−

-λ ψt dxdt − Θ

Q

(

λ ) − β(θ ∗ ) β(θε,σ ε,σ Δψdxdt − λ

Q

uψdxdt +

= Q

(

-λ (ψ1 )t dxdt + Φ

− Q

(

-λ ψt dxdt. Φ

Q

(6.129)

(

(

(

αv ψdσ dt, Σ

-λ · ∇ψ1 dxdt ∇Φ

(6.130)

Q

( +

Q

( =

λ 2 λ ∗ ∗ 2 -λ ψ1 dxdt ((ϕε,σ ) + ϕε,σ ϕε,σ + ϕε,σ ) − 1)Φ

-λ Θ

λ ∗ Q θε,σ θε,σ

ψ1 dxdt.

Now, we remark that .

λ ) − β(θ ∗ ) β(θε,σ ε,σ -λ , = β ' ( θ λ )Θ λ

∗ and .θ λ , where .θ λ is a measurable function taking intermediate values between .θε,σ ε,σ ∗ a.e. in Q. Moreover, due to (6.103), we have that .θ λ → θε,σ strongly in .L2 (Q) as

6.4 Optimality Conditions

181

λ → 0. Therefore, by the Lipschitz continuity of some restriction of .β to a bounded subset of .(0, +∞), we deduce that

.

∗ β ' (θ λ ) → β ' (θε,σ ) strongly in L2 (Q), as λ → 0,

.

whence .

λ ) − β(θ ∗ ) β(θε,σ ε,σ ∗ )Y weakly in L1 (Q), as λ → 0, → β ' (θε,σ λ

and also weakly in .L2 (Q) due to the boundedness of .β ' (θ λ ) in .L∞ (Q) and to the Lebesgue dominated convergence theorem. Analogously, in view of (6.103), we have that λ 2 λ ∗ ∗ 2 ∗ 2 - → (3(ϕε,σ - and ((ϕε,σ ) +ϕε,σ ϕε,σ +(ϕε,σ ) −1)Φ ) −1)Φ

.

-λ Y Θ → ∗ 2 weakly in L2 (Q), as λ → 0. λ ∗ θε,σ θε,σ (θε,σ ) Now, we can pass to the limit in (6.129)–(6.130) and find out that ( .

(

-ψt dxdt − Y

−

β

Q

Q

'

∗ (θε,σ )Y Δψdxdt

uψdxdt + Q

( −

- 1 )t dxdt + Φ(ψ

Q

(

αv ψdσ dt,

Q

= Q

Q

Σ

- · ∇ψ1 dxdt + ∇Φ

(

- t dxdt Φψ

−

(

( =

.

(

( Q

Y ∗ )2 (θε,σ

∗ 2 - 1 dxdt (3(ϕε,σ ) − 1)Φψ

ψ1 dxdt,

-, Φ - yield a solution to (6.106)–(6.110) (see (6.111)–(6.112)). which means that .Y - = Y, Φ - = Φ. Since this solution is unique, we obtain .Y n We denote 1 ∗ σ ∗ ∗ I1,ε := λ1 (θε,σ − θf ) + (ξε,σ − ηε,σ ), ε

.

.

(6.131)

σ ∗ ∗ ∗ ∗ := λ2 (ϕε,σ − ηε,σ ), ξε,σ := j ' (θε,σ ), . I2,ε

(6.132)

1 σ ∗ ∗ ∗ I3,ε := −λ2 (ϕε,σ − ηε,σ ) + (θc − θε,σ ) ε

(6.133)

σ , I σ , I σ , ξ ∗ ∈ L∞ (Q). and observe that .I1,ε ε,σ 2,ε 3,ε

182

6 An Optimal Control Problem for a Phase Transition Model

We also recall that the projection of a point .ζ on a set .[a, b] is defined by ⎧ ⎨ a, if ζ < a .Proj[a,b] (ζ ) = ζ, if ζ ∈ (a, b) ⎩ b, if ζ > b. Let us denote by .pε,σ and .qε,σ the dual variables and introduce the dual system 1 σ q = −I1,ε , in Q, . ∗ )2 ε,σ (θε,σ

(6.134)

∗ 2 σ ) − 1)qε,σ + (pε,σ )t = −I2,ε , in Q, . (qε,σ )t + Δqε,σ − (3(ϕε,σ

(6.135)

∂pε,σ ∂qε,σ + α(x)pε,σ = 0, = 0, on Σ.. ∂ν ∂ν

(6.136)

pε.σ (T ) = 0, qε,σ (T ) = 0, in Ω.

(6.137)

∗ (pε,σ )t + β ' (θε,σ )Δpε,σ +

.

Proposition 6.9 Assume (6.25) and (6.28). Then, the dual system (6.134)–(6.137) has a unique solution pε,σ , qε,σ ∈ L2 (0, T ; H 2 (Ω)) ∩ L∞ (0, T ; V ) ∩ W 1,2 (0, T ; L2 (Ω)).

.

(6.138)

Moreover, the estimates .

|| || ||(pε,σ )t ||2 2 L

||2 || + ||(pε,σ )t ||L∞ (0,T ;V ) + (Q)

(

T

0

|| || ||Δpε,σ (t)||2 2 dt ≤ C, . L (Q) (6.139)

|| || ||(qε,σ )t ||2 2

L (Q)

||2 || + ||(qε,σ )t ||L∞ (0,T ;V ) +

( 0

T

|| || ||Δqε,σ (t)||2 2 dt ≤ C L (Q)

(6.140)

hold independently of .σ > 0. Proof First, we perform the variable transformation .t ' = T − t in (6.134)–(6.137). Then, the thesis follows from Proposition 6.7, by setting in the transformed system ∗ a = β ' (θε,σ ), b = −

.

1 σ , ω = I1,ε , ∗ )2 (θε,σ

∗ 2 σ c = (3(ϕε,σ ) − 1), d = −1, g = I2,ε , and γ = 0.

The estimates (6.139)–(6.140) can be obtained by standard computations, testing (6.134) by .pε,σ (t) and (6.135) by .qε,σ (t), integrating, combining the resulting equalities, and so on. Finally, a comparison of terms in (6.134) and (6.135) yields the desired estimates also for .Δpε,σ and .Δqε,σ . n

6.4 Optimality Conditions

183

Proposition 6.10 Under the assumptions (6.25) and 6.28), the optimality conditions for .(Pε,σ ) read u∗ε,σ = ProjK1 (−pε,σ + u∗ε ), a.e. on Q, .

(6.141)

∗ vε,σ = ProjK2 (−αpε,σ + vε∗ ), a.e. on Σ, .

(6.142)

( ) ∗ = Proj[−1,1] χε,σ , a.e. on Q, ηε,σ

(6.143)

.

χε,σ =

) ( 1 1 ∗ ∗ − (θc − θε,σ ) . λ2 ϕε,σ λ2 + 1 ε

Proof One can take .ψ = pε,σ in (6.111) and .ψ1 = qε,σ in (6.112), add the equalities, and integrate by parts to obtain ( { .

−

(pε,σ )t + β Q

'

∗ (θε,σ )Δpε,σ

+

}

1 ∗ )2 (θε,σ

qε,σ

Y dxdt

( { } ∗ 2 ) − 1)qε,σ + (pε,σ )t Φdxdt (qε,σ )t + Δqε,σ − (3(ϕε,σ

−

Q

( =

( u pε,σ dxdt +

Q

αv pε,σ dσ dt. Σ

Therefore, using (6.134) and (6.135), we obtain ( .

Q

( σ I1,ε Y dxdt +

( Q

σ I2,ε Φdxdt =

( upε,σ dxdt + Q

αv pε,σ dσ dt.

(6.144)

Σ

Now, we write the optimality condition ∗ ∗ Jε,σ (u∗ε,σ , vε,σ , ηε,σ ) ≤ Jε,σ (< u,< v, < η), for any (< u,< v, < η) ∈ K1 × K2 × K[−1,1] .

.

λ , < λ , making some computations, In particular, taking .< u = uλε,σ , < v = vε,σ η = ηε,σ dividing by .λ, and letting .λ go to 0, we arrive to the inequality

( λ1

.

Q

∗ (θε,σ − θf )Y dxdt + λ2

( Q

∗ ∗ (ϕε,σ − ηε,σ )(Φ − η)dxdt

( 1 ∗ ∗ (ξ ∗ Y + η θc − ηθε,σ − ηε,σ Y )dxdt + ε Q ε,σ ( ( ∗ ∗ ∗ + (uε,σ − uε )udxdt + (vε,σ − vε∗ )v dσ dt (

Q

+ Q

Σ

∗ (ηε,σ − ηε∗ )ηdxdt ≥ 0.

184

6 An Optimal Control Problem for a Phase Transition Model

With the previous notation, this yields (

( .

σ I1,ε Y dxdt +

Q

( ( σ σ I2,ε Φdxdt + I3,ε ηdxdt + (u∗ε,σ − u∗ε )udxdt Q ( Q Q ( ∗ ∗ + (vε,σ − vε∗ )v dσ dt + (ηε,σ − ηε∗ )ηdxdt ≥ 0. Σ

Q

(6.145) Looking at (6.144) and (6.145), we obtain ( .

Q

u(pε,σ + u∗ε,σ

− u∗ε )dxdt

( + (

Σ

+ Q

∗ v (αpε,σ + vε,σ − vε∗ )dσ dt σ ∗ η(I3,ε + ηε,σ − ηε∗ )dxdt ≥ 0.

Recalling (6.102), we finally deduce ( .

Q

) ( (u∗ε,σ − u)(− pε,σ + u∗ε,σ − u∗ε )dxdt ( + Σ

( + Q

( ( )) ∗ ∗ (vε,σ − v) − αpε,σ + vε,σ − vε∗ dσ dt )) ( ( σ ∗ ∗ (ηε,σ − η) − I3,ε + ηε,σ − ηε∗ dxdt ≥ 0,

for any .(u, v, η) ∈ K1 × K2 × K[−1,1] . This implies .

− (pε,σ + u∗ε,σ − u∗ε ) ∈ ∂IK1 (u∗ε,σ ), a.e. on Q, .

(6.146)

∗ ∗ −(αpε,σ + vε,σ − vε∗ ) ∈ ∂IK2 (vε,σ ), a.e. on Σ, .

(6.147)

σ ∗ ∗ −(I3,ε + ηε,σ − ηε∗ ) ∈ ∂I[−1,1] (ηε,σ ), a.e. on Q.

(6.148)

Now, for a set .K, we recall that .∂IK (ζ ) = NK (ζ ), the latter being the normal cone at .ζ to .K, and the relation f ∈ (I + NK )(ζ )

.

implies that ζ = (I + NK )−1 f = PK (f ).

.

It follows by a simple algebra that (6.146)-(6.148) imply (6.141)-(6.143), as claimed. n

6.4 Optimality Conditions

185

6.4.3 Optimality Conditions for (Pε ) Theorem 6.11 Assume (6.25), (6.28), and let {(u∗ε , vε∗ , ηε∗ ), (θε∗ , ϕε∗ )} be optimal in (Pε ). Then, the optimality conditions for (Pε ) read ⎧ ∗ on {(t, x) ∈ Q; pε (t, x) > 0} ⎨ uε (t, x) = um , . um ≤ u∗ε (t, x) ≤ uM , on {(t, x) ∈ Q; pε (t, x) = 0} ⎩ ∗ uε (t, x) = uM , on {(t, x) ∈ Q; pε (t, x) < 0}, ⎧ ∗ on {(t, x) ∈ Σ; pε (t, x) > 0} ⎨ vε (t, x) = vm , vm ≤ vε∗ (t, x) ≤ vM , on {(t, x) ∈ Σ; pε (t, x) = 0} ⎩ ∗ vε (t, x) = vM , on {(t, x) ∈ Σ; pε (t, x) < 0},

.

.

⎧ ∗ on {(t, x) ∈ Q; ξε∗ (t, x) < −1} ⎨ ηε (t, x) = −1, ∗ ∗ η (t, x) = ξε (t, x), on {(t, x) ∈ Q; ξε∗ (t, x) ∈ (−1, 1)} ⎩ ε∗ ηε (t, x) = 1, on {(t, x) ∈ Q; ξε∗ (t, x) > 1},

(6.149)

(6.150)

(6.151)

where ξε∗ = ϕε∗ −

.

1 (θc − θε∗ ) ελ2

and (pε , qε ) is the solution to the problem 1 qε = −I1,ε , in Q, . (θε∗ )2

(6.152)

∗ 2 (qε )t + Δqε − (3(ϕε,σ ) − 1)qε + (pε )t = −I2,ε , in Q, .

(6.153)

(pε )t + β ' (θε∗ )Δpε +

.

∂pε ∂qε + α(x)pε = 0, = 0, on Σ, . ∂ν ∂ν pε (T ) = 0, qε (T ) = 0, in Ω,

(6.154) (6.155)

and where 1 I1,ε = λ1 (θε∗ − θf ) + (ξε∗ − ηε∗ ), with ξε∗ ∈ ∂j (θε∗ ) a.e. in Q, ε 1 I2,ε = λ2 (ϕε∗ − ηε∗ ), I3,ε = −λ2 (ϕε∗ − ηε∗ ) + (θc − θε∗ ). ε

.

∗ , η∗ ) with the Proof Under the hypotheses, (Pε,σ ) has a minimizer (u∗ε,σ , vε,σ ε,σ ∗ ∗ corresponding state (θε,σ , ϕε,σ ) solution to (6.1)–(6.6). We pass to the limit in (Pε,σ ). According to Proposition 6.6, we have the convergences in (6.69)–(6.71)

186

6 An Optimal Control Problem for a Phase Transition Model

and (6.74)–(6.75) where the actual limits are θε∗ and ϕε∗ . By the estimates (6.139)– (6.140) in Proposition 6.9, by selecting a subsequence, we have pε,σ → pε , qε,σ → qε weakly in L2 (0, T ; H 2 (Ω)) ∩ W 1,2 (0, T ; L2 (Ω)),

.

weak* in L∞ (0, T ; V ), and strongly in L2 (0, T ; V ), as σ → 0. Then, recalling (6.146) and passing to the limit, we find that .

) ( − pε,σ + u∗ε,σ − u∗ε → −pε strongly in L2 (Q), as σ → 0,

which, along with (6.69), yields .

− pε ∈ ∂IK1 (u∗ε ) = NK1 (u∗ε ).

(6.156)

This relation is due to the fact that ∂IK1 is maximal monotone (see Theorem 7.29 in the Appendix) and so strongly–weakly closed. For the other two controllers, we apply the same argument in (6.147) and (6.148), and we obtain .

− αpε ∈ ∂IK2 (vε∗ ) = NK2 (vε∗ ),

(6.157)

− I3,ε ∈ ∂IK[−1,1] (ηε∗ ),

(6.158)

and .

where we took into account the convergence .

1 σ ∗ ∗ ∗ ∗ ∗ −(I3,ε + ηε,σ − ηε∗ ) = λ2 (ϕε,σ − ηε,σ ) − (θc − θε,σ ) − (ηε,σ − ηε∗ ) ε 1 → λ2 (ϕε∗ − ηε∗ ) − (θc − θε∗ ) =: −I3,ε strongly in L2 (Q), as σ → 0. ε

Also, we deduce that σ ∗ ∗ I2,ε = λ2 (ϕε,σ − ηε,σ ) → λ2 (ϕε∗ − ηε∗ ) =: I2,ε strongly in L2 (Q), as σ → 0.

.

∗ ) , it turns If ξε∗ denotes the weak* limit in L∞ (Q) of some subsequence of (ξε,σ σ ∗ ∗ ∗ ' ∗ out that ξε ∈ ∂j (θε ) a.e. in Q. Indeed, recalling that ξε,σ = jσ (θε,σ ), we can write

( .

Q

∗ (jσ (θε,σ ) − jσ (z))dxdt

( ≤ Q

∗ ∗ jσ' (θε,σ )(θε,σ − z)dxdt

for any z ∈ L2 (Q) and pass to the limit as σ → 0 taking (6.76) into account. Thus, we have

6.4 Optimality Conditions

( .

Q

187

(j (θε∗ ) − j (z))dxdt ≤

( Q

ξε∗ (θε∗ − z)dxdt,

which implies ξε∗ ∈ ∂j (θε∗ ) a.e. in Q. Consequently, we have that 1 ∗ σ ∗ ∗ I1,ε =λ1 (θε,σ − θf ) + (ξε,σ − ηε,σ ) ε 1 →λ1 (θε∗ − θf ) + (ξε∗ − ηε∗ ) =: I1,ε , weakly in L2 (Q), as σ → 0. ε

.

All the above arguments prove that the solution to (6.134)–(6.137) converges to the solution to (6.152)–(6.155) as σ → 0. Actually, due to the uniform boundedness properties ensured by (6.25) and (6.28), we also point out that ∗ β ' (θε,σ ) → β ' (θε∗ ),

.

1 1 → ∗ 2, ∗ )2 (θε,σ (θε )

∗ 2 (3(ϕε,σ ) − 1) → (3(ϕε∗ )2 − 1)

weak* in L∞ (Q) and strongly in L2 (Q), as σ → 0. We remark that the selection ξε∗ from ∂j (θε∗ ) = S(θε∗ ), which is present in I1,ε , is not uniquely determined unless θε∗ /= θc a.e. in Q. On the other hand, the pair (pε , qε ) turns out to be the unique solution of the problem (6.152)–(6.155) once ξε∗ is fixed in I1,ε . Relation (6.158) implies 1 1 λ2 ϕε∗ − (θc − θε∗ ) ∈ λ2 (I + N[−1,1] )(ηε∗ ), ε λ2

.

which gives ( ) 1 ηε∗ = Proj[−1,1] ξε∗ , with ξε∗ = ϕε∗ − (θc − θε∗ ). ελ2

.

Then, (6.156), (6.157), and (6.159) imply (6.149)-(6.151), as claimed.

(6.159) n

These results are useful also for the numerical computation of an optimal control in (P ). Due to the convergence of (Pε ) to (P ) proved in Theorem 6.5, an approximating sequence of optimal controls in (Pε ) approximates an optimal control in (P ). However, an optimal control in (Pε ) given by (6.149)-(6.151) is not uniquely determined. But, Proposition 6.6 shows that a certain optimal control (u∗ε , vε∗ , ηε∗ ) in (Pε ) is approximated by a strongly convergence of an optimal control ∗ , η∗ ) in (P sequence (u∗ε,σ , vε,σ ε,σ ). Thus, it is sufficient to compute numerically ε,σ ∗ ∗ ∗ (uε,σ , vε,σ , ηε,σ ) by the simple expressions (6.146)–(6.148).

Chapter 7

Appendix

In this appendix, there are included some notation, definitions, and results of functional analysis, operator theory, and convex analysis often used in the book. For more details about these, we refer the reader, e.g., to [13, 17, 21] and [28]. Let .R be the space of real numbers with the norm denoted .|·| , and .RN the N dimensional real space with the Euclidian norm .|·|N , for .N > 1. Let .(X, ||·||X ) be a real normed vector space with the norm denoted .||·||X . The dual of .X, denoted .X' , is defined as X' = {f : X → R; f linear and continuous}.

.

(7.1)

The space .X' is a Banach space (complete normed vector space) even if X is not complete. We have .

|f (x)| ≤ Mf ||x||X ,

(7.2)

where .Mf is a number depending in general on .f. The norm in the dual space is defined by .

||f ||X' := sup |f (x)| . ||x||X ≤1

(7.3)

The value of .f ∈ X' at .x ∈ X is indicated either by .f (x) or by the notation ' .X ' ,X which is called the pairing between X and .X . The scalar product on a Hilbert space X is denoted by .(·, ·)X .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Marinoschi, Dual Variational Approach to Nonlinear Diffusion Equations, PNLDE Subseries in Control 102, https://doi.org/10.1007/978-3-031-24583-1_7

189

190

7 Appendix

7.1 Lp Spaces, Sobolev Spaces, and Vectorial Spaces In the subsequent part, .Ω is an open bounded subset of .RN (N ≥ 1), with the boundary .| := ∂Ω assumed sufficiently smooth. By .C k (Ω), .0 ≤ k ≤ ∞, we denote the set of real-valued functions defined in .Ω that have continuous partial derivatives of an order up to and including .k. If k 0 .k = 0, we use the notation .C(Ω) instead of .C (Ω). The space .C (Ω) represents the 0 functions belonging to .C k (Ω) with compact support included in .Ω. The support of a function is defined by supp .u := {x ∈ Ω; u(x) /= 0}. Lp Spaces We consider the equivalence classes of real-valued functions that coincide almost everywhere (a.e.) on the Lebesgue measurable subset .Ω ⊂ RN . For .p ∈ R, 1 ≤ p < ∞, we define

.

{ |f (x)|p dx < ∞}

Lp (Ω) := {f : Ω → R; f is Lebesgue measurable and

.

Ω

and denote [{ .

||f ||Lp (Ω) :=

]1/p |f (x)| dx p

.

Ω

For .p = ∞, we define .

L∞ (Ω) := {f : Ω → R; f measurable and there exists C > 0, such that |f (x)| ≤ C a.e. on Ω}

and set .

||f ||L∞ (Ω) := inf{C; |f (x)| ≤ C a.e. on Ω} = ess sup |f (x)| . x∈Ω

If D is an open bounded subset of .Ω, with .D ⊂ Ω, let p

Lloc (Ω) := {f : Ω → R; f is measurable, f |D ∈ Lp (D), ∀p ∈ [1, +∞)}.

.

Theorem 7.1 For .1 ≤ p ≤ ∞, Lp (Ω) is a Banach space. If .1 < p < ∞, ' the space .Lp is reflexive and separable and .(Lp (Ω))' = Lp (Ω), where .p' is 1 1 1 the conjugate of .p, i.e., . p + p' = 1. The space .L is separable, but it is not reflexive and .(L1 (Ω))' = L∞ (Ω). The space .L∞ (Ω) is nor reflexive or separable and .(L∞ (Ω))' ⊃ L1 (Ω). For simplicity, the norm on the spaces .Lp (Ω) may be denoted by .||·||p for .p ∈ ' [1, ∞], and the pairing between the spaces .Lp and the dual .Lp is indicated by one of the following notations:

7.1 Lp Spaces, Sobolev Spaces, and Vectorial Spaces

191

{ f (g) = p' ,p = (f, g)p' ,p =

'

f (x)g(x)dx for f ∈ Lp (Ω), g ∈ Lp (Ω).

.

Ω

If there is no danger of confusion, the function { arguments in the integrands are skipped sometimes for simplicity, that is, . Q f (t, x)dxdt is simply written { . Q f dxdt. '

Theorem 7.2 (Hölder’s Inequality) Let .f ∈ Lp (Ω), .g ∈ Lp (Ω), .1 ≤ p ≤ ∞. Then .fg ∈ L1 (Ω) and { .

Ω

|f (x)g(x)| dx ≤ ||f ||p' ||g||p .

(7.4)

We also remind the Young’s inequality '

bp ap + ' , for a ≥ 0, b ≥ 0, .ab ≤ p p and a consequence of Hölder’s inequality: Let .fi ∈ Lpi (Ω), i = 1, 2, . . . , k, with . p1 = p .f = f1 f2 · · · fk ∈ L (Ω) and .

1 p1

+

1 p2

+ ... +

|| || ||f ||p ≤ ||f1 ||p1 ||f2 ||p2 . . . ||fp ||p . k

(7.5)

1 pk

≤ 1. Then,

(7.6)

Theorem 7.3 (Dunford–Pettis) Let .Ω be an open bounded subset of .RN , and let 1 .F be a bounded subset of .L (Ω). Then .F is relatively compact .(weakly sequentially 1 {compact.) in .L (Ω) if and only if for any .ε > 0 there exists .δ > 0 such that . A |f (x)| dx < ε, ∀A ⊂ Ω with meas.(A) < δ, ∀f ∈ F. Theorem 7.4 (Egorov) Assume that .Ω is a measure space with finite measure. Let (fn )n be a sequence of measurable functions on .Ω such that .fn (x) → f (x) a.e. on .Ω, where .|f (x)| < ∞ a.e. Then, for any .ε, there exists A measurable such that meas.(Ω\A) < ε and .fn → f uniformly on .A.

.

Sobolev Spaces Denote by .D(Ω) the space .C0∞ (Ω) of real-valued infinitely differentiable functions defined in .Ω, with compact support, equipped with the inductive limit topology. The space .D' (Ω) is the dual space of .D(Ω), that is, the space of all linear functionals defined on .D(Ω) with values in .R that are continuous with respect to the inductive limit topology of .D(Ω). The elements of .D' (Ω) are called scalar distributions defined on .Ω. A partial derivative of a distribution .u ∈ D' (Ω) with respect to .xj is given by ∂φ ∂u ∂ . ∂xj (φ) := −u( ∂xj ) for all .φ ∈ D(Ω). If .Di = ∂xi , i = 1, 2, . . . , N , then the α distribution .D u, called the derivative of order .α of .u ∈ D' (Ω), is defined by D α u(ϕ) := (−1)|α| u(D α ϕ), ∀ϕ ∈ D(Ω),

.

192

7 Appendix

αN where .D α u(x) = D1α1 . . . DN u(x), x ∈ Ω, α = (α1 , . . . , αN ) ∈ N = {1, 2, . . .}, |α| = α1 + . . . + αN . For .1 ≤ p ≤ ∞ and .k ≥ 1, we introduce the Sobolev spaces

W k,p (Ω) := {f : Ω → R; f is measurable and D α f ∈ Lp (Ω), |α| ≤ k}

.

endowed with the norm ⎛

.

||f ||W k,p (Ω)

⎞1/p Σ || || ||D α f ||p ⎠ := ⎝ if 1 ≤ p < ∞ p |α|≤k

and .

|| || ||f ||W k,∞ (Ω) := max ||D α f ||∞ , |α| ≤ k. |α|≤k

Here .D α f is considered in the sense of distributions. The space .W k,p (Ω) is a Banach space under the norms defined before. The space 1,p .W (Ω) is reflexive for .1 < p < ∞ and separable for .1 ≤ p < ∞. We denote by k,p k k,p (Ω). .W 0 (Ω) the completion of .C0 (Ω) in the norm of .W k k,2 By convenience, one denotes .H (Ω) := W (Ω) and .W0k,2 (Ω) := H0k (Ω). These are both Hilbert spaces with respect to the scalar product (u, v)H k (Ω) =

Σ{

.

D α u(x)D α v(x)dx.

|α|≤k Ω

The dual of .H0k (Ω) is denoted by .H −k (Ω). If .Ω is an open subset of .RN of class .C 1 , with the boundary .|, then each .u ∈ C(Ω) is well -defined on .Ω. The restriction of u to .| is called the trace of u to .|, and it is denoted by .γ0 (u). If any confusion is avoided, it can be denoted by u as well. Theorem 7.5 (Trace Theorem) Let .Ω be an open subset of .RN , of class .C 1 with 1 compact boundary .| or .Ω = RN + . For .u ∈ H (Ω), there exists a mapping .γ0 that extends the classical trace, and there is .cT r > 0 such that .

||γ0 (u)||L2 (|) ≤ cT r ||u||H 1 (Ω) , ∀u ∈ H 1 (Ω).

We have .H01 (Ω) = {u ∈ H 1 (Ω); γ0 (u) = 0}. Theorem 7.6 (Rellich–Kondrachov) Let .Ω be an open subset of .RN of class .C 1 with compact boundary .|. Then

7.1 Lp Spaces, Sobolev Spaces, and Vectorial Spaces

193

if p < N, W 1,p (Ω) ⊂ Lq (Ω), ∀q ∈ [1, p∗ ),

.

1 1 1 = − , p∗ p N

if p = N, W 1,p (Ω) ⊂ Lq (Ω), ∀q ∈ [p, +∞), if p > N, W 1,p (Ω) ⊂ C(Ω), with compact injections. Fractional Sobolev Spaces Let .σ ∈ (0, 1) and .p ∈ [1, +∞). We define the fractional Sobolev spaces { W

.

σ,p

(Ω) = z ∈ L (Ω); p

| | |z(x) − z(x ' )| |x − x ' |σ +N/p

} ∈ L (Ω × Ω) p

(7.7)

equipped with the natural norm (see, e.g., [28], p. 314). For .s ∈ R, s > 1, not an integer, .s = m + σ, m being the integer part of .s, one defines { } W s,p (Ω) = z ∈ W m,p (Ω); D α z ∈ W σ,p (Ω), ∀α with |α| = m .

.

(7.8)

If .z ∈ W 1,r (Ω), with .r > 1, it follows that the trace of z on .|, denoted by .γ (z), is well-defined, 1

γ (z) ∈ W 1− r ,r (|), ||γ (z)||

.

1

W 1− r ,r (|)

≤ C ||z||W 1,r (Ω) ,

(7.9) 1

and the operator .z → γ (z) is surjective from .W 1,r (Ω) onto .W 1− r ,r (|) (see, e.g., [28], p. 315). We also have .γ (z) ∈ L1 (|) for .z ∈ W 1,1 (Ω). Theorem 7.7 (Alaoglu) The closed unit ball of the dual space of a normed vector space is sequentially compact in the weak* topology. Corollary 7.8 Let X be a separable Banach space, and let .(fn )n be a bounded sequence in its dual .X' . Then, .(fn )n is weak* compact in .X' , that is, there exists a subsequence .(fnk ) that converges weak* (see [28], p. 76, Corollary 3.30). Vectorial Distributions and .W k,p Spaces Consider now .[0, T ] a fixed real interval, and let X be a Banach space. We denote by .D' (0, T ; X) the space of all linear and continuous operators from .D(0, T ) to .X. Here, .D(0, T ) = C0∞ (0, T ) is the space of all infinitely differentiable functions .ϕ : (0, T ) → R with compact support. An element of .D' (0, T ; X) is called a vectorial distribution on .(0, T ). We denote

194

7 Appendix

Lp (0, T ; X) .

{ } { T p ||f (t)||X dt < ∞ , := f : (0, T ) → X a.a. t; f measurable, 0 { }

L∞ (0, T ; X):= f : (0, T ) → X a.a. t; f measurable, ess sup ||f (t)||X < ∞ . t∈(0,T )

These are Banach spaces with the norms ({ .

||f ||Lp (0,T ;X) :=

T 0

)1/p p ||f (t)||X

dt

, if p ∈ [1, ∞),

and .

||f ||L∞ (0,T ;X) := ess sup ||f (t)||X , if p = ∞, t∈(0,T )

respectively. For k be a positive integer, .W k,p (0, T ; X) is the space of all vectorial distributions .u ∈ D' (0, T ; X) with the property that { } dj u p W k,p (0, T ; X) = u ∈ D' (0, T ; X); ∈ L (0, T ; X), j = 0, 1, . . . , k dt j

.

j

with . ddt ju (denoted also .(u(j ) )) the derivative in the sense of distributions ( j ) dj u d ϕ j , ∀ϕ ∈ D(0, T ). . (ϕ) = (−1) u j dt dt j The space .W k,p (0, T ; X) is a Banach space with the norm

.

||u||W k,p (0,T ;X)

⎧ ⎫1/p k || j ||p ⎨Σ ⎬ || d u || || || := , if 1 ≤ p < ∞, || dt j || p ⎩ L (0,T ;X) ⎭ j =0

and .

||u||W k,∞ (0,T ;X)

|| j || || d u || || := max || , if p = ∞. 0≤j ≤k || dt j ||L∞ (0,T ;X)

If X is a Banach space, let .

C k ([a, b]; X) = {f : [a, b] → R; f is k times differentiable with continuous k-th derivative} .

7.1 Lp Spaces, Sobolev Spaces, and Vectorial Spaces

195

For .k = 0, we denote .C([a, b]; X). It is a Banach space with respect to the norm .

||u||C([a,b];X) = sup {||u(t)||X }. t∈[a,b]

An X-valued function .x(t) defined on .[0, T ] is said to Σ be absolutely continuous on [0, T ] if for each .ε > 0, there exists .δ(ε) > 0 such that . n ||x(βn ) − x(αn )||X ≤ ε, Σ whenever . n |βn − αn | ≤ δ(ε) and .(αn , βn ) ∩ (αm , βm ) = ∅ for .m /= n. We introduce .Ak,p (0, T ; X), the space of all absolutely continuous functions dj u .u : [0, T ] → X whose derivatives . (defined almost everywhere) are absodt j lutely continuous and belong to .Lp (0, T ; X), for .j = 1, 2, . . . , k. In particular, 1,p (0, T ; X) consists of all absolutely continuous functions .u : [0, T ] → X .A with the property that the function .t → du dt exists a.e. on .(0, T ) and belongs to du p .L (0, T ; X). Here, . is the strongly derivative of u defined as dt .

.

du u(t + h) − u(t) (t) = lim strongly in X. h→0 dt h

Let X be a Banach space and .u ∈ Lp (0, T ; X), .1 ≤ p ≤ ∞. Then, the following conditions are equivalent: (i) .u ∈ W k,p (0, T ; X). (ii) there is .u1 ∈ Ak,p (0, T ; X) such that .u(t) = u1 (t) a.e. on .(0, T ). In particular, W 1,p (0, T ; X) ⊂ C([0, T ]; X).

.

Lemma 7.9 Let .X1 , X2 , X3 be three Banach spaces, .X1 and .X3 reflexive, .X1 ⊂ X2 ⊂ X3 with dense and continuous inclusions, and the inclusion .X1 ⊂ X2 is compact. Let .1 ≤ p1 , p2 ≤ ∞ and { } du ∈ Lp3 (0, T ; X3 ) . W := u ∈ Lp1 (0, T ; X1 ); dt

.

(i) If .p1 < ∞, then the embedding of W in .Lp1 (0, T ; X2 ) is compact. (ii) If .p1 = ∞ and .p3 > 1, then the embedding of W in .C([0, T ]; X) is compact. This result is called the Aubin–Lions–Simon lemma. The first point (i) is due to Aubin and Lions (see [5], see [56], Theorem 5.1, p. 58), and point (ii) was completed by Simon (see [73]). Theorem 7.10 (Arzelà) A subset .X0 of .C([a, b]) is compact if and only if it is bounded and equicontinuous, i.e., if and only if: (i) There exists a constant M such that .||f ||L∞ (a,b) ≤ M for all .f ∈ X0 .

196

7 Appendix

(ii) For each .ε > 0, there is .δ > 0 that .|f (x) − f (y)| < ε for all .f ∈ X0 and for all .x, y ∈ [a, b] such that .|x − y| < δ (where .δ may depend on .ε, but it is independent of f and of .x, y). Theorem 7.11 (Arzelà–Ascoli) Let X be a Banach space and .M ⊂ C([0, T ]; X) be a family of functions such that: (i) .||u(t)||X ≤ M, ∀t ∈ [0, T ], u ∈ M. (ii) .M is equi-uniformly continuous, i.e., for any .ε, there is .δ(ε) such that .

||u(t) − u(s)||X ≤ ε if |t − s| ≤ δ(ε), ∀u ∈ M.

(iii) For each .t ∈ [0, T ], the set .{u(t); u ∈ M} is compact in .X. Then, .M is compact in .C([0, T ]; X). Definition 7.12 Let X be a Banach space and .y : [a, b] → X. The total variation of f on .[a, b] is defined by n Σ ||y(si ) − y(si−1 )||X , V (y; [a, b]) = sup P ∈P i=1

.

(7.10)

where .P = {P = (s0 , . . . , sn ); P is a partition of .[a, b]} is the set of all partitions of .[a, b]. If .V (y; [a, b]) < ∞, the function y is said to be with bounded variation on [a, b], and the space of such functions is denoted by .BV ([a, b]; X).

.

Theorem 7.13 (Helly) Let X be a reflexive separable Banach space with a separable dual .X' . Let .(wn )n ⊂ BV ([a, b]; X) be such that .||wn (t)||X ≤ C for all .t ∈ [a, b] and Var.(wn ; [a, b]) ≤ C for all .n, C being a constant. Then, there exists a subsequence .(wnk ) ⊂ (wn ) and a function .w ∈ BV ([0, T ]; X), such that wnk (t) → w(t) strongly in X, as k → ∞, for all t ∈ [a, b].

.

The strong version of the Helly Theorem is due to Foias and asserts that if .X is a general Banach space, and for each .t ∈ [a, b], the family .(wn (t)) is compact in .X, then the sequence .(wnk ) can be chosen strongly convergent on .[a, b] (see e.g., [21], p. 48, Remark 1.127).

7.2 Operators in Banach Spaces We recall some definitions and results related to maximal monotone operators in Banach spaces. For more details, see, e.g., [13, 17].

7.2 Operators in Banach Spaces

197

We consider X and Y two normed vector spaces and .X × Y their Cartesian product space. An element of .X × Y will be denoted .[x, y] for .x ∈ X and .y ∈ Y. Definition 7.14 A multivalued operator .A from X to Y is a subset of .X × Y. We define D(A) := {x ∈ X; Ax /= ∅}, the domain of A,

.

R(A) :=

||

.

Ax, the range of A,

x∈D(A)

A−1 := {[y, x]; [x, y] ∈ A}, the inverse of A,

.

which can be multivalued. Let X and .X' be a Banach space and its dual, respectively. We define the duality mapping J of the space X by || ||2 < > ' J : X → 2X , J (x) = {x ' ∈ X' ; x ' , x X' ,X = ||x||2X = ||x ' ||X' }.

.

(7.11)

Generally, it is a multivalued operator. In the case when X is a Hilbert space identified with its dual, then .J (x) = x. If the Hilbert space X is not identified with its dual .X' , then the duality mapping ' ' .J : X → X is the canonical isomorphism of X onto .X (see [13], p. 3). Next, we give a sequence of definitions related to the properties of the operators. Definition 7.15 An operator A defined from X to .X' is called monotone if .

X' ,X ≥ 0, ∀[xi , yi ] ∈ A, i = 1, 2.

(7.12)

The operator A is called maximal monotone if it is not properly contained in any other monotone subset of .X × X' . The operator A .: X → X' is said to be strongly monotone if .

X' ,X ≥ ρ ||x1 − x2 ||2X ,

(7.13)

for any .[xi , yi ] ∈ A, i = 1, 2 with .ρ > 0 fixed. If A is single-valued, then .yi ∈ Axi is replaced in all definitions and results by Axi for .xi ∈ D(A). The operator A is bounded if it maps every bounded subset of X into a bounded subset of .X' . The operator A is said to be injective if

.

Ax ∩ Ay /= ∅ implies x = y.

.

(7.14)

198

7 Appendix

The operator A is called closed if [xn , yn ] ∈ A and xn → x, yn → y imply [x, y] ∈ A.

.

(7.15)

A subset A of .X × X' is called demiclosed if it is strongly–weakly closed in ' .X × X , i.e., for .[xn , yn ] ∈ A, xn → x strongly in X and yn → y weakly in X' imply [x, y] ∈ A.

.

(7.16)

The single-valued operator .A : D(A) = X → X' is said to be hemicontinuous if A(x + λy) → Ax weakly in X' , as λ → 0, ∀x, y ∈ X.

.

(7.17)

The single-valued operator .A : X → X' is called demicontinuous if it is strongly–weakly continuous from X to .X' , i.e., Axn → Ax weakly in X' for any xn → x strongly in X.

.

(7.18)

The (multivalued) operator .A : X → X' is coercive if there exists .x0 ∈ X such that .

< yn , xn − x0 >X' ,X = +∞ n→∞ ||xn ||X lim

(7.19)

for all .[xn , yn ] ∈ A such that .limn→∞ ||xn ||X = +∞. Proposition 7.16 Let .A ⊂ X × X' be a maximal monotone operator. Then, A is weakly–strongly closed on .X × X' , that is, if yn = Axn , xn → x weakly in X, yn → y strongly in X' , then, y ∈ Ax.

.

Moreover, .A−1 is maximal monotone on .X' ×X, and so A is strongly–weakly closed, too (see [13], p. 21, Proposition 2.1). Theorem 7.17 Let X and .X' be reflexive, and let .J : X → X' be the duality mapping of .X. Let A be a monotone operator of .X × X' . Then A is maximal monotone in .X × X' if and only if, for any .λ > 0 (equivalently for some .λ > 0), ' .R(A + λJ ) = X . Theorem 7.18 Let X be reflexive, and let A be monotone, everywhere defined, and hemicontinuous from X to .X' . Then A is maximal monotone. If A is coercive, then A is surjective, i.e., .R(A) = X' . Definition 7.19 The multivalued operator .A : X → X is called accretive if for any [xi , yi ] ∈ A, .i = 1, 2, there exists .f ∈ J (x1 − x2 ) such that

.

(y1 − y2 , f )X' ≥ 0.

.

(7.20)

7.2 Operators in Banach Spaces

199

An accretive operator is said to be maximal accretive if it is not properly contained in any accretive subset of .X × X. An accretive operator is called m-accretive if .R(I + A) = X. The operator .A ⊂ X × X is said to be quasi-m-accretive, or .ω-m-accretive if .λI + A is accretive for .λ > ω, λ > 0. The accretiveness can be equivalently expressed by .

||x1 − x2 ||X ≤ ||x1 − x2 + λ(y1 − y2 )||X' ,

(7.21)

for all .λ > 0, [xi , yi ] ∈ A, i = 1, 2. Definition 7.20 Let A be an m-accretive operator. Let .λ > 0. We define .

Jλ x = (I + λA)−1 x, for x ∈ R(I + λA) Aλ x = λ−1 (x − Jλ x), for x ∈ R(I + λA).

(7.22)

The operator .Jλ is called the resolvent of A, and .Aλ is called the Yosida approximation of .A. The properties of .Aλ can be found, e.g., in [13], p. 99. Now, we give a result related to the extension of linear operators on Banach spaces. Let .A0 : D(A0 ) ⊂ X → X be a linear closed, densely defined operator on a Banach space X, with .D(A0 ) a reflexive space. We assume that .A0 is injective and surjective. Let .A∗0 : D(A∗0 ) ⊂ X' → X' be its adjoint, defined by .

< ∗ ∗ > < > A0 x , ψ X' ,X = x ∗ , A0 ψ X' ,X for all x ∗ ∈ D(A∗0 ), ψ ∈ D(A0 ).

(7.23)

The operator .A0 can be extended to an operator A from X to .(D(A∗0 ))' , the dual of the space .D(A∗0 ), by .

A : D(A) = X ⊂ (D(A∗0 ))' → (D(A∗0 ))' , > < (D(A∗ ))' ,D(A∗ ) = y, A∗0 x ∗ X,X' for y ∈ X, x ∗ ∈ D(A∗0 ). 0

(7.24)

0

The adjoint of A is .

X' ,X

A∗ : D(A∗0 ) ⊂ X' → X' , ∗ ∗ D(A∗0 ),(D(A∗0 ))' , for x ∈ D(A0 ), y ∈ D(A)=X. (7.25)

=

Lemma 7.21 If .A0 : D(A0 ) ⊂ X → X with the properties mentioned before is injective and surjective, then its extension .A : X → (D(A∗0 ))' is surjective, that is, ∗ ' .R(A) = (D(A )) . 0

200

7 Appendix

Proof Let us prove first that .R(A) is closed in .(D(A∗0 ))' , that is, if .yn ∈ X, yn → y in X and .Ayn = fn , we have .limn→∞ fn = Ay. Indeed, we have .

< > > > < < fn , x ∗ (D(A∗ ))' ,D(A∗ ) = Ayn , x ∗ (D(A∗ ))' ,D(A∗ ) = yn , A∗0 x ∗ X,X' 0 0 0 0 > > < < → y, A∗0 x ∗ X,X' = Ay, x ∗ (D(A∗ )),D(A∗ ) , for all x ∗ ∈ D(A∗0 ), 0

0

which implies the conclusion. Then, by the closed range theorem (see [83], p. 205), it follows that R(A) = N ⊥ (A∗ ),

.

where .N ⊥ (A∗ ) is the orthogonal of the kernel .N(A∗ ), that is, N ⊥ (A∗ ) = {x ∈ (D(A∗0 ))' ; (D(A∗ ))' ,D(A∗ ) = 0 for all f ∈ D(A∗0 )}. 0 0 (7.26) In order to show that .R(A) = R(A) = (D(A∗0 ))' (since .R(A) is closed), it is sufficient to prove that .N(A∗ ) = {0}. Let .x ∗ ∈ N (A∗ ) ⊂ D(A∗0 ) ⊂ X' . Then, .A∗ x ∗ = 0, and by the previous definitions, we successively have .

< > < > < > 0 = A∗ x ∗ , y X' ,X = x ∗ , Ay D(A∗ ),(D(A∗ ))' = A∗0 x ∗ , y X' ,X 0 0 < > = x ∗ , A0 y X' ,X , for all y ∈ D(A0 ) ⊂ X.

.

We recall that .A0 is surjective, and so the previous equality takes place for all .z = A0 y ∈ X. Therefore, .x ∗ = 0, that is, .N(A∗ ) = {0}, which implies that .R(A) = (D(A∗0 ))' , as claimed. n

7.3 Convex Functions and Subdifferential Mappings We denote .R = R ∪ {+∞}. Definition 7.22 Let X be a real Banach space and .X' its dual. The function .ϕ : X → R is called convex if ϕ (λx + (1 − λ)y) ≤ λϕ(x) + (1 − λ)ϕ(y),

.

(7.27)

for .λ ∈ [0, 1] and .x, y ∈ X. A convex function .ϕ : X → R is said to be proper if .(−∞) ∈ / ϕ(X) and .ϕ(X) /= {+∞}. The set D(ϕ) := {x ∈ X; ϕ(x) < +∞}

.

7.3 Convex Functions and Subdifferential Mappings

201

is called the effective domain, and the set Sλ (ϕ) := {x ∈ X; ϕ(x) ≤ λ}

.

(7.28)

is called the level set of .ϕ. Definition 7.23 A function .ϕ : X → R is called strongly (weakly, respectively) lower semicontinuous (denoted for short l.s.c.) if for any .λ ∈ R the level set is strongly (weakly, respectively) closed. Note that a proper, convex, l.s.c. function .ϕ is continuous on the interior of .D(ϕ). Also, we recall that a convex set is closed if and only if it is weakly closed. Proposition 7.24 The function .ϕ : X → R is strongly lower semicontinuous on X if it is strongly sequentially lower semicontinuous, i.e., for any sequence .(xn )n≥1 that converges strongly to .x, we have .

lim inf ϕ(xn ) ≥ ϕ(x), ∀x ∈ X.

xn →x

(7.29)

The function .ϕ : X → R is weakly lower semicontinuous on X if it is weakly sequentially lower semicontinuous, i.e., .

lim inf ϕ(xn ) ≥ ϕ(x), if xn → x weakly in X.

xn →x

(7.30)

Proposition 7.25 Any convex function .ϕ : X → R is strongly lower semicontinuous if and only if it is weakly lower semicontinuous. Any strongly sequentially convex l.s.c. function is weakly sequentially l.s.c. If the function is not convex, the above assertion does not function in both senses, namely, the weakly lower semicontinuity is a stronger property and implies the strongly lower semicontinuity, but the reverse is not true. Proposition 7.26 Let .ϕ : X → R be a weakly lower semicontinuous function such that every level set .{x; ϕ(x) ≤ λ} is weakly compact. Then .ϕ attains its infimum on X. In particular, if X is reflexive and .ϕ is a l.s.c. proper convex function on X such that .lim ϕ(x) → ∞ as .||x||X → ∞, then there exists .x0 ∈ X such that .ϕ(x0 ) = infx∈X {ϕ(x)}. Proposition 7.27 Any proper l.s.c. convex function on X is bounded below by an affine function, i.e., there exist .x ∗ ∈ X' and .μ ∈ R such that < > ϕ(x) ≥ x ∗ , x X' ,X + μ, ∀x ∈ X.

.

Definition 7.28 Let .ϕ be a proper convex function on X and .x ∈ X. Then, the set < > ∂ϕ(x) := {x ∗ ∈ X' ; ϕ(x) − ϕ(y) ≤ x ∗ , x − y X' ,X , ∀y ∈ X}

.

(7.31)

is called the subdifferential of .ϕ at x, and its elements are called subgradients of .ϕ at .x.

202

7 Appendix

Theorem 7.29 (Rockafellar) If .ϕ : X → (−∞, ∞] is a proper convex function, then .∂ϕ is a monotone operator from X to .X' . If .ϕ is still l.s.c., then .∂ϕ is maximal monotone. Corollary 7.30 Let .ϕ be a l.s.c. proper convex function on .X. Then .D(∂ϕ) is a dense subset of .D(ϕ). Definition 7.31 Let .f : X → R. The mapping .f ' : X × X → R defined by f (x + λy) − f (x) λ→0 λ

f ' (x, y) = lim

.

(if exists) is called the directional derivative of .f. The function .f : X → R is said to be Gâteaux differentiable at x if there exists the Gâteaux derivative denoted ' ' .∇f (x) ∈ X such that .f (x, y) = X ' ,X , for all .y ∈ X. Let .ϕ be Gâteaux differentiable at .x. Then .∂ϕ(x) consists of a single element, namely the Gâteaux derivative of .ϕ at .x. Definition 7.32 The normal cone to K at x is the set .NK (x) ⊂ X' , defined by < > NK (x) := {x ∗ ∈ X' ; x ∗ , x − y X' ,X ≥ 0, ∀y ∈ K}.

.

(7.32)

Some Examples Let .ϕ : X → (−∞, ∞], ϕ(x) = 12 ||x||2X . The function .ϕ is a proper, convex, l.s.c. function, and .∂ϕ = J, the duality mapping of .X. Let K be a closed convex subset of X, and let .IK : X → (−∞, ∞] be the indicator function defined by { IK (x) :=

.

0, if x ∈ K +∞, otherwise.

(7.33)

Then, .IK is convex and l.s.c. on X, .D(∂IK ) = D(IK ) = K and .∂IK (x) = {0} for x ∈ int. K (interior of .K).

.

If x ∈ ∂K, then ∂IK (x) = NK (x).

.

(7.34)

Any set .K ⊂ X is convex (respectively, closed) if and only if .IK is convex (respectively, l.s.c.). Definition 7.33 Let .ϕ : X → R be a proper convex l.s.c function. For .λ > 0, the function { } ||x − u||2X .ϕλ (x) = inf (7.35) + ϕ(u) u∈X 2λ is called the Moreau regularization of .ϕ.

7.3 Convex Functions and Subdifferential Mappings

203

The properties of .ϕλ can be found, e.g., in [13], p. 48, Theorem 2.9. Namely, we have: Theorem 7.34 Let X be a reflexive and strictly convex Banach space with strictly convex dual. Let .ϕ : X → R be a l.s.c. convex, proper function, and let .A = ∂ϕ. Then the function .ϕλ is convex, continuous, Gâteaux differentiable, and .∇ϕλ = Aλ . Moreover, for all .x ∈ X, we have ϕλ (x) =

.

||x − Jλ x|| + ϕ(Jλ x), 2λ

lim ϕλ (x) = ϕ(x),

λ→∞

(7.36)

ϕ(Jλ x) ≤ ϕλ (x) ≤ ϕ(x). Proposition 7.35 Let X be reflexive and .ϕ : X → (−∞, ∞] be a proper convex l.s.c. function on .X. Then, the following conditions are equivalent: ϕ(x) (i) . ||x|| = +∞ as ||x||X → ∞ with x ∈ D(ϕ). X (ii) .R(∂ϕ) = X' and (∂ϕ)−1 is bounded on bounded sets.

Definition 7.36 The conjugate of .ϕ is the function .ϕ ∗ : X' → R defined by ϕ ∗ (w) = sup {X' ,X − ϕ(x)}

.

(7.37)

x∈X

(also called Fenchel–Legendre conjugate, see, e.g., [48], p. 28). Now, we recall some properties of the conjugate functions .ϕ and .ϕ ∗ (see, e.g., [13], p. 8–9, Proposition 15 and Proposition 1.9) and of the convex integrands (see, e.g., [13], p. 56–57, Proposition 2.7, see also [69]). Proposition 7.37 Let .ϕ : X → (−∞, ∞] be a proper convex l.s.c. function on X. Then, its conjugate .ϕ ∗ is proper, convex, and l.s.c. on .X' . Moreover, .ϕ and .ϕ ∗ satisfy the Legendre–Fenchel relations < > ϕ(x) + ϕ ∗ (x ∗ ) − x ∗ , x X' ,X ≥ 0,

.

< > ϕ(x) + ϕ ∗ (x ∗ ) − x ∗ , x X' ,X = 0 if and only if x ∗ ∈ ∂ϕ(x).

.

(7.38) (7.39)

In particular, ∂ϕ ∗ = (∂ϕ)−1 and (ϕ ∗ )∗ = ϕ.

.

Let .j : Ω × Rm → R, 1 ≤ m < ∞, having the following properties: (i) .y → j (x, y) is a proper, convex, and l.s.c. function a.e. .x ∈ Ω ⊂ RN . (ii) .x → j (x, y) is measurable for each .y ∈ Rm .

(7.40)

204

7 Appendix

(iii) int. D(j (x, ·)) /= ∅ for every .x ∈ Ω, where int. D(j (x, ·)) is the interior of the domain of .j. Such a function is called a normal convex integrand. Let us define .ϕ : (Lp (Ω))m → R by ⎧{ ⎨ j (x, y(x))dx, if g(x, y) ∈ L1 (Ω), .ϕ(y) = ⎩ Ω +∞, otherwise. Proposition 7.38 Let g satisfy assumptions (i)–(iii). Then the function .ϕ is convex, lower semicontinuous, and proper and ∂ϕ(x) = {w ∈ (Lp (Ω))m ; w(x) = ∂j (x, y(x)) a.e. x ∈ Ω}.

.

Moreover, .ϕ ∗ the conjugate of .ϕ is proper, convex, and lower semicontinuous, too. Finally, we recall some results for the case .m = +∞, regarding the convex functionals defined on .L∞ (Q) and on the dual space .(L∞ (Q))' , as characterized in Rockafellar’s paper [70]. First, let us recall the Lebesgue decomposition that establishes that every linear functional w on .(L∞ (Q))' can be uniquely expressed as the sum between an absolutely continuous component .wa ∈ L1 (Q) and a singular component .ws ∈ (L∞ (Q))' , w = wa + ws .

(7.41)

.

The absolutely continuous linear part .wa is expressed by an element .u∗ ∈ L1 (Q) (the density of .wa ), in the sense that > < ∗ .wa (u) = u, u = L∞ (Q),L1 (Q)

{

uu∗ dxdt

Q

for any .u ∈ L∞ (Q). In the following, we shall identify the absolutely continuous part .wa by .u∗ , and this is the reason for which .wa ∈ L1 (Q). Let .IK be the indicator function of a closed convex set .K ⊂ L∞ (Q). The conjugate of .IK , denoted by .IK∗ , is defined by IK∗ (w) = sup {w(ψ)}, ∀ w ∈ (L∞ (Q))' .

.

(7.42)

ψ∈K

This function is convex, lower semicontinuous, and positively homogeneous. Now, let .j (t, x, ·) be a convex lower semicontinuous function on .R and .j ∗ (t, x, ·) its conjugate. We define ϕ : L∞ (Q) → (−∞, ∞], ϕ(y) =

{ j (t, x, y(t, x))dxdt,

.

Q

(7.43)

7.3 Convex Functions and Subdifferential Mappings

205

with the effective domain D(ϕ) = {y ∈ L∞ (Q); ϕ(y) < +∞}.

.

(7.44)

Theorem 7.39 If .j (t, x, y(t, x)) is majorized by a summable function of .(t, x) for at least one .y ∈ L∞ (Q) and .j ∗ (t, y, z(t, x)) is majorized by a summable function of .(t, x) for at least one .z ∈ L1 (Q), then .ϕ is well-defined, convex, and lower semicontinuous on .L∞ (Q). Also, its conjugate .ϕ ∗ : (L∞ (Q))' → (−∞, ∞] is well-defined on .(L∞ (Q))' , and it is expressed by ϕ ∗ (w) =

{

.

Q

∗ j ∗ (t, x, wa (t, x))dxdt + ID(ϕ) (ws ), ∀ w ∈ (L∞ (Q))' .

This result is due to Rockafellar (see Theorem 1 in [70]).

(7.45)

References

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Index

D Diffusion coefficient, 1 fast, 2 slow, 1 Distribution vectorial, 193 Duality mapping, 197 F Function absolutely continuous, 195 convex, 200 effective domain of, 201 level set of, 201 proper, 200 directional derivative of, 202 Gâteaux derivative, 202 lower semicontinuous (l.s.c.), 201 strongly lower semicontinuous, 201 subdifferential of, 201 subgradient of, 201 weakly lower semicontinuous, 201 H Heaviside graph, 82 I Indicator function, 202 Inequality Hölder, 191 Young, 191

L Legendre–Fenchel relations, 203 Lemma Aubin–Lions–Simon, 195 Level set, 39

M Mollifier, 128 Moreau approximation, 54

N Normal cone, 202

O Operator accretive, 198 adjoint, 199 bounded, 197 closed, 6, 198 coercive, 198 demiclosed, 198 demicontinuous, 198 hemicontinuous, 198 injective, 197 m-accretive, 199 maximal monotone, 197 monotone, 197 multivalued, 197 quasi m-accretive, 199 strongly monotone, 197

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Marinoschi, Dual Variational Approach to Nonlinear Diffusion Equations, PNLDE Subseries in Control 102, https://doi.org/10.1007/978-3-031-24583-1

211

212 P Pairing, 189 S Signum graph, 68 Space dual, 189 fractional Sobolev, 193 functions with bounded variation, 196 T Theorem

Index Alaoglu, 193 Arzelà, 195 Arzelà–Ascoli, 13, 61, 196 Dunford–Pettis, 46, 191 Egorov, 191 Helly, 27 Rellich–Kondrachov, 192 trace, 192 Trace, 192

Y Yosida approximation, 53