The Neumann Problem for the Cauchy-Riemann Complex. (AM-75), Volume 75 9781400881529

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Annals of Mathematics Studies Number 75

THE NEUMANN PROBLEM FOR THE CAUCHY-RIEMANN COMPLEX BY

G. B. FOLLAND AND J. J. KOHN

PRINCETON U N IV ERSITY PRESS AND U N IV ERSIT Y OF T O K YO PRESS PRINCETON, NEW JERSEY 1972

Copyright © 1972 by Princeton University Press

All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means including information storage and retrieval systems without permission in writing from the publisher, except by a reviewer who may quote brief passages in a review. LC Card: 72-1984 ISBN: 0-691-08120-4 AMS 1970: primary 35N15 secondary 58G05, 35H05, 35D10, 35F15

Published in Japan exclusively by University of Tokyo Press in other parts of the world by Princeton University Press

Printed in the United States of America

FOREWORD This book is based on the notes from lectures given by the second author at Princeton University in the year 1970-71, which were subse­ quently expanded and revised by the first author. Our aim has been to provide a thorough and coherent account of the solution of the d-Neumann problem and certain of its applications and ramifications, and we have taken the opportunity afforded by the monograph format to employ a some­ what more leisurely style than is common in the original journal articles. It is our hope that this book may thereby be accessible to a fairly wide audience and that it may also provide a sort of working introduction to some of the recent techniques in partial differential equations. In keeping with this philosophy, we have tried to make the book as self-contained as possible. On the geometrical side, we assume the read­ er is familiar with differentiable manifolds and their native flora and fauna: vector fields, differential forms, partitions of unity, etc. On the analytical side, we assume only an elementary knowledge of functional analysis and Fourier analysis. In the body of the text we also assume an acquaintance with Sobolev spaces and pseudodifferential operators, but we have included an appendix which develops these theories as far as they are needed. G. B. F. J . J-

JUNE, 1972

K.

TABLE OF CONTENTS FOREWORD

v

CHAPTER I: Formulation of the Problem 1. Introduction 2. Almost-complex manifolds and differential operators 3. Operators on Hilbert space

3 6 11

CHAPTER II: The Main Theorem 1. Statement of the theorem 2. Estimates and regularity in the in interior the interior 3. Elliptic regularization 4. Estimates Estimates at at the the boundary boundary 5. Proof Proof of of the the Main Main Theorem Theorem

19 19 24 24 31 31 o5 35 44 44

CHAPTER III: Interpretation of the Main Theorem 1. Existence Existence and and regularity regularity theorems theorems for the for the d complex d complex 2. Pseudoconvexity Pseudoconvexity and and the the basic basic estimate estimate

47 47 56 56

CHAPTER IV: Applications 1. The Newlander-Nirenberg theorem 2. The Levi problem 3. Remarks on d cohomology 4. Multiplier operators on holomorphic functions

70 73 76 79

CHAPTER V: The Boundary Complex 1. Duality theorems 2. The induced boundary complex 3. d-closed extensions of forms 4. The abstract model

82 86 88 92

CHAPTER VI: Other Methods and Results 1. The method of weight functions 2. Holder and L^ estimates for d 3. Miscellaneous remarks and questions vii

105 108 111

viii

TABLE OF CONTENTS

APPENDIX: The Functional Analysis of Differential Operators 1. Sobolev norms on Euclidean space 2. Sobolev norms on manifolds 3. Tangential Sobolev norms 4. Difference operators 5. Operators constructed from Asand Aj 5

114 122 125 126 130

REFERENCES

136

TERMINOLOGICAL INDEX

143

NOTATIONAL INDEX

145

The Neumann Problem for the Cauchy-Riemann Complex

CHAPTER I FORMULATION OF THE PROBLEM 1.

Introduction In the nineteenth century two approaches to the theory of functions of

a complex variable were initiated by Weierstrass and Riemann, respectively. The first was to study power series, canonical products, and such, staying within the analytic category; the second was to work in the C°° category, using the differential equations and associated variational problems aris­ ing from the situation.

The first approach was generalized to functions of

several variables by K. Oka, H. Cartan, and others, and it is along these lines that the modern theory of several complex variables has largely de­ veloped.

The second approach has been used with great success in the

case of compact complex manifolds (the Hodge theory, cf. Weil [46]), and more recently these methods have been extended to open manifolds. This extension, however, poses rather delicate analytical problems. In particu­ lar, it leads to a non-coercive boundary value problem for the complex Laplacian, the d-Neumann problem. It is our purpose here to present a detailed solution of this problem for domains with smooth boundary satis­ fying certain pseudoconvexity conditions and to indicate its applications to complex function theory. By way of introduction, let us consider functions (and, more generally, differential forms) on a bounded domain M in Cn with smooth boundary bM. The Cauchy-Riemann operator d defined on functions by df = V _ Sj dz| extends naturally to yield the Dolbeault complex

0 — > AP>°(M) - X AP’ ^M ) 3

AP>n(M)— , 0

4

THE NEUMANN PROBLEM FOR THE d COMPLEX

where AP,C^(M) is the space of smooth forms of type (p,q) on M. The holomorphic functions are precisely the solutions of the homogeneous equation df =

0.

The inhomogeneous equation df = cf> is also of interest.

Consider

the following version of the Levi problem', given p e bM, is there a holo­ morphic function on M that blows up at p? In general, the answer is no: for example, if M is the region between two concentric spheres, Hartogs’ theorem [18] says that any holomorphic function on M extends holomorphically to the interior of the inner sphere.

However, if M is strongly

convex at p (meaning that there is a neighborhood U of p such that for any q e (M U bM) fl U, the line segment between p and q lies in M), a classical construction of E. E. Levi guarantees the existence of a neighborhood V of p and a holomorphic function w on M fl V which blows up only at p. Now suppose the equation di = 0 (where cf> satis­ fies the compatibility condition d = 0 ) is always solvable in M in such a way that f is smooth up to bM (i.e., can be extended smoothly across bM) whenever is.

Then we can solve the Levi

problem. Indeed, let if/ be a smooth function with support in V which is identically one near p. Then iffw is defined on all of M and is smooth away from p; since d(i//w) =

0

near p, d(i//w) is smooth up to

the boundary. Therefore there is a function f, smooth up to the boundary, which satisfies di = d(if/w). Finally,

( —if/w is holomorphic in M and

blows up at p. (We shall discuss this construction in greater detail in §4.2.) Let us consider the equation di =

where f and cf> are supposed

square-integrable. If a solution f exists, it is determined only modulo the space K = }g e L 2 (M): .

( . . )

(Note that (1.1.2) reduces to (1.1.1) when dcf> = 0, for then

9

55*5i = o

(73*59,50) = o = > (5*5i9f 3*30) = o =*► 3*50 = o .)

The equation (1.1.2) is a boundary value problem in disguise, for is as

To compute & we use the fact that has compact support. (Proof: above, and if/ =

2

1H

i/r

1H

dz*

a

d z^ e AP,q - 1 (M) has compact support,

= 2p+q( - l) p ^

X

, 2P*q(. i ) P +i 2

IH

( 2

kH f kH

^

,J H ^3 1 , * m ) .

(The annoying factor of 2P+q comes from the fact that = 25-j.) Thus ^

.

2 C-i)p+

'2

I H J k « J H ^ l l d z' , d * H .

On a general manifold M these formulas remain valid if we replace d z 1 , . . . , d z n by a local orthonormal basis

, ..., oon for A 0 , 1 (M), ex­

cept for some error terms resulting from the fact that the forms coj are not d-closed and the coefficients of the metric are not constant.

These

10

THE NEUMANN PROBLEM FOR THE d COMPLEX

terms, however, do not involve differentiation of the coefficients of the forms. Thus we have, for cf> e Ap,q(M), (1.2.3)

dcf> = (—1)^

klJK ^ k j

+ terms of order zero,

(1.2.4) dcf) = (—1)P+1

where

>\ajP + terms of order zero,

is the vector field dual to c o (There is no factor of 2 here

because the coj^s are orthonormal!) In general, if E and F are vector bundles on a manifold M and D: T(E)

T(F) is a differential operator of order k, where T denotes

spaces of sections, we define for each

77

^ 0 in T*M the symbol

a(D, 77): E x ->FX of D at r/ as follows. Let p be a function defined near x with p(x) =

0

and dpx = 77, and for each

0 6

E x let 0 be a

local section of E extending 0. Then o(D, -q)Q = -j— D(p^0)|x. It is easily verified that cr(D, rf)d is independent of the choice of p and d. Moreover, if D' is the formal adjoint of D with respect to Hermitian structures on E and F, T(F) is elliptic.

For details on these matters, see, e.g., Palais [38]. Let us compute the symbols of d and # on an integrable Hermitian almost-complex manifold M. With rj,p as above and

x = d(pcf>)x = [dpA]x = n Q ^ A0 X, so cr(d, 77) = II 0 ^7 7 a (•)• Therefore, a(&, 77) = - n Q ^

v

(•) where = . (The

reader who wishes to have the formula for a($, 77) involve ^

Qrj instead

of n Q xi7 must adjust the definition of v to make peace with the conju­ gate linearity lurking in these duality relations.)

FORMULATION OF THE PROBLEM

11

Since rj is real, I I 0 ^rj ^ 0 whenever rj ^ 0. Thus if 9 is a (p, q)covector at x satisfying (Hq ^rf) a basis for CT*M of which IIq

a

^ 77

6=

0,

by expressing 6 in terms of

is a member, we see that 6 is d iv isi­

ble by n Q 1 ?7, i.e ., 0 = n Q x17 a if/ for some (p, q—l)-covector if/. There­ fore:

(1.2.5) P R O P O S IT IO N .

The symbol sequence for d is exact, i.e., the

d complex ( 1 . 2 . 2 ) is elliptic.

One could also show this by verifying that