Mahboob khan , Ousmania university. Adesh Pawan Tripathi , Attended ddu gkp university. Feng Li , Student at Macquarie University. Show More. No Downloads. Views Total views. Actions Shares. Embeds 0 No embeds. No notes for slide. Partial differential equations and complex analysis 1. Gavosto and Marco M. Includes bibliographical references p.

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ISBN 1. Differential equations. Functions of a complex variable. Mathematical analysis. Functions of several complex variables. K9 Reprinted material is quoted with permission, and sources are indicated. A wide varietyof references are listed. Every reasonable effort has been made to give reliable dataand information, but the author and the publisher cannot assume responsibility for thevalidity of all materials or for the consequences of their use. All rights reserved. This book, or any parts thereof, may not be reproduced in any formwithout written consent from the publisher.

Box ,Champaign, IL To the memory of my grandmother,Eda Crisafulli. PrefaceThe subject of partial differential equations is perhaps the broadest and deepestin all of mathematics.

It is difficult for the novice to gain a foothold in thesubject at any level beyond the most basic. At the same time partial differentialequations are playing an ever more vital role in other branches of mathematics. This assertion is particularly true in the subject of complex analysis.

It is my experience that a new subject is most readily learned when presentedin vitro. Thus this book proposes to present many of the basic elements of linearpartial differential equations in the context of how they are applied to the study ofcomplex analysis. We shall treat the Dirichlet and Neumann problems for ellipticequations and the related Schauder regularity theory. Both the classical point ofview and the pseudodifferential operators approach will be covered. Then weshall see what these results say about the boundary regularity of biholomorphicmappings.

We shall study the a-Neumann problem, then consider applications tothe complex function theory of several variables and to the Bergman projection. The book culminates with applications of the a-Neumann problem, includinga proof of Feffermans theorem on the boundary behavior of biholomorphicmappings. There is also a treatment of the Lewy unsolvable equation fromseveral different points of view.

We shall explore some partial differential equations that are of current interestand that exhibit some surprises. These include the Laplace-Beltrami operatorfor the Bergman metric on the ball. Along the way, we shall give a detailedtreatment of the Bergman kernel and associated metric, the Szego kernel, andthe Poisson-Szego kernel. Some of this material, particularly that in Chapter 6,may be considered ancillary and may be skipped on a first reading of this book.

Complete and self-contained proofs of all results are provided. Some of theseappear in book form for the first time. Our treatrlent of the a-Neumann problemparallels some classic treatments, but since we present the problem in a concretesetting we are able to provide more detail and a more leisurely pace. Background required to read this book is a basic grounding in real and com-plex analysis. The book Function Theory of Several Complex Variables by thisauthor will also provide useful background for many of the ideas seen here.

Acquaintance with measure theory will prove helpful. For motivation, exposure 8. All other needed ideas aredeveloped here. A word of warning to the reader unversed in reading tracts on partial differ-ential equations: the metier of this subject is estimates. To keep track of theconstants in these estimates would be both wasteful and confusing. Although incertain aspects of stability and control theory it is essential to name and catalogall constants, that is not the case here.

Thus we denote most constants by Gor G; the values of these constants may change from line to line, even thoughthey are denoted with the same letter. In some contexts we shall use the nowpopular symbol ;S to mean "is less than or equal to a constant times It is a pleasure to thank Estela Gavosto andMarco Peloso who wrote up the notes from my lectures. They put in a lot ofextra effort to correct my omissions and clean up my proofs and presentations.

### Table of Contents

I also thank the other students who listened to my thoughts and provided usefulremarks. When the Euclidean plane is studied as a real analytic object, it is convenientto study differential equations using the partial differential operators a a ax and ay. This is so at least in part because each of these operators has a null space namely the functions depending only on y and the functions depending only onx, respectively that plays a significant role in our analysis think of the methodof guessing solutions to a linear differential equation having the form u x v y.

Finally, the Laplacian is written in complex notation as1.

## Complex Analysis and Differential Equations

When X is an open set, this notion is self-explanatory. When X is an arbitrary set, it is rather complicated, but possible,to obtain a complete understanding see [STSI]. For the purposes of this book, we need to understand the case when X isa closed set in Euclidean space. In this circumstance we say that f is C kon X if there is an open neighborhood U of X and a C k function j on U 1such that the restriction of to X equals f. We write f E Ck X. This definition is equivalent to allother reasonable definitions of C k for a non-open set. We shall present a moredetailed discussion of this matter in Section 3.

Now let us formulate the Dirichlet problem on the disc D.

## Partial Differential Equations and Complex Analysis - CRC Press Book

Let U bean open set with nontrivial intersection with S see Figure 1. Here v denotes the unit normal direction at z E S. Notice that the solution to the Dirichlet problem posed above is unique: ifUl and U2 both solve the problem, then Ul - U2 is a hannonic function havingzero boundary values on D. In particular, in the Dirichlet problem the specifying of boundary values alsouniquely determines the normal derivative of the solution function u. However, in order to obtain uniqueness in the Cauchy problem, we mustspecify both the value of U on S and the normal derivative of u on S.

How canthis be? The reason is that the Dirichlet problem is posed with a simple closedboundary curve; the Cauchy problem is instead a local one. Questions of whenfunction theory reflects algebraic topology are treated, for instance, by the deRham theorem and the Atiyah-Singer index theorem.

It follows from 1. In fact, the sum in 1.

This last formula allows one to do the estimates to check for uniform conver-gence, and thus to justify the change of order of the sum and the integral. We might hope that u is then the solution of theDirichlet problem with data f.

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In fact, the proof shows that the convergence is uniform in. This shows that u is continuousat the boundary it is obviously continuous elsewhere and completes the proof. Lipschitz Spaces This is a venerable question in thetheory of partial differential equations and will be a recurring theme throughoutthis book. In order to formulate the question precisely and give it a carefulanswer, we need suitable function spaces. The most naive function spaces for studying the question formulated in thelast paragraph are the C k spaces, mentioned earlier. However, these spacesare not the most convenient for our study.

The reason, which is of centralimportance, is as follows: We shall learn later, by a method of Hormander[H03], that the boundary regularity of the Dirichlet problem is equivalent to theboundedness of certain singular integral operators see [STSI] on the boundary. Singular integral operators, central to the understanding of many problems inanalysis, are not bounded on the C k spaces. This fact explains the mysteriouslyimprecise formulation of regularity results in many books on partial differentialequations.

It also means that we shall have to work harder to get exact regularityresults. Because of the remarks in the preceding paragraph, we now introduce the scaleof Lipschitz spaces. They will be somewhat familiar, but there will be somenew twists to which the reader should pay careful attention. A comprehensivestudy of these spaces appears in [KR2]. See [KR3] for a discussion ofthese matters. However, singular integraloperators are not bounded on this space. Construct an analogous example, foreach positive integer k, of a function in A k Lip k If U is a bounded open set with smooth boundary and if 9 E Aex U thendoes it follow that 9 extends to be in A ex U?

I Let us now discuss the definition of C k spaces in some detail. However, the analogous statement for Lipschitz spaces is true-see [KR2]. Categories : Complex analysis Differential equations. Hidden categories: Webarchive template wayback links CS1 maint: multiple names: authors list. Namespaces Article Talk. Views Read Edit View history. By using this site, you agree to the Terms of Use and Privacy Policy. Chapter 4 of the book ODEs and Dynamical systems by Gerald Teschl note that at this link it can be downloaded free of charge.

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Home Questions Tags Users Unanswered. Complex differential equations Ask Question. Asked 1 year, 1 month ago. Active 1 year, 1 month ago. Viewed times. Amir Sagiv Amir Sagiv 1, 12 12 silver badges 31 31 bronze badges. One of the best is E. Ince, Ordinary differential equations multiple editions. Others are E. Coddington and N.