**Mathematical Surveys and Monographs**

Volume: 51;
1997;
291 pp;
Hardcover

MSC: Primary 31; 35; 46;

**Print ISBN: 978-0-8218-0335-6
Product Code: SURV/51**

List Price: $102.00

AMS Member Price: $81.60

MAA Member Price: $91.80

**Electronic ISBN: 978-1-4704-1282-1
Product Code: SURV/51.E**

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# Fine Regularity of Solutions of Elliptic Partial Differential Equations

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*Jan Malý; William P. Ziemer*

The primary objective of this book is to give a comprehensive
exposition of results surrounding the work of the authors concerning
boundary regularity of weak solutions of second-order elliptic
quasilinear equations in divergence form. The structure of these
equations allows coefficients in certain \(L^{p}\) spaces, and thus it
is known from classical results that weak solutions are locally
Hölder continuous in the interior. Here it is shown that weak
solutions are continuous at the boundary if and only if a Wiener-type
condition is satisfied. This condition reduces to the celebrated
Wiener criterion in the case of harmonic functions. The work that
accompanies this analysis includes the "fine" analysis of Sobolev
spaces and a development of the associated nonlinear potential
theory. The term "fine" refers to a topology of \(\mathbf R^{n}\) which is
induced by the Wiener condition.

The book also contains a complete development of regularity of solutions of
variational inequalities, including the double obstacle problem, where the
obstacles are allowed to be discontinuous. The regularity of the solution is
given in terms involving the Wiener-type condition and the fine topology. The
case of differential operators with a differentiable structure and
\(\mathcal C^{1,\alpha}\) obstacles is also developed. The book concludes with a chapter
devoted to the existence theory, thus providing the reader with a complete
treatment of the subject ranging from regularity of weak solutions to the
existence of weak solutions.

#### Readership

Graduate students and research mathematicians interested in the theory of regularity of weak solutions of elliptic differential equations, Sobolev space theory, and potential theory.

#### Reviews & Endorsements

Very well written and may be read at different levels. Some parts may be used in a postgraduate course in advanced PDEs but for sure it is useful for all researchers who study regularity of solutions of elliptic PDEs via real analysis techniques.

-- Zentralblatt MATH

This book does a superb job of placing into perspective the regularity devlopments of the past four decades for weak solutions \(u\) to general divergence structure quasilinear second-order elliptic partial differential equations in arbitrary bound domains \(\mathbf \Omega\) of \(n\)-space, that is \(\text{div} A(x, u, \Delta u)= B(x, u, \Delta u)\). The book begins with an excellent preface, and each chapter concludes with historical notes—very welcome sections. There are two notations guides: one at the beginning, for basic notation, and one at the end, a notation index.

-- Bulletin of the London Mathematical Society

#### Table of Contents

# Table of Contents

## Fine Regularity of Solutions of Elliptic Partial Differential Equations

- Contents vii8 free
- Preface ix10 free
- Basic Notation xiii14 free
- Chapter 1. Preliminaries 116 free
- Chapter 2. Potential Theory 6378
- 2.1 Capacity 6378
- 2.2 Laplace equation 92107
- 2.3 Regularity of minimizers 109124
- 2.3.1 Abstract minimization 110125
- 2.3.2 Minimizers and weak solutions 110125
- 2.3.3 Higher regularity 115130
- 2.3.4 The De Giorgi method 118133
- 2.3.5 Moser's iteration technique 122137
- 2.3.6 Removable singularities 126141
- 2.3.7 Estimates of supersolutions 128143
- 2.3.8 Estimates of energy minimizers 131146
- 2.3.9 Dirichlet problem 135150
- 2.3.10 Application of thinness: the Wiener criterion 139154

- 2.4 Fine topology 143158
- 2.5 Fine Sobolev spaces 148163
- 2.6 Historical notes 157172

- Chapter 3. Quasilinear Equations 161176
- Chapter 4. Fine Regularity Theory 185200
- Chapter 5. Variational Inequalities – Regularity 233248
- Chapter 6. Existence Theory 253268
- References 273288
- Index 283298
- Notation Index 289304