# Fine Regularity of Solutions of Elliptic Partial Differential Equations

American Mathematical Soc., 1997 - Mathematics - 291 pages
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 Holder 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.

### What people are saying -Write a review

We haven't found any reviews in the usual places.

### Contents

 Chapter 1 Preliminaries 1 Chapter 2 Potential Theory 63 Chapter 3 Quasilinear Equations 161 Chapter 4 Fine Regularity Theory 185 Chapter 5 Variational Inequalities Regularity 233
 Chapter 6 Existence Theory 253 References 273 Index 283 Notation Index 289 Copyright