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Fine Regularity of Solutions of Elliptic Partial Differential Equations
Jan Malý, Charles University, Prague, Czech Republic, and William P. Ziemer, Indiana University, Bloomington
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Mathematical Surveys and Monographs
1997; 291 pp; hardcover
Volume: 51
ISBN-10: 0-8218-0335-2
ISBN-13: 978-0-8218-0335-6
List Price: US$91 Member Price: US$72.80
Order Code: SURV/51

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.

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

"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