This book deals with the tangential oblique derivative problem for second order linear and non-linear elliptic and parabolic operators. In a large survey a lot of the most interesting results obtained during the last 30 years are proposed. Historically, the problem was stated first by Poincar? when studying the tides, but the same problem arises in the theory of Brownian motion, too. The main difficulties in investigating this problem are due to the fact that at the points of tangency between the vector field, representing the boundary operator, and the boundary of the domain the Lopatinskii condition is failed and boundary value problems with infinite dimensional kernel or cokernel can appear. By using subelliptic type estimates for pseudodifferential operators in Sobolev and H?lder spaces many interesting results have been proved for linear problems during the last 30 years. The authors propose for the first time an investigation of the degenerate oblique derivative problem for semilinear elliptic and parabolic operators. To do this, they use subelliptic estimates (Egorov, H?rmander, Tr?ves, Winzell, Guan, Sawyer) and the Leray-Schauder fixed point principle. In this way theorems on existence, uniqueness and regularity of the classical solutions in H?lder classes are derived. In a lot of cases considered the coefficients are not infinitely smooth, and the set of degeneration of the problem is a rather massive one, i.e., it is not obliged to be a submanifold of the boundary and can have positive measure.

Boundary problems constitute an essential field of common mathematical interest, they lie in the center of research activities both in analysis and geometry. This book encompasses material from both disciplines, and focuses on their interactions which are particularly apparent in this field. Moreover, the survey style of the contributions makes the topics accessible to a broad audience with a background in analysis or geometry, and enables the reader to get a quick overview.

The Oblique Derivative Problem (ODP), introduced and first studied by Henry Poincaré, is one of the classical problems not only in the theory of Partial Differential Equations but also in Mathematical Physics. This is the first monograph, written by one of the leading scientists in this area, which is completely devoted to the ODP. All main results in this field are described with full proofs based on modern techniques. The book contains a lot of results that have been unknown to a wide audience till now. A special chapter containing extensive material from geometry, functional analysis and differential equations, which is used in the proofs, makes the book self–contained to a large extent. A short Appendix containig open problems will stimulate the reader to further research in this area.

This book is an easy-to-read reference providing a link between functional analysis and diffusion processes. More precisely, the book takes readers to a mathematical crossroads of functional analysis (macroscopic approach), partial differential equations (mesoscopic approach), and probability (microscopic approach) via the mathematics needed for the hard parts of diffusion processes. This work brings these three fields of analysis together and provides a profound stochastic insight (microscopic approach) into the study of elliptic boundary value problems. The author does a massive study of diffusion processes from a broad perspective and explains mathematical matters in a more easily readable way than one usually would find. The book is amply illustrated; 14 tables and 141 figures are provided with appropriate captions in such a fashion that readers can easily understand powerful techniques of functional analysis for the study of diffusion processes in probability. The scope of the author’s work has been and continues to be powerful methods of functional analysis for future research of elliptic boundary value problems and Markov processes via semigroups. A broad spectrum of readers can appreciate easily and effectively the stochastic intuition that this book conveys. Furthermore, the book will serve as a sound basis both for researchers and for graduate students in pure and applied mathematics who are interested in a modern version of the classical potential theory and Markov processes. For advanced undergraduates working in functional analysis, partial differential equations, and probability, it provides an effective opening to these three interrelated fields of analysis. Beginning graduate students and mathematicians in the field looking for a coherent overview will find the book to be a helpful beginning. This work will be a major influence in a very broad field of study for a long time.

The Sixth International Colloquium on Differential Equations was organized by the Institute for Basic Science of Inha University, the International Federation of Nonlinear Analysts, the Mathematical Society of Japan, the Pharmaceutical Faculty of the Medical University of Sofia, the University of Catania, and UNESCO, with the cooperation of a number of international mathematical organizations, and was held at the Technical University of Plovdiv, Bulgaria, from 18 to 23 August 1995. This proceedings volume contains selected talks which deal with various aspects of differential and partial differential equations.

This mathematical monograph is a study of interior regularity of weak solutions of second order linear divergence form equations with degenerate ellipticity and rough coefficients. The authors show that solutions of large classes of subelliptic equations with bounded measurable coefficients are H lder continuous. They present two types of results f

This 3rd edition provides an insight into the mathematical crossroads formed by functional analysis (the macroscopic approach), partial differential equations (the mesoscopic approach) and probability (the microscopic approach) via the mathematics needed for the hard parts of Markov processes. It brings these three fields of analysis together, providing a comprehensive study of Markov processes from a broad perspective. The material is carefully and effectively explained, resulting in a surprisingly readable account of the subject. The main focus is on a powerful method for future research in elliptic boundary value problems and Markov processes via semigroups, the Boutet de Monvel calculus. A broad spectrum of readers will easily appreciate the stochastic intuition that this edition conveys. In fact, the book will provide a solid foundation for both researchers and graduate students in pure and applied mathematics interested in functional analysis, partial differential equations, Markov processes and the theory of pseudo-differential operators, a modern version of the classical potential theory.