An Introduction To Maximum Principles And Symmetry In Elliptic Problems

Advanced text, originally published in 2000, on differential equations, with plentiful supply of exercises all with detailed hints.

Maximum principles are bedrock results in the theory of second order elliptic equations. This principle, simple enough in essence, lends itself to a quite remarkable number of subtle uses when combined appropriately with other notions. Intended for a wide audience, the book provides a clear and comprehensive explanation of the various maximum principles available in elliptic theory, from their beginning for linear equations to recent work on nonlinear and singular equations.

The main goal of this book is to present results pertaining to various versions of the maximum principle for elliptic and parabolic systems of arbitrary order. In particular, the authors present necessary and sufficient conditions for validity of the classical maximum modulus principles for systems of second order and obtain sharp constants in inequalities of Miranda-Agmon type and in many other inequalities of a similar nature. Somewhat related to this topic are explicit formulas for the norms and the essential norms of boundary integral operators. The proofs are based on a unified approach using, on one hand, representations of the norms of matrix-valued integral operators whose target spaces are linear and finite dimensional, and, on the other hand, on solving certain finite dimensional optimization problems. This book reflects results obtained by the authors, and can be useful to research mathematicians and graduate students interested in partial differential equations.

A collection of self contained state-of-the art surveys. The authors have made an effort to achieve readability for mathematicians and scientists from other fields, for this series of handbooks to be a new reference for research, learning and teaching. - written by well-known experts in the field - self contained volume in series covering one of the most rapid developing topics in mathematics

This book deals with a systematic study of a dynamical system approach to investigate the symmetrization and stabilization properties of nonnegative solutions of nonlinear elliptic problems in asymptotically symmetric unbounded domains. The usage of infinite dimensional dynamical systems methods for elliptic problems in unbounded domains as well as finite dimensional reduction of their dynamics requires new ideas and tools. To this end, both a trajectory dynamical systems approach and new Liouville type results for the solutions of some class of elliptic equations are used. The work also uses symmetry and monotonicity results for nonnegative solutions in order to characterize an asymptotic profile of solutions and compares a pure elliptic partial differential equations approach and a dynamical systems approach. The new results obtained will be particularly useful for mathematical biologists.

This book provides the reader with a gentle path through the multifaceted theory of vector fields, starting from the definitions and the basic properties of vector fields and flows, and ending with some of their countless applications, in the framework of what is nowadays called Geometrical Analysis. Once the background material is established, the applications mainly deal with the following meaningful settings:

This book provides a comprehensive introduction to the mathematical theory of nonlinear problems described by elliptic partial differential equations. These equations can be seen as nonlinear versions of the classical Laplace equation, and they appear as mathematical models in different branches of physics, chemistry, biology, genetics, and engineering and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on the calculus of variations and functional analysis. Concentrating on single-valued or multivalued elliptic equations with nonlinearities of various types, the aim of this volume is to obtain sharp existence or nonexistence results, as well as decay rates for general classes of solutions. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including bifurcation, stability, asymptotic analysis, and optimal regularity of solutions. The method of presentation should appeal to readers with different backgrounds in functional analysis and nonlinear partial differential equations. All chapters include detailed heuristic arguments providing thorough motivation of the study developed later on in the text, in relationship with concrete processes arising in applied sciences. A systematic description of the most relevant singular phenomena described in this volume includes existence (or nonexistence) of solutions, unicity or multiplicity properties, bifurcation and asymptotic analysis, and optimal regularity. The book includes an extensive bibliography and a rich index, thus allowing for quick orientation among the vast collection of literature on the mathematical theory of nonlinear phenomena described by elliptic partial differential equations.

During the week of August 31 - September 4, 1998, a conference in honour of Vladimir Maz'ya was held in Rostock as a satellite meeting of the World Congress of Mathematicians. It was sponsored by the German Research Founda tion (Deutsche Forschungsgemeinschaft) and the Ministry of Education and Cul tural Affairs of the land Mecklenburg-Vorpommern. During his forty year career Maz'ya contributed to so many areas of mathematical analysis that such a broad topic of the conference as "Functional Analysis, Partial Differential Equations and Applications" sounds quite nat~al. The conference was organized by the Depart ment of Mathematics of the University of Rostock and the Weierstrass Institute of Applied Analysis and Stochastics in Berlin on the occasion of his 60th birth day. For many years Maz'ya was connected with mathematicians from Berlin and Rostock through his work in potential theory, in differential and pseudodifferen tial equations and in approximation theory. In 1990 he was awarded an honorary doctorate by the University of Rostock. Shortly before the meeting, one of its organizers, an outstanding mathematician and Maz'ya's dear friend, Siegfried Pr6Bdorf died. This was a heavy loss for the and for the conference in particular. During the German mathematical community meeting the rector of the University of Rostock, Prof. Wildenhain, the director of the Weierstrass Institute, Prof. Sprekels, and Prof. Maz'ya remembered S. Pr6Bdorf very warmly. The conference was attended by 109 mathematicians from 21 countries, and the program included 24 invited lectures and 63 short communications.