Emmanuele DiBenedetto

Ugo Pietro Gianazza

Vincenzo Vespri

Degenerate and singular parabolic equations have been the subject of extensive research for the last 25 years. Despite important achievements, the issue of the Harnack inequality for non-negative solutions to these equations, both of p-Laplacian and porous medium type, while raised by several authors, has remained basically open. Recently considerable progress has been made on this issue, to the point that, except for the singular sub-critical range, both for the p-laplacian and the porous medium equations, the theory is reasonably complete. It seemed therefore timely to trace a comprehensive overview, that would highlight the main issues and also the problems that still remain open. The authors give a comprehensive treatment of the Harnack inequality for non-negative solutions to p-laplace and porous medium type equations, both in the degenerate (p/i”2 or im/i”1) and in the singular range (1“ip/i2 or 0“im/i

Harnack's Inequality for Degenerate and Singular Parabolic Equations

This text is meant to be a self-contained, elementary introduction to Partial Differential Equations, assuming only advanced differential calculus and some basic LP theory. Although the basic equations treated in this book, given its scope, are linear, we have made an attempt to approach them from a nonlinear perspective. Chapter I is focused on the Cauchy-Kowaleski theorem. We discuss the notion of characteristic surfaces and use it to classify partial differential equations. The discussion grows out of equations of second order in two variables to equations of second order in N variables to p.d.e.'s of any order in N variables. In Chapters II and III we study the Laplace equation and connected elliptic theory. The existence of solutions for the Dirichlet problem is proven by the Perron method. This method clarifies the structure ofthe sub(super)harmonic functions and is closely related to the modern notion of viscosity solution. The elliptic theory is complemented by the Harnack and Liouville theorems, the simplest version of Schauder's estimates and basic LP -potential estimates. Then, in Chapter III, the Dirichlet and Neumann problems, as well as eigenvalue problems for the Laplacian, are cast in terms of integral equations. This requires some basic facts concerning double layer potentials and the notion of compact subsets of LP, which we present.

Partial Differential Equations

Gui-Qiang Chen

This volume is a collection of original research papers and expository articles stemming from the scientific program of the Nonlinear PDE Emphasis Year held at Northwestern University (Evanston, IL) in March 1998. The book offers a cross-section of the most significant recent advances and current trends and directions in nonlinear partial differential equations and related topics. The book's contributions offer two perspectives. There are papers on general analytical treatment of the theory and papers on computational methods and applications originating from significant realistic mathematical models of natural phenomena. Also included are articles that bridge the gap between these two perspectives, seeking synergistic links between theory and modeling and computation. The volume offers direct insight into recent trends in PDEs. This volume is also available on the Web. Those who purchase the print edition can gain free access by going to www.ams.org/conm/.

International Conference on Nonlinear Partial Differential Equations and Applications, March 21-24, 1998, Northwestern University

Nonlinear Partial Differential Equations

The second edition of this classic textbook presents a rigorous and self-contained introduction to real analysis with the goal of providing a solid foundation for future coursework and research in applied mathematics. Written in a clear and concise style, it covers all of the necessary subjects as well as those often absent from standard introductory texts. Each chapter features a “Problems and Complements” section that includes additional material that briefly expands on certain topics within the chapter and numerous exercises for practicing the key concepts. The first eight chapters explore all of the basic topics for training in real analysis, beginning with a review of countable sets before moving on to detailed discussions of measure theory, Lebesgue integration, Banach spaces, functional analysis, and weakly differentiable functions. More topical applications are discussed in the remaining chapters, such as maximal functions, functions of bounded mean oscillation, rearrangements, potential theory, and the theory of Sobolev functions. This second edition has been completely revised and updated and contains a variety of new content and expanded coverage of key topics, such as new exercises on the calculus of distributions, a proof of the Riesz convolution, Steiner symmetrization, and embedding theorems for functions in Sobolev spaces. Ideal for either classroom use or self-study, Real Analysis is an excellent textbook both for students discovering real analysis for the first time and for mathematicians and researchers looking for a useful resource for reference or review. Praise for the First Edition: “[This book] will be extremely useful as a text. There is certainly enough material for a year-long graduate course, but judicious selection would make it possible to use this most appealing book in a one-semester course for well-prepared students.” —Mathematical Reviews

Real Analysis

This graduate text in real analysis is a solid building block for research in analysis, PDEs, the calculus of variations, probability, and approximation theory. It covers all the core topics, such as a basic introduction to functional analysis, and it discusses other topics often not addressed including Radon measures, the Besicovitch covering Theorem, the Rademacher theorem, and a constructive presentation of the Stone-Weierstrass Theoroem.

Emmanuele DiBenedetto

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