Partial Differential Equations: (214 Graduate Texts in Mathematics)

This textbook is intended for students who wish to obtain an introduction to the theory of partial di?erential equations (PDEs, for short), in particular, those of elliptic type. Thus, it does not o?er a comprehensive overview of the whole ?eld of PDEs, but tries to lead the reader to the most important methods and central results in the case of elliptic PDEs. The guiding qu- tion is how one can ?nd a solution of such a PDE. Such a solution will, of course, depend on given constraints and, in turn, if the constraints are of the appropriate type, be uniquely determined by them. We shall pursue a number of strategies for ?nding a solution of a PDE; they can be informally characterized as follows: (0) Write down an explicit formula for the solution in terms of the given data (constraints). This may seem like the best and most natural approach, but this is possible only in rather particular and special cases. Also, such a formula may be rather complicated, so that it is not very helpful for detecting qualitative properties of a solution. Therefore, mathematical analysis has developed other, more powerful, approaches. (1) Solve a sequence of auxiliary problems that approximate the given one, and show that their solutions converge to a solution of that original pr- lem. Di?erential equations are posed in spaces of functions, and those spaces are of in?nite dimension.

Table of Contents:
Introduction: What Are Partial Differential Equations?.- The Laplace Equation as the Prototype of an Elliptic Partial Differential Equation of Second Order.- The Maximum Principle.- Existence Techniques I: Methods Based on the Maximum Principle.- Existence Techniques II: Parabolic Methods. The Heat Equation.- Reaction-Diffusion Equations and Systems.- The Wave Equation and its Connections with the Laplace and Heat Equations.- The Heat Equation, Semigroups, and Brownian Motion.- The Dirichlet Principle. Variational Methods for the Solution of PDEs (Existence Techniques III).- Sobolev Spaces and L2 Regularity Theory.- Strong Solutions.- The Regularity Theory of Schauder and the Continuity Method (Existence Techniques IV).- The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash.

Review :
From the reviews of the second edition: "Because of the nice global presentation, I recommend this book to students and young researchers who need the now classical properties of these second-order partial differential equations. Teachers will also find in this textbook the basis of an introductory course on second-order partial differential equations." - Alain Brillard, Mathematical Reviews "Beautifully written and superbly well-organised, I strongly recommend this book to anyone seeking a stylish, balanced, up-to-date survey of this central area of mathematics." - Nick Lord, The Mathematical Gazette “It is an expanded translation by the author of the German original. … The range of methods is wide, covering integral kernels, maximum principles, variational principles, gradient descents, weak derivatives and Sobolev spaces. … the proof are clear and pleasant, provided the reader has a good command in integration theory. … This book is an interesting introduction to the multiple facets of partial differential equations –– especially to regularity theory –– for the reader who has already a good background in analysis.” (Jean Van Schaftingen, Bulletin of the Belgian Mathematical Society, 2007)