# Partial Differential Equations

Springer Science & Business Media, Aug 12, 2002 - Mathematics - 325 pages
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.

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### Contents

 What Are Partial Differential Equations? 1 The Maximum Principle 31 Methods Based on the Maximum 51 Parabolic Methods The Heat 77 The Wave Equation and Its Connections with the Laplace 113 Through the Darboux Equation 117 The Heat Equation Semigroups and Brownian Motion 127 The Dirichlet Principle Variational Methods for the Solu 157
 Sobolev Spaces and L2 Regularity Theory 193 Strong Solutions 243 The Regularity Theory of Schauder and the Continuity 255 The Moser Iteration Method and the Regularity Theorem 275 Appendix Banach and Hilbert Spaces The LpSpaces 309 Index 323 Copyright