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Unformatted text preview: Partial Differential Equations T. W. K¨ orner after Joshi and Wassermann October 12, 2002 Small print These notes are a digest of much more complete notes by M. S. Joshi and A. J. Wassermann which are also being issued for this course. I should very much appreciate being told of any corrections or possible improvements and might even part with a small reward to the first finder of particular errors. This document is written in L A T E X2e and stored in the file labelled ~twk/IIB/PDE.tex on emu in (I hope) read permitted form. My email address is twk@dpmms . Contents 1 Introduction 2 1.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Ordinary differential equations 6 2.1 The contraction mapping theorem . . . . . . . . . . . . . . . . 6 2.2 Vector fields, integral curves and flows . . . . . . . . . . . . . 8 2.3 First order linear and semilinear PDEs . . . . . . . . . . . . . 10 2.4 First order quasilinear PDEs . . . . . . . . . . . . . . . . . . 11 3 Distributions 13 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 The support of a distribution . . . . . . . . . . . . . . . . . . 17 3.3 Fourier transforms and the Schwartz space . . . . . . . . . . . 19 3.4 Tempered distributions . . . . . . . . . . . . . . . . . . . . . . 21 4 Convolution and fundamental solutions 23 4.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2 Fundamental solutions . . . . . . . . . . . . . . . . . . . . . . 27 4.3 Our first fundamental solution . . . . . . . . . . . . . . . . . . 28 4.4 The parametrix . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 4.5 Existence of the fundamental solution . . . . . . . . . . . . . . 30 5 The Laplacian 32 5.1 A fundamental solution . . . . . . . . . . . . . . . . . . . . . . 32 5.2 Identities and estimates . . . . . . . . . . . . . . . . . . . . . 33 5.3 The dual Dirichlet problem for the unit ball . . . . . . . . . . 36 6 Dirichlet’s problem for the ball and Poisson’s formula 38 7 The wave equation 40 8 The heat equation 45 8.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 8.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 9 References 48 1 Introduction 1.1 Generalities When studying ordinary (i.e. not partial) differential equations we start with linear differential equations with constant coefficients. We then study linear differential equations and then plunge into a boundless ocean of non linear differential equations. The reader will therefore not be surprised if most of a first course on the potentially much more complicated study of partial differential equations should limit itself essentially to linear partial differential equations....
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This note was uploaded on 02/18/2012 for the course MATH 533 taught by Professor Drewarmstrong during the Fall '11 term at University of Miami.
 Fall '11
 DrewArmstrong
 Equations, The Land

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