## Theory and Applications of Partial Differential EquationsThis book is a product of the experience of the authors in teaching partial differential equations to students of mathematics, physics, and engineering over a period of 20 years. Our goal in writing it has been to introduce the subject with precise and rigorous analysis on the one hand, and interesting and significant applications on the other. The starting level of the book is at the first-year graduate level in a U.S. university. Previous experience with partial differential equations is not required, but the use of classical analysis to find solutions of specific problems is not emphasized. From that perspective our treatment is decidedly theoretical. We have avoided abstraction and full generality in many situations, however. Our plan has been to introduce fundamental ideas in relatively simple situations and to show their impact on relevant applications. The student is then, we feel, well prepared to fight through more specialized treatises. There are parts of the exposition that require Lebesgue integration, distributions and Fourier transforms, and Sobolev spaces. We have included a long appendix, Chapter 8, giving precise statements of all results used. This may be thought of as an introduction to these topics. The reader who is not familiar with these subjects may refer to parts of Chapter 8 as needed or become somewhat familiar with them as prerequisite and treat Chapter 8 as Chapter O. |

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### Contents

Introduction to Partial Differential Equations | 1 |

Wave Equation | 11 |

Heat Equation | 53 |

Copyright | |

17 other sections not shown

### Common terms and phrases

applied arbitrary Banach space boundary conditions boundary point boundary values Cauchy problem Chapter characteristic choose compact conservation law consider constant continuous converges convex Corollary cp(x curve define denote density depends derivatives Dirichlet problem discontinuity eigenvalues elliptic entropy entropy condition estimate example Exercise exists exterior finite fixed follows formula given grad harmonic function heat equation hence Hilbert space Hint holds implies inequality initial data initial value problem integral equation Jr Jr Laplace's equation Lemma linear Lipschitz maximum principle neighborhood Neumann problem norm obtain operator Partial Differential Equations proof of Theorem Proposition Prove radius result Riemann problem satisfies scalar Section sequence shock smooth sphere subset sufficiently small Suppose surface Theorem 2.1 uniformly vanishes variables vector wave equation weak solution x e Q zero