Partial Differential Equations

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modem as weIl as the classical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathematical Sci ences (AMS) series, which will focus on advanced textbooks and research level monographs.

Texte du rabat
This text provides an introduction to the theory of partial differential equations. It introduces basic examples of partial differential equations, arising in continuum mechanics, electromagnetism, complex analysis and other areas, and develops a number of tools for their solution, including particularly Fourier analysis, distribution theory, and Sobolev spaces. These tools are applied to the treatment of basic problems in linear PDE, including the Laplace equation, heat equation, and wave equation, as well as more general elliptic, parabolic, and hyperbolic equations. Companion texts, which take the theory of partial differential equations further, are AMS volume 116, treating more advanced topics in linear PDE, and AMS volume 117, treating problems in nonlinear PDE. This book is addressed to graduate students in mathematics and to professional mathematicians, with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis.

Contenu
Basic Theory of ODE and Vector Fields.- A Nonsmooth vector fields.- References.- 2 The Laplace Equation and Wave Equation.- References.- 3 Fourier Analysis, Distributions, and Constant-Coefficient Linear PDE.- References.- 4 Sobolev Spaces.- References.- 5 Linear Elliptic Equations.- Spaces of generalized functions on manifolds with boundary.- The Mayer-Vietoris sequence in deRham cohomology.- References.- 6 Linear Evolution Equations.- Some Banach spaces of harmonic functions.- The stationary phase method.- References.- A Outline of Functional Analysis.- References.- B Manifolds, Vector Bundles, and Lie Groups.- References.