Partial Differential Equations Basic Theory

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.

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.