Partial differential equations 1. Foundations and integral representations. With consideration of lectures by E. Heinz. (Q411317)

The revised and enlarged version of this volume gives a special emphasis to geometric and complex variable methods involving integral representations. The new second edition of this volume has been thoroughly revised and a new section on the boundary behavior of Cauchy's integral has been added. The main topics treated in this book are the following: (i) integration and differentiation on manifolds; (ii) foundations of functional analysis; (iii) Brouwer's mapping degree; (iv) generalized analytic functions; (v) potential theory and spherical harmonics; and (vi) linear partial differential equations. Chapter 1 deals with the differentiation and integration on manifolds, where the improper Riemannian integral is used. Chapter 2 is concerned with some fundamental techniques in functional analysis. The author presents Daniell's integral and is interested both in the Riemann integral and in the Lebesgue integration theory. In Chapter 3 topological properties of mappings in \({\mathbb R}^n\) are studied, and nonlinear systems of equations are solved. In this context a basic tool is the topological Brouwer's degree of a mapping. In Chapter 4 the author develops the theory of holomorphic functions in one and several complex variables. Chapter 5 is devoted to the potential theory in \({\mathbb R}^n\). Chapter 6 deals with the qualitative theory of linear partial differential equations. A basic tool in the arguments is played by the maximum principle. This unique book will become a standard reference and it should be found in every mathematical library.