Published:
Berlin, Heidelberg: Springer Berlin Heidelberg, 1963
Published in:Die Grundlehren der Mathematischen Wissenschaften, In Einzeldarstellungen mit Besonderer Berücksichtigung der Anwendungsgebiete ; 116 Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics ; 116 SpringerLink ; Bücher Springer eBook Collection ; Mathematics and Statistics
Description:
I: Functional analysis -- I. Distribution theory -- II. Some special spaces of distributions -- II: Differential operators with constant coefficients -- III. Existence and approximation of solutions of differential equations -- IV. Interior regularity of solutions of differential equations -- V. The Cauchy problem (constant coefficients) -- III: Differential operators with variable coefficients -- VI. Differential equations which have no solutions -- VII. Differential operators of constant strength -- VIII. Differential operators with simple characteristics -- IX. The Cauchy problem (variable coefficients) -- X. Elliptic boundary problems -- Appendix. Some algebraic lemmas -- Index of notations.
The aim of this book is to give a systematic study of questions con cerning existence, uniqueness and regularity of solutions of linear partial differential equations and boundary problems. Let us note explicitly that this program does not contain such topics as eigenfunction expan sions, although we do give the main facts concerning differential operators which are required for their study. The restriction to linear equations also means that the trouble of achieving minimal assumptions concerning the smoothness of the coefficients of the differential equations studied would not be worth while; we usually assume that they are infinitely differenti able. Functional analysis and distribution theory form the framework for the theory developed here. However, only classical results of functional analysis are used. The terminology employed is that of BOURBAKI. To make the exposition self-contained we present in Chapter I the elements of distribution theory that are required. With the possible exception of section 1.8, this introductory chapter should be bypassed by a reader who is already familiar with distribution theory.