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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.
List of contents
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.
About the author
Lars Hörmander, born 1931 in Sweden, did his secondary schooling as well as his undergraduate and doctoral studies in Lund. His principle teacher and adviser at the University of Lund was Marcel Riesz until he returned, then Lars Gårding. In 1956 he worked in the USA, at the universities of Chicago, Kansas, Minnesota and New York, before returning to a chair at the University of Stockholm. He remained a frequent visitor to the US, particularly to Stanford and was Professor at the IAS, Princeton from 1964 to 1968. In 1968 he accepted a chair at the University of Lund, Sweden, where, today, he is Emeritus Professor.§§Hörmander's lifetime work has been devoted to the study of partial differential equations and its applications in complex analysis. In 1962 he was awarded the Fields Medal for his contributions to the general theory of linear partial differential operators.
