Linear Partial Differential Operators

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. Front Matter....Pages I-VII Distribution theory....Pages 1-33 Some special spaces of distributions....Pages 33-62 Existence and approximation of solutions of differential equations....Pages 63-96 Interior regularity of solutions of differential equations....Pages 96-114 The Cauchy problem (constant coefficients)....Pages 114-155 Differential equations which have no solutions....Pages 156-170 Differential operators of constant strength....Pages 170-180 Differential operators with simple characteristics....Pages 180-230 The Cauchy problem (variable coefficients)....Pages 230-241 Elliptic boundary problems....Pages 242-274 Back Matter....Pages 275-287