This book is an outgrowth of a course which we have given almost periodically over the last eight years. It is addressed to beginning graduate students of mathematics, engineering, and the physical sciences. Chapter I presents all the elementary Hilbert space theory that is needed for the book. Chapter II is an introduction to distributions and Sobolev spaces. Chapter III is an exposition of the theory of linear elliptic boundary value problems in variational form. Chapter IV is an exposition of the generation theory of linear semigroups of contractions and its applications to solve initial-boundary value problems for partial differential equations. Chapter VII begins with some reflections on Chapter~III and develops into an elementary alternative treatment of certain elliptic boundary value problems by the classical Dirichlet principle. |