This book is intended to be both a thorough introduction to contemporary research in optimization theory for elliptic systems with its numerous applications and a textbook at the undergraduate and graduate level for courses in pure or applied mathematics or in continuum mechanics. Various processes of modern technology and production are described by el- liptic partial differential equations. Optimization of these processes reduces to op- timization problems for elliptic systems. The numerical solution of such problems is associated with the solution of the following questions. 1. The setting...