Existence and relaxation problems in optimal shape design

Abstract

A general abstract theorem on existence of solutions to optimal shape design problems for systems governed by partial differential equations, or variational inequalities or hemivariational inequalities is formulated and two main properties (conditions) responsible for the existence are discussed. When one of them fails one have to make "relaxation" in order to get some generalized optimal shapes. In particular, some relaxation "in state", based on $\Gamma$ convergence, is presented in details for elliptic, parabolic and hyperbolic PDEs (and then for optimal shape design problems), while the relaxation "in cost functional" is discussed for some special classes of functionals.