Relaxation in Optimization Theory and Variational Calculus

Abstract

Background generalities: order and topology linear and convex analysis optimization theory functions and measure spaces means of continuous functions some differential and integral equations non-cooperative game theory. Theory of convex compactifications: convex compactifications canonical form of convex compactifications approximation of convex compactifications extension of mappings. Young measures and their generalizations: classical Young measures various generalizations approximation theory extensions of Nemytskii mappings. Relaxation in optimization theory: abstract optimization problems optimization problems on Lebesgue spaces example - optimal control of dynamical systems example - elliptic optimal control problems example - parabolic optimal control problems example - optimal control of integral equations. relaxation in variational calculus I: convex compactifications of Sobolev spaces relaxation of variational problems - p > 1 optimality conditions for relaxed problems relaxation of variational problems - p = 1 convex approximation of relaxed problems. relaxation in variational calculus II: prerequisites around quasiconvexity gradient generalized Young functionals relaxation scheme and its FEM-approximation further approximation - an inner case further approximation - an outer case double-well problem - sample calculations. Relaxation in game theory: abstract game-theoretical problems games on Lebesgue spaces example - games with dynamical systems example - elliptic games bibliography list of symbols. (Part contents).