Lower semicontinuity, optimization and regularity of variational problems under general PDE constraints

Abstract

We investigate variational properties of integral functionals defined on spaces of measures satisfying a general PDE constraint. The study of these properties is motivated by the following three problems: existence of solutions, optimality conditions of variational solutions, and regularity of optimal design problems. After the introduction, each chapter of this dissertation corresponds to one of the topics listed above. <br /> The first chapter is introductory, we state the main results of this work and discuss how their different subjects relate to each other. In this chapter we also discuss the historical background in which our work originated. <br /> The second chapter, on the study of existence, focuses in providing sufficient and necessary conditions for the weak* lower semicontinuity of a general class of integral functionals defined for PDE constrained spaces of measures. We provide a characterization based on recent developments on the structure of PDE-constrained measures and their relation to a convexity class (quasiconvexity); our methods rely on blow-up techniques, rigidity arguments, and the study of generalized Young measures. <br /> The second chapter is dedicated to the analysis and derivation of saddle-point conditions of minimizers of convex integral functionals defined on spaces of PDE-constrained measures (even in higher generality than in the first chapter). The analysis is carried out by means of convex analysis and duality methods. <br /> Lastly, the fourth chapter discusses the regularity properties of a general model in optimal design. Our variational model involves a Dirichlet energy term (defined for a general class of elliptic operators) and a perimeter term (often associated to the design). In this work, we use Gamma-convergence techniques and derive a monotonicity formula to show a standard lower bound on the density of the perimeter of optimal designs. The conclusion of the results then follows from standard geometric measure theory arguments.