Optimization problems for solutions of elliptic equations and stability issues

Introduction: In this thesis we address some problems related to two topics in the Calculus of Variations which have attracted a growing interest in recent decades. In a general simpli cation we can refer to those two elds as optimization problems for shapes and for solutions of elliptic equations and quantitative stability problems for geometric and functional inequalities. It is worth bearing in mind that the historical and mathematical development of these two classes of problems have merged and looking at them as separate elds may not be the best approach to adopt. On the other hand a division of the works presented may simplify the reading, for this reason the thesis is divided into two main parts. In the rst, Part I, containing three chapters, we deal with optimization problems related to the shape optimization eld. In Part II, we address the quantitative stability of three problems, the rst one regarding a class of isoperimetric inequalities, the second one about a spectral optimization problem and the third one concerning a class of functional inequalities.