Two problems in Shape Optimization

Shape optimization is an old classical field of research, which has inspired deep mathematical theories as well as numerous crucial applications in industry and engineering. A shape optimization problem consists on finding the shape which minimizes a certain cost functional while satisfying given constraints. We want to find a solution $\Omega^*$ on $\mathcal{U}$ of $$\min\limits_{\Omega \in \mathcal{U}} \mathcal{J}(\Omega),$$ where $\mathcal{U}$ is a class of subsets of $\mathbb{R}^n$, called set of the admissible shapes, and $\mathcal{J}$ is a functional with values in $\mathbb{R}$ that depends on the solution of a given partial differential equation, that varies with respect to the domain. We want to treat two different kinds of optimization problem based on the way the solution of the given partial differential equation varies with respect to the domain. In the first part of the thesis we describe the general theory for shape optimization problems with Dirichlet condition on the free boundary. Given a bounded open set $\Omega \subseteq \mathbb R^n$ and an open subset $\omega \subseteq \Omega$ belonging to the set of the possible shapes $\mathcal{U}$, we have the elliptic equation $$\begin{cases} -\Delta y = f \\ y|_{\partial{\omega}} = 0, \end{cases}$$ where $f$ is a fixed function in $H^{-1}(\Omega)$. Our aim is to compute $$\min_{\omega \in \mathcal U} \mathcal{J}(\omega),$$ where the functional $\mathcal{J}$ depends on the solution $y_{\omega}$ of this equation. We want to study our problem under rather general assumptions using the theory of Calculus of Variations. First of all we describe different notions of convergence on domains of $\mathbb R^n$ and various results on the continuity of the solution, presenting examples and counterexamples on the existence of a minimum. Furthermore, we introduce the technique of relaxation and we write optimality conditions. In the second part of the work we have a functional $\mathcal{J}$ that depends on the solution $y_{\omega}$ of an elliptic equation with source ...