Uniform ball condition and existence of optimal shapes for geometric functionals involving boundary-value problems

In this article, we are interested in shape optimization problems where the functionals are defined on the boundary of the domain, involving the geometry of the associated surface and the boundary values of the solution to a state equation posed on the inner domain enclosed by the shape. Hence, we pursue here the study initiated in a previous work by considering a specific class admissible shapes. Given $\varepsilon > 0$ and a fixed non-empty large bounded open hold-all $B \subset \mathbb{R}^{n}$, $n \geqslant 2$, we define $\mathcal{O}_{\varepsilon}(B)$ as the class of open sets $\Omega \subset B$ satisfying a $\varepsilon$-ball condition, which has an equivalent characterization in terms of uniform $C^{1,1}$-regularity of the boundary $\partial \Omega$. The main contribution of this paper is to prove the existence of a minimizer in the class $\mathcal{O}_{\varepsilon}(B)$ for problems of the form: \[ \inf_{\Omega \in \mathcal{O}_{\varepsilon}(B)} \int_{\partial \Omega} j \left[ u_{\Omega} \left( \mathbf{x} \right), \nabla u_{\Omega} \left( \mathbf{x} \right), \mathbf{x}, \mathbf{n} \left( \mathbf{x} \right), H \left(\mathbf{x} \right) \right] dA \left( \mathbf{x} \right) , \] where $u_{\Omega}$ denotes to the solution of the Dirichlet Laplacian posed on the domain $\Omega$ or to the one associated with a Neumann or Robin boundary condition, where $\mathbf{n}$ is the unit outward normal vector, and where $H$ can refer either to the the scalar mean curvature, to the Gaussian curvature, or more generally to any of the symmetric polynomials in the principal curvatures. We only assume here the continuity of $j$ with respect to the set of variables, convexity with respect to the last variable, and quadratic growth regarding the first two variables. We give various applications in the field of partial differential equations such as existence for: \[ \inf_{\Omega \in \mathcal{O}_{\varepsilon}(B)} \int_{\Omega} j \left[ \mathbf{x}, u_{\Omega} \left( \mathbf{x} \right) , \nabla u_{\Omega} \left( \mathbf{x} \right) , \mathrm{Hess}~u_{\Omega} \left( \mathbf{x} \right) \right] dV \left( \mathbf{x} \right), \] and boundary shape identifications in the area of inverse and control problems: \[ \inf_{\substack{\Omega \in \mathcal{O}_{\varepsilon}(B) \\ \Gamma_{0} \subseteq \partial \Omega}} \int_{\Gamma_{0}} \left[ \left( \partial_{n}u_{\Omega} - f_{0} \right)^{2} + \left( u_{\Omega} - g_{0} \right)^{2} \right] dA. \]