Using Self-adjoint Extensions in Shape Optimization

Self-adjoint extensions of elliptic operators are used to model the solution of a partial differential equation defined in a singularly perturbed domain. The asymptotic expansion of the solution of a Laplacian with respect to a small parameter e is first performed in a domain perturbed by the creation of a small hole. The resulting singular perturbation is approximated by choosing an appropriate self-adjoint extension of the Laplacian, according to the previous asymptotic analysis. The sensitivity with respect to the position of the center of the small hole is then studied for a class of functionals depending on the domain. A numerical application for solving an inverse problem is presented. Error estimates are provided and a link to the notion of topological derivative is established.