Oberseminar "Mathematik des Maschinellen Lernens und Angewandte Analysis" - Dr. Eloi Martinet

We are interested in the shape optimization problem which consists of maximizing the eigenvalues of the Laplacian with homogeneous Neumann conditions. These eigenvalues are particularly relevant to acoustic and thermal problems, and in particular to the "hot spot conjecture". Unlike Dirichlet eigenvalues, those associated with the Neumann problem are rather unstable in nature, making the optimization problem difficult. We'll see how some numerical explorations of the problem for domains in the plane and the sphere allowed to reveal certain properties of the optima.