Shape optimization for interior Neumann and transmission eigenvalues
For researchers in spectral shape optimization, this work provides numerical methods for problems where theoretical results are scarce or nonexistent, though the approach is incremental.
This paper presents numerical shape optimization for interior Neumann and transmission eigenvalues, achieving maximization of Neumann eigenvalues via boundary integral equations and a nonlinear eigenvalue solver, and providing the first numerical results for minimization of interior transmission eigenvalues.
Shape optimization problems for interior eigenvalues is a very challenging task since already the computation of interior eigenvalues for a given shape is far from trivial. For example, a concrete maximizer with respect to shapes of fixed area is theoretically established only for the first two non-trivial Neumann eigenvalues. The existence of such a maximizer for higher Neumann eigenvalues is still unknown. Hence, the problem should be addressed numerically. Better numerical results are achieved for the maximization of some Neumann eigenvalues using boundary integral equations for a simplified parametrization of the boundary in combination with a non-linear eigenvalue solver. Shape optimization for interior transmission eigenvalues is even more complicated since the corresponding transmission problem is non-self-adjoint and non-elliptic. For the first time numerical results are presented for the minimization of interior transmission eigenvalues for which no single theoretical result is yet available.