  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.
