Shape optimization for interior Neumann and transmission eigenvalues

TL;DR

This paper explores the complex problem of shape optimization for interior Neumann and transmission eigenvalues, presenting numerical methods and results for cases lacking theoretical solutions, highlighting the challenges and advancements in this area.

Contribution

It introduces numerical approaches for optimizing interior transmission eigenvalues and extends shape optimization techniques to non-self-adjoint, non-elliptic problems.

Findings

Numerical results for maximization of Neumann eigenvalues using boundary integral equations.

First numerical results for minimization of interior transmission eigenvalues.

Challenges in theoretical understanding of higher Neumann eigenvalue maximizers.

Abstract

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.