Shape Optimization for Interior Neumann and Transmission Eigenvalues

Kleefeld, Andreas

doi:10.1007/978-3-030-16077-7_15

Contribution to a book

FZJ-2019-03904

Shape Optimization for Interior Neumann and Transmission Eigenvalues

Kleefeld, A. (Corresponding author)FZJ*

2019
Springer International Publishing Cham
ISBN: 978-3-030-16076-0

Integral Methods in Science and Engineering Cham : Springer International Publishing 185-196 (2019) [10.1007/978-3-030-16077-7_15]

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Please use a persistent id in citations: http://hdl.handle.net/2128/22510 doi:10.1007/978-3-030-16077-7_15

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.

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