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TY  - THES
AU  - Bundrock, Lukas
TI  - The Robin eigenvalue in exterior domains
PB  - RWTH Aachen University
VL  - Dissertation
CY  - Aachen
M1  - RWTH-2023-10726
SP  - 1 Online-Ressource : Illustrationen
PY  - 2023
N1  - Veröffentlicht auf dem Publikationsserver der RWTH Aachen University
N1  - Dissertation, RWTH Aachen University, 2023
AB  - In the present thesis we consider eigenvalues of the Robin Laplacian, the Laplace operator with Robin boundary conditions, in the exterior of a compact set. In contrast to the Robin Laplacian in bounded domains, the essential spectrum is not empty. D. Krejcirik and V. Lotoreichik show that there is a discrete eigenvalue below the essential spectrum if and only if α, the parameter of the boundary condition, is smaller than a constant α<sup>*</sup> which depends on the domain.Using the results of G. Auchmuty and Q. Han, we show that α<sup>*</sup> coincides with the first eigenvalue of an appropriate Steklov eigenvalue problem. In addition, we characterize the existence of exactly k discrete eigenvalues of the Robin Laplacian depending on α. If the first eigenvalue of the Robin Laplacian λ<sub>1</sub> is discrete, we consider a geometric optimization problem. We follow the ideas of C. Bandle and A. Wagner and show that the ball is a local maximizer of λ<sub>1</sub>(\mathbbR<sup>n</sup> \―Ω) among all smooth bounded domains Ω ⊆ \mathbbR<sup>n</sup>  with fixed volume. If we exclude translations, the ball is a strict local maximizer. In addition, we show that the exterior problem can be approximated by spherical shells and illustrate the transition from a discrete spectrum to an essential spectrum. Furthermore, we show that the ball is a local maximizer of the first Steklov eigenvalue in exterior domains.
LB  - PUB:(DE-HGF)11
DO  - DOI:10.18154/RWTH-2023-10726
UR  - https://publications.rwth-aachen.de/record/973341
ER  -