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Optimization of Steklov-Neumann eigenvalues

by H. Ammari and K. Imeri and N. Nigam

(Report number 2019-37)

Abstract
This paper examines the Laplace equation with mixed boundary conditions, the Neumann and Steklov boundary conditions. This models a container with holes in it, like a pond filled with water but partly covered by immovable pieces on the surface. The main objective is to determine the right extent of the covering pieces, so that any shock inside the container yields a resonance. To this end, an algorithm is developed which uses asymptotic formulas concerning perturbations of the partitioning of the boundary pieces. Proofs for these formulas are established. Furthermore, this paper displays some results concerning bounds and examples with regards to the governing problem.

Keywords: Steklov eigenvalue problem, boundary integral operators, mixed boundary conditions, perturbations formulas.

@Techreport{AIN19_841,
  author = {H. Ammari and K. Imeri and N. Nigam},
  title = {Optimization of Steklov-Neumann eigenvalues},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-37},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-37.pdf },
  year = {2019}
}

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