Optimization of Steklov-Neumann eigenvalues. (English) Zbl 1453.35056
Summary: 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.
MSC:
| 35J08 | Green’s functions for elliptic equations |
| 35B20 | Perturbations in context of PDEs |
| 35P05 | General topics in linear spectral theory for PDEs |
| 76B10 | Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing |
| 65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
Keywords:
Steklov eigenvalue problem; boundary integral operators; mixed boundary conditions; perturbations formulasReferences:
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