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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
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References:

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