Optimization of the first Steklov eigenvalue in domains with holes: a shape derivative approach |
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Authors: | Julián Fernández Bonder Pablo Groisman Julio D Rossi |
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Institution: | (1) Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria (1428), Buenos Aires, Argentina;(2) Instituto de Cálculo, FCEyN, Universidad de Buenos Aires, Pabellón II, Ciudad Universitaria (1428), Buenos Aires, Argentina;(3) Consejo Superior de Investigaciones Científicas (CSIC), Serrano 117, Madrid, Spain |
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Abstract: | The best Sobolev trace constant is given by the first eigenvalue of a Steklov-like problem. We deal with minimizers of the
Rayleigh quotient ‖u‖2
H
1
(Ω)
2/‖u‖2
L
2
(∂Ω) for functions that vanish in a subset A⊂ Ω, which we call the hole. We look for holes that minimize the best Sobolev trace constant among subsets of Ω with prescribed
volume. First, we find a formula for the first variation of the first eigenvalue with respect to the hole. As a consequence
of this formula, we prove that when Ω is a ball the symmetric hole (a centered ball) is critical when we consider deformations
that preserves volume but is not optimal. Finally, we prove that by the Finite Element Method we can approximate the optimal
configuration and, by means of the shape derivative, we design an algorithm to compute the discrete optimal holes.
Mathematics Subject Classification (2000) 35P15, 49K20, 49M25, 49Q10 |
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Keywords: | Steklov eigenvalues Sobolev trace embedding Shape derivative |
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