Optimization of the first Steklov eigenvalue in domains with holes: A shape derivative approach

J. FERNÁNDEZ BONDER; P. GROISMAN; J. D. ROSSI

ANNALI DI MATEMATICA PURA ED APPLICATA

SPRINGER HEIDELBERG

Año: 2007 vol. 186 p. 341 - 358

0373-3114

The best Sobloev trace constant is given by the first eigenvalue of a Steklov-like problem. We deal with minimizers of the Rayleigh quotient $|u|_{H^1(U)} / |u|_{L^2(partial U)}$ for functions that vanish in a subset $Asubset U$ ­, which we call the hole. We look for holes that minimize the best Sobolev trace constant among subsets of ­ 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 ­ 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 and algorithm to compute the discrete optimal holes.for functions that vanish in a subset $Asubset U$ ­, which we call the hole. We look for holes that minimize the best Sobolev trace constant among subsets of ­ 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 ­ 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 and algorithm to compute the discrete optimal holes.