Annali di Matematica Pura ed Applicata

, Volume 186, Issue 2, pp 341–358

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

  • Julián Fernández Bonder
  • Pablo Groisman
  • Julio D. Rossi

DOI: 10.1007/s10231-006-0009-y

Cite this article as:
Bonder, J.F., Groisman, P. & Rossi, J.D. Annali di Matematica (2007) 186: 341. doi:10.1007/s10231-006-0009-y


The best Sobolev trace constant is given by the first eigenvalue of a Steklov-like problem. We deal with minimizers of the Rayleigh quotient ‖u2H1(Ω)2/‖u2L2(∂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.


Steklov eigenvaluesSobolev trace embeddingShape derivative

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Julián Fernández Bonder
    • 1
  • Pablo Groisman
    • 2
  • Julio D. Rossi
    • 3
  1. 1.Departamento de Matemática, FCEyNUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.Instituto de Cálculo, FCEyNUniversidad de Buenos AiresBuenos AiresArgentina
  3. 3.Consejo Superior de Investigaciones Científicas (CSIC)MadridSpain