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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 H1(Ω)/||u|| 2 L2(∂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. © 2006 Springer-Verlag.

Registro:

Documento: Artículo
Título:Optimization of the first Steklov eigenvalue in domains with holes: A shape derivative approach
Autor:Bonder, J.F.; Groisman, P.; Rossi, J.D.
Filiación:Departamento de Matemática, Universidad de Buenos Aires, Ciudad Universitaria (1428), Buenos Aires, Argentina
Instituto de Cálculo, Universidad de Buenos Aires, Ciudad Universitaria (1428), Buenos Aires, Argentina
Consejo Superior de Investigaciones Científicas (CSIC), Serrano 117, Madrid, Spain
Palabras clave:Shape derivative; Sobolev trace embedding; Steklov eigenvalues
Año:2007
Volumen:186
Número:2
Página de inicio:341
Página de fin:358
DOI: http://dx.doi.org/10.1007/s10231-006-0009-y
Título revista:Annali di Matematica Pura ed Applicata
Título revista abreviado:Ann. Mat. Pura Appl.
ISSN:03733114
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03733114_v186_n2_p341_Bonder

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Citas:

---------- APA ----------
Bonder, J.F., Groisman, P. & Rossi, J.D. (2007) . Optimization of the first Steklov eigenvalue in domains with holes: A shape derivative approach. Annali di Matematica Pura ed Applicata, 186(2), 341-358.
http://dx.doi.org/10.1007/s10231-006-0009-y
---------- CHICAGO ----------
Bonder, J.F., Groisman, P., Rossi, J.D. "Optimization of the first Steklov eigenvalue in domains with holes: A shape derivative approach" . Annali di Matematica Pura ed Applicata 186, no. 2 (2007) : 341-358.
http://dx.doi.org/10.1007/s10231-006-0009-y
---------- MLA ----------
Bonder, J.F., Groisman, P., Rossi, J.D. "Optimization of the first Steklov eigenvalue in domains with holes: A shape derivative approach" . Annali di Matematica Pura ed Applicata, vol. 186, no. 2, 2007, pp. 341-358.
http://dx.doi.org/10.1007/s10231-006-0009-y
---------- VANCOUVER ----------
Bonder, J.F., Groisman, P., Rossi, J.D. Optimization of the first Steklov eigenvalue in domains with holes: A shape derivative approach. Ann. Mat. Pura Appl. 2007;186(2):341-358.
http://dx.doi.org/10.1007/s10231-006-0009-y