Optimization of the first Steklov eigenvalue in domains with holes: A shape derivative approach
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 constan...
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2007
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03733114_v186_n2_p341_Bonder http://hdl.handle.net/20.500.12110/paper_03733114_v186_n2_p341_Bonder |
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paper:paper_03733114_v186_n2_p341_Bonder |
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paper:paper_03733114_v186_n2_p341_Bonder2023-06-08T15:37:52Z Optimization of the first Steklov eigenvalue in domains with holes: A shape derivative approach Shape derivative Sobolev trace embedding Steklov eigenvalues 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. 2007 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03733114_v186_n2_p341_Bonder http://hdl.handle.net/20.500.12110/paper_03733114_v186_n2_p341_Bonder |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Shape derivative Sobolev trace embedding Steklov eigenvalues |
| spellingShingle |
Shape derivative Sobolev trace embedding Steklov eigenvalues Optimization of the first Steklov eigenvalue in domains with holes: A shape derivative approach |
| topic_facet |
Shape derivative Sobolev trace embedding Steklov eigenvalues |
| description |
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. |
| title |
Optimization of the first Steklov eigenvalue in domains with holes: A shape derivative approach |
| title_short |
Optimization of the first Steklov eigenvalue in domains with holes: A shape derivative approach |
| title_full |
Optimization of the first Steklov eigenvalue in domains with holes: A shape derivative approach |
| title_fullStr |
Optimization of the first Steklov eigenvalue in domains with holes: A shape derivative approach |
| title_full_unstemmed |
Optimization of the first Steklov eigenvalue in domains with holes: A shape derivative approach |
| title_sort |
optimization of the first steklov eigenvalue in domains with holes: a shape derivative approach |
| publishDate |
2007 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03733114_v186_n2_p341_Bonder http://hdl.handle.net/20.500.12110/paper_03733114_v186_n2_p341_Bonder |
| _version_ |
1768544505519144960 |