Some remarks on the optimization of eigenvalue problems involving the p-Laplacian
Given a bounded domain \(\Omega \subset \mathbb{R}^n\), numbers \(p \gt 1\), \(\alpha \geq 0\) and \(A \in [0,|\Omega |]\), consider the optimization problem: find a subset \(D \subset \Omega \), of measure \(A\), for which the first eigenvalue of the operator \(u\mapsto -\text{div} (|\nabla u|^{p-2...
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| Format: | Article |
| Language: | English |
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AGH Univeristy of Science and Technology Press
2008-01-01
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| Series: | Opuscula Mathematica |
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| Online Access: | http://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2841.pdf |
| _version_ | 1819050661689425920 |
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| author | Wacław Pielichowski |
| author_facet | Wacław Pielichowski |
| author_sort | Wacław Pielichowski |
| collection | DOAJ |
| description | Given a bounded domain \(\Omega \subset \mathbb{R}^n\), numbers \(p \gt 1\), \(\alpha \geq 0\) and \(A \in [0,|\Omega |]\), consider the optimization problem: find a subset \(D \subset \Omega \), of measure \(A\), for which the first eigenvalue of the operator \(u\mapsto -\text{div} (|\nabla u|^{p-2}\nabla u)+ \alpha \chi_D |u|^{p-2}u \) with the Dirichlet boundary condition is as small as possible. We show that the optimal configuration \(D\) is connected with the corresponding positive eigenfunction \(u\) in such a way that there exists a number \(t\geq 1\) for which \(D=\{u \leq t\}\). We also give a new proof of symmetry of optimal solutions in the case when \(\Omega \) is Steiner symmetric and \(p = 2\). |
| first_indexed | 2024-12-21T11:51:36Z |
| format | Article |
| id | doaj.art-184b4cf197e04bac9e1d1d8b34bcea1d |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-12-21T11:51:36Z |
| publishDate | 2008-01-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-184b4cf197e04bac9e1d1d8b34bcea1d2022-12-21T19:05:02ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742008-01-012845615662841Some remarks on the optimization of eigenvalue problems involving the p-LaplacianWacław Pielichowski0Cracow University of Technology, Institute of Mathematics, ul. Warszawska 24, 31-155 Cracow, PolandGiven a bounded domain \(\Omega \subset \mathbb{R}^n\), numbers \(p \gt 1\), \(\alpha \geq 0\) and \(A \in [0,|\Omega |]\), consider the optimization problem: find a subset \(D \subset \Omega \), of measure \(A\), for which the first eigenvalue of the operator \(u\mapsto -\text{div} (|\nabla u|^{p-2}\nabla u)+ \alpha \chi_D |u|^{p-2}u \) with the Dirichlet boundary condition is as small as possible. We show that the optimal configuration \(D\) is connected with the corresponding positive eigenfunction \(u\) in such a way that there exists a number \(t\geq 1\) for which \(D=\{u \leq t\}\). We also give a new proof of symmetry of optimal solutions in the case when \(\Omega \) is Steiner symmetric and \(p = 2\).http://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2841.pdf\(p\)-Laplacianthe first eigenvalueSteiner symmetry |
| spellingShingle | Wacław Pielichowski Some remarks on the optimization of eigenvalue problems involving the p-Laplacian Opuscula Mathematica \(p\)-Laplacian the first eigenvalue Steiner symmetry |
| title | Some remarks on the optimization of eigenvalue problems involving the p-Laplacian |
| title_full | Some remarks on the optimization of eigenvalue problems involving the p-Laplacian |
| title_fullStr | Some remarks on the optimization of eigenvalue problems involving the p-Laplacian |
| title_full_unstemmed | Some remarks on the optimization of eigenvalue problems involving the p-Laplacian |
| title_short | Some remarks on the optimization of eigenvalue problems involving the p-Laplacian |
| title_sort | some remarks on the optimization of eigenvalue problems involving the p laplacian |
| topic | \(p\)-Laplacian the first eigenvalue Steiner symmetry |
| url | http://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2841.pdf |
| work_keys_str_mv | AT wacławpielichowski someremarksontheoptimizationofeigenvalueproblemsinvolvingtheplaplacian |