Shape optimization problems involving nonlocal and nonlinear operators
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914833500209152 |
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| author | Dussel, Ignacio Ceresa |
| author_facet | Dussel, Ignacio Ceresa |
| contents | In this research, we investigate a general shape optimization problem in which the state equation is expressed using a nonlocal and nonlinear operator. We prove the existence of a minimum point for a functional $F$ defined on the family of all 'quasi-open' subsets of a bounded open set $Ω$ in $\mathbb{R}^n$. This is ensured under the condition that $F$ demonstrates decreasing behavior concerning set inclusion and is lower semicontinuous with respect to a suitable topology associated with the fractional $p$-Laplacian under Dirichlet boundary conditions. Moreover, we study the asymptotic behavior of the solutions when $s\to1$ and extend this result to the anisotropic case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08579 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Shape optimization problems involving nonlocal and nonlinear operators Dussel, Ignacio Ceresa Analysis of PDEs In this research, we investigate a general shape optimization problem in which the state equation is expressed using a nonlocal and nonlinear operator. We prove the existence of a minimum point for a functional $F$ defined on the family of all 'quasi-open' subsets of a bounded open set $Ω$ in $\mathbb{R}^n$. This is ensured under the condition that $F$ demonstrates decreasing behavior concerning set inclusion and is lower semicontinuous with respect to a suitable topology associated with the fractional $p$-Laplacian under Dirichlet boundary conditions. Moreover, we study the asymptotic behavior of the solutions when $s\to1$ and extend this result to the anisotropic case. |
| title | Shape optimization problems involving nonlocal and nonlinear operators |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.08579 |