Topics in elliptic problems: from semilinear equations to shape optimization
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914719335448576 |
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| author | Tavares, Hugo |
| author_facet | Tavares, Hugo |
| contents | In this paper, which corresponds to an updated version of the author's Habilitation lecture in Mathematics, we do an overview of several topics in elliptic problems. We review some old and new results regarding the Lane-Emden equation, both under Dirichlet and Neumann boundary conditions, then focus on sign-changing solutions for Lane-Emden systems. We also survey some results regarding fully nontrivial solutions to gradient elliptic systems with mixed cooperative and competitive interactions. We conclude by exhibiting results on optimal partition problems, with cost functions either related to Dirichlet eigenvalues or to the Yamabe equation. Several open problems are referred along the text. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_08997 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Topics in elliptic problems: from semilinear equations to shape optimization Tavares, Hugo Analysis of PDEs 35B09, 35B33, 35B38, 35B45, 35B65, 35J20, 35J50, 35J61, 35R35, 49Q05, 58J05, 58E05 In this paper, which corresponds to an updated version of the author's Habilitation lecture in Mathematics, we do an overview of several topics in elliptic problems. We review some old and new results regarding the Lane-Emden equation, both under Dirichlet and Neumann boundary conditions, then focus on sign-changing solutions for Lane-Emden systems. We also survey some results regarding fully nontrivial solutions to gradient elliptic systems with mixed cooperative and competitive interactions. We conclude by exhibiting results on optimal partition problems, with cost functions either related to Dirichlet eigenvalues or to the Yamabe equation. Several open problems are referred along the text. |
| title | Topics in elliptic problems: from semilinear equations to shape optimization |
| topic | Analysis of PDEs 35B09, 35B33, 35B38, 35B45, 35B65, 35J20, 35J50, 35J61, 35R35, 49Q05, 58J05, 58E05 |
| url | https://arxiv.org/abs/2311.08997 |