Non-Positivity of the heat equation with non-local Robin boundary conditions
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arXiv
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| Format: | Preprint |
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2024
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| author | Glück, Jochen Mui, Jonathan |
| author_facet | Glück, Jochen Mui, Jonathan |
| contents | We study heat equations $\partial_t u - \operatorname{div}(A\nabla u) = 0$ on bounded Lipschitz domains $Ω$, where $-\operatorname{div}(A\nabla\,\cdot\,)$ is a second-order uniformly elliptic operator with generalised Robin boundary conditions. These boundary conditions are formally given by $ν\cdot A\nabla u + Bu=0$, where $B\in\mathcal{L}(L^2(\partialΩ))$ is a general operator. In contrast to large parts of the literature on non-local Robin boundary conditions, we also allow for operators $B$ that destroy the positivity preserving property of the solution semigroup. Nevertheless, we obtain ultracontractivity of the semigroup under quite mild assumptions on $B$. For a certain class of operators $B$ we demonstrate that the semigroup is in fact eventually positive rather than positivity preserving. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_15114 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-Positivity of the heat equation with non-local Robin boundary conditions Glück, Jochen Mui, Jonathan Analysis of PDEs Spectral Theory Primary: 35J25, 35P05, Secondary: 46B42, 47B65 We study heat equations $\partial_t u - \operatorname{div}(A\nabla u) = 0$ on bounded Lipschitz domains $Ω$, where $-\operatorname{div}(A\nabla\,\cdot\,)$ is a second-order uniformly elliptic operator with generalised Robin boundary conditions. These boundary conditions are formally given by $ν\cdot A\nabla u + Bu=0$, where $B\in\mathcal{L}(L^2(\partialΩ))$ is a general operator. In contrast to large parts of the literature on non-local Robin boundary conditions, we also allow for operators $B$ that destroy the positivity preserving property of the solution semigroup. Nevertheless, we obtain ultracontractivity of the semigroup under quite mild assumptions on $B$. For a certain class of operators $B$ we demonstrate that the semigroup is in fact eventually positive rather than positivity preserving. |
| title | Non-Positivity of the heat equation with non-local Robin boundary conditions |
| topic | Analysis of PDEs Spectral Theory Primary: 35J25, 35P05, Secondary: 46B42, 47B65 |
| url | https://arxiv.org/abs/2404.15114 |