Ultracontractivity of Heat semigroups in $\mathrm{L}^{2}\left( Ω\right)$ with non-local Robin boundary conditions using Nash's inequality
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916049008459776 |
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| author | Schwerdt, Christoph |
| author_facet | Schwerdt, Christoph |
| contents | We study heat equations $\frac{\partial u}{\partial t} - \operatorname{div} \left( A \nabla u \right) = 0$ on bounded Lipschitz domains $Ω$ in $\mathbb{R}^{d}$ for $d \in \mathbb{N}$, where $-\operatorname{div} \left( A \nabla \cdot \right)$ 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 $ν$ is the outer unit normal on $\partialΩ$ and $B \in \mathcal{L} \left( \mathrm{L}^{2}\left( \partial Ω\right) \right)$ is a general operator which is allowed to destroy the positivity preserving property of the solution semigroup. Ultracontractivity of the solution semigroup is shown by using Nash's inequality on the Sobolev space $H^{1}( Ω)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_13413 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ultracontractivity of Heat semigroups in $\mathrm{L}^{2}\left( Ω\right)$ with non-local Robin boundary conditions using Nash's inequality Schwerdt, Christoph Analysis of PDEs Functional Analysis We study heat equations $\frac{\partial u}{\partial t} - \operatorname{div} \left( A \nabla u \right) = 0$ on bounded Lipschitz domains $Ω$ in $\mathbb{R}^{d}$ for $d \in \mathbb{N}$, where $-\operatorname{div} \left( A \nabla \cdot \right)$ 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 $ν$ is the outer unit normal on $\partialΩ$ and $B \in \mathcal{L} \left( \mathrm{L}^{2}\left( \partial Ω\right) \right)$ is a general operator which is allowed to destroy the positivity preserving property of the solution semigroup. Ultracontractivity of the solution semigroup is shown by using Nash's inequality on the Sobolev space $H^{1}( Ω)$. |
| title | Ultracontractivity of Heat semigroups in $\mathrm{L}^{2}\left( Ω\right)$ with non-local Robin boundary conditions using Nash's inequality |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2605.13413 |