Abstract:We study heat equations $\frac{\partial u}{\partial t} - \operatorname{div} \left( A \nabla u \right) = 0$ on bounded Lipschitz domains $\Omega$ 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 $\nu \cdot A \nabla u + Bu = 0$ where $\nu$ is the outer unit normal on $\partial\Omega$ and $B \in \mathcal{L} \left( \mathrm{L}^{2}\left( \partial \Omega \right) \right)$ 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 $H^{1}( \Omega )$.
| Subjects: | Analysis of PDEs (math.AP); Functional Analysis (math.FA) |
| Cite as: | arXiv:2605.13413 [math.AP] |
| (or arXiv:2605.13413v3 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2605.13413 arXiv-issued DOI via DataCite |
From: Christoph Schwerdt [view email]
[v1]
Wed, 13 May 2026 12:09:11 UTC (13 KB)
[v2]
Mon, 18 May 2026 22:02:05 UTC (1 KB) (withdrawn)
[v3]
Tue, 26 May 2026 12:33:28 UTC (15 KB)
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