Ultracontractivity of Heat semigroups in $\mathrm{L}^{2}\left( \Omega \right)$ with non-local Robin boundary conditions using Nash's inequality

TL;DR

This paper investigates the ultracontractivity of heat semigroups associated with elliptic operators under non-local Robin boundary conditions on Lipschitz domains, using Nash's inequality.

Contribution

It establishes ultracontractivity results for heat semigroups with general non-local Robin boundary conditions, extending previous work to more general boundary operators.

Findings

Ultracontractivity holds under mild assumptions on boundary operator B.

Nash's inequality is effectively used to prove ultracontractivity.

Results apply to elliptic operators with non-local boundary conditions on Lipschitz domains.

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 $d>2$, 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 $H^{1}( \Omega )$ under quite mild assumptions on $B$.