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

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 $H^{1}( \Omega )$.