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 $\frac{\partial u}{\partial t}-\mathrm{div}\left(A\nabla u\right)=0$ on bounded Lipschitz domains $\Omega$ in ${\mathbb{R}}^d$ for $d\in \mathbb{N}$, where $-\mathrm{div}\left(A\nabla \cdot \right)$ 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 $H^1(\Omega )$.