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

We study heat equations $\frac{\mathrm{\partial }u}{\mathrm{\partial }t}-\mathrm{div}\left(A\mathrm{\nabla }u\right)=0$ on bounded Lipschitz domains $\mathrm{\Omega }$ in ${\mathbb{R}}^d$ for $d\in \mathbb{N}$, where $-\mathrm{div}(A\mathrm{\nabla }\cdot )$ is a second-order uniformly elliptic operator with generalised Robin boundary conditions. These boundary conditions are formally given by $\nu \cdot A\mathrm{\nabla }u+Bu=0$ where $\nu$ is the outer unit normal on $\mathrm{\partial }\mathrm{\Omega }$ and $B\in \mathcal{L}\left({\mathrm{L}}^2\left(\mathrm{\partial }\mathrm{\Omega }\right)\right)$ 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 $H^1(\mathrm{\Omega })$.