Robin Laplacians with non-local boundary conditions

Abstract

We study semigroups $(e^{t\mathrm{\Delta }}{)}_{t\ge 0}$ generated by Laplacians associated to generalised Robin boundary conditions $$\begin{array}{rl}-\mathrm{\Delta }u& =f\text{in }\mathrm{\Omega }\\ \frac{\partial u}{\partial \nu }+Bu& =0\text{on }\partial \mathrm{\Omega }\end{array}$$ on a bounded Lipschitz domain $\mathrm{\Omega }$, where $B$ is a bounded linear operator on $L^2(\partial \mathrm{\Omega }).$ The classical case is when $B$ is a multiplication operator given by a function $\beta \in L^{\mathrm{\infty }}(\partial \mathrm{\Omega })$, but the general setting above also includes non-local boundary conditions, for instance if $B$ is an integral operator given by a kernel function. We take an abstract approach to the analysis, using tools from the theory of sesquilinear forms, Banach lattices, and positive semigroups. Under natural assumptions on the operator $B$, we show that the associated semigroup is ultracontractive (that is, the operator $e^{t\mathrm{\Delta }}$ maps $L^2$ into $L^{\mathrm{\infty }}$ for every $t>0$), a result which is well-known in the classical case. We also obtain sufficient conditions for eventual positivity of the semigroup.

This is joint work with Jochen Glück, and is supported by the Deutsche Forschungsgemeinschaft (DFG Project 515394002).