Generalized nonlocal Robin Laplacian on arbitrary domains

Abstract

In this paper, we prove that it is always possible to define a realization of the Laplacian $\Delta _{\kappa ,\theta }$ on $L^2(\Omega )$ subject to nonlocal Robin boundary conditions with general jump measures on arbitrary open subsets of ${\mathbb {R}}^N$. This is made possible by using a capacity approach to define an admissible pair of measures $(\kappa ,\theta )$ that allows the associated form ${\mathcal {E}}_{\kappa ,\theta }$ to be closable. The nonlocal Robin Laplacian $\Delta _{\kappa ,\theta }$ generates a sub-Markovian $C_0$-semigroup on $L^2(\Omega )$ which is not dominated by the Neumann Laplacian semigroup unless the jump measure $\theta$ vanishes.