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 $Δ_{κ,θ}$ on $L^2(Ω)$ 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 admissible pair of measures $(κ,θ)$ that allows the associated form $\mathcal E_{κ,θ}$ to be closable. The nonlocal Robin Laplacian $Δ_{κ,θ}$ generates a sub-Markovian $C_0-$semigroup on $L^2(Ω)$ which is not dominated by Neumann Laplacian semigroup unless the jump measure $θ$ vanishes. Finally, the convergence of semigroups sequences $e^{-tΔ_{κ_n,θ_n}}$ is investigated in the case of vague convergence and $γ-$convergence of admissible pair of measures $(κ_n,θ_n)$.