Generalized Nonlocal Robin Laplacian on Arbitrary Domains

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 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 Neumann Laplacian semigroup unless the jump measure \(\theta\) vanishes. Finally, the convergence of semigroups sequences \(e^{-t\Delta_{\kappa_n,\theta_n}}\) is investigated in the case of vague convergence and \(\gamma-\)convergence of admissible pair of measures \((\kappa_n,\theta_n)\).