Diffusion with nonlocal Robin boundary conditions

Abstract

We investigate a second order elliptic differential operator $A_{\beta, \mu}$ on a bounded, open set $\Omega\subset\mathbb{R}^{d}$ with Lipschitz boundary subject to a nonlocal boundary condition of Robin type. More precisely we have $0\leq \beta\in L^{\infty}(\partial\Omega)$ and $\mu\colon \partial \Omega \to \mathscr{M}(\overline{\Omega})$, and boundary conditions of the form \[ \partial_{\nu}^{\mathscr{A}}u(z)+\beta(z)u(z)=\int_{\overline{\Omega}}u(x)\mu(z)(dx),\ z\in\partial\Omega, \] where $\partial_{\nu}^{\mathscr{A}}$ denotes the weak conormal derivative with respect to our differential operator. Under suitable conditions on the coefficients of the differential operator and the function $\mu$ we show that $A_{\beta, \mu}$ generates a holomorphic semigroup $T_{\beta,\mu}$ on $L^{\infty}(\Omega)$ which enjoys the strong Feller property. In particular, it takes values in $C(\overline{\Omega})$. Its restriction to $C(\overline{\Omega})$ is strongly continuous and holomorphic. We also establish positivity and contractivity of the semigroup under additional assumptions and study the asymptotic behavior of the semigroup.