Diffusion with nonlocal Robin boundary conditions
  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. 
Wolfgang Arendt, Stefan Kunkel, Markus Kunze
math.FA math.AP