Diffusion with nonlocal Robin boundary conditions

We investigate a second order elliptic differential operator A(beta,mu) on a bounded, open set Omega subset of R-d with Lipschitz boundary subject to a nonlocal boundary condition of Robin type. More precisely we have 0 <= beta is an element of L-infinity(partial derivative Omega) and mu : partial derivative Omega -> Mu((Omega) over bar), and boundary conditions of the form partial derivative(A)(V) u(z) + beta(z)u(z) = integral((Omega) over bar)u(x) mu(z)(dx), z is an element of partial derivative Omega, where partial derivative(A)(V) 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 semi group T-beta,T-mu on L-infinity (Omega) which enjoys the strong Feller property. In particular, it takes values in C((Omega) over bar). Its restriction to C((Omega) over bar) is strongly continuous and holo-morphic. We also establish positivity and contractivity of the semigroup under additional assumptions and study the asymptotic behavior of the semigroup.