Diffusion with nonlocal Robin boundary conditions

Abstract

We investigate a second order elliptic differential operator 𝐴_𝛽,𝜇 on a bounded, open set Ω ⊂ ℝ^𝑑 with Lipschitz boundary subject to a nonlocal boundary condition of Robin type. More precisely we have 0 ≤ 𝛽 ∈ 𝐿^∞(∂Ω) and 𝜇: ∂Ω→ℳ(\overline{Ω}), and boundary conditions of the form \[ \partial_{\nu}^{\mathscr{A}}u(z)+\beta(z)u(z)=\int_{\overline{\Omega}}u(x)\mu(z)(\mathrm{d}x), \quad z\in\partial\Omega, \] where ∂_𝜈^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 𝜇 we show that 𝐴_𝛽,𝜇 generates a holomorphic semigroup 𝑇_𝛽,𝜇 on 𝐿^∞(Ω) which enjoys the strong Feller property. In particular, it takes values in 𝐶(\overline{Ω}). Its restriction to 𝐶(\overline{Ω}) 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.