Diffusion with nonlocal boundary conditions

Abstract

We consider second order differential operators $A_{\mu }$ on a bounded, Dirichlet regular set $\Omega \subset {\mathbb{R}}^d$, subject to the nonlocal boundary conditions \[ u(z) = \int_Ωu(x)\, μ(z, dx)\quad \mbox{for } z \in \partial Ω. \] Here the function $\mu :\partial \Omega \to {\mathcal{M}}^{+}(\Omega )$ is $\sigma (\mathcal{M}(\Omega ),C_b(\Omega ))$-continuous with $0\le \mu (z,\Omega )\le 1$ for all $z\in \partial \Omega$. Under suitable assumptions on the coefficients in $A_{\mu }$, we prove that $A_{\mu }$ generates a holomorphic positive contraction semigroup $T_{\mu }$ on $L^{\infty }(\Omega )$. The semigroup $T_{\mu }$ is never strongly continuous, but it enjoys the strong Feller property in the sense that it consists of kernel operators and takes values in $C(\bar{\Omega })$. We also prove that $T_{\mu }$ is immediately compact and study the asymptotic behavior of $T_{\mu }(t)$ as $t\to \infty$.

18 pages, no figures; comments of the referees incorporated

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