Wentzell boundary conditions in the context of Dirichlet forms

Abstract

Let $X$ be a locally compact space, $m$ a Radon measure on $X$,\, ${{\mathfrak h}}$ a regular Dirichlet form in $L_2(X,m)$. For a Radon measure $\mu$ we interpret ${{\mathfrak h}}$ as a regular Dirichlet form $\tau$ in $L_2(m+\mu)$. We show that $\mu$ decomposes as $\mu_r+\mu_s$, where $\mu_r$ is coupled to ${{\mathfrak h}}$ and $\mu_s$ decouples from ${{\mathfrak h}}$. Additionally to this `space perturbation', a second perturbation is introduced by a measure $\nu$ describing absorption. The main object of the paper is to apply this setting to a study of the Wentzell boundary condition \[ -\alpha Au + n\cdot a\nabla u + \gamma u = 0 \quad \text{on}\ \partial\Omega \] for an elliptic operator $A=-\nabla{\hspace{-0.1em}}\cdot(a\nabla)$, where $\Omega\subseteq{\mathbb{R}}^d$ is open, $n$ the outward normal, and $\alpha$, $\gamma$ are suitable functions. It turns out that the previous setting can be applied with $\mu=\alpha dS$, $\nu=\gamma dS$, under suitable conditions. Besides the description of the $d$-dimensional case we give a more detailed analysis of the one-dimensional case. As a further topic in the general setting we study the question whether mass conservation carries over from the unperturbed form ${{\mathfrak h}}$ to the space perturbed form $\tau$. In the Appendix we extend a known closability criterion from the minimal to the maximal form.