The Wentzell Laplacian via forms and the approximative trace

Abstract

We use form methods to define suitable realisations of the Laplacian on a domain $\Omega$ with Wentzell boundary conditions, i.e. such that $\partial_{\mathrm{n}}u + \beta u + \Delta u = 0$ holds in a suitable sense on the boundary of $\Omega$. For those realisations, we study their semigroup generation properties. Using the approximative trace, we give a unified treatment that in part allows irregular and even fractal domains. Moreover, we admit $\beta$ to be merely essentially bounded and complex-valued. If the domain is Lipschitz, we obtain a kernel continuous up to the boundary.