The Wentzell Laplacian via forms and the approximative trace

Abstract

<p style='text-indent:20px;'>We use form methods to define suitable realisations of the Laplacian on a domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> with Wentzell boundary conditions, i.e. such that <inline-formula><tex-math id="M2">\begin{document}$ \partial_ { \rm{{n}}} u + \beta u + \Delta u = 0 $\end{document}</tex-math></inline-formula> holds in a suitable sense on the boundary of <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. For those realisations, we study their semigroup generation properties. Using the approximative trace, we give a unified treatment that in part allows irregular and even fractal domains. Moreover, we admit <inline-formula><tex-math id="M4">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> to be merely essentially bounded and complex-valued. If the domain is Lipschitz, we obtain a kernel continuous up to the boundary.</p>