The Wentzell Laplacian via forms and the approximative trace
Wolfgang Arendt, Manfred Sauter · Discrete and Continuous Dynamical Systems - S · 2022
We use form methods to define suitable realisations of the Laplacian on a domain \begin{document}$ \Omega $\end{document} with Wentzell boundary conditions, i.e. such that \begin{document}$ \partial_ { \rm{{n}}} u + \beta u + \Delta u = 0 $\end{document} holds in a suitable sense on the boundary of \begin{document}$ \Omega $\end{document} . For those realisations, we study their semigroup generation properties. Using the approximative trace, we give a unified treatment that in part allows irregular and even fractal domains. Moreover, we admit \begin{document}$ \beta $\end{document} to be merely essentially bounded and complex-valued. If the domain is Lipschitz, we obtain a kernel continuous up to the boundary.