The Wentzell Laplacian via forms and the approximative trace

We use form methods to define suitable realisations of the Laplacian on a domain $Ω$ with Wentzell boundary conditions, i.e. such that $\partial_{\mathrm{n}}u + βu + Δu = 0$ holds in a suitable sense on the boundary of $Ω$. 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 $β$ to be merely essentially bounded and complex-valued. If the domain is Lipschitz, we obtain a kernel continuous up to the boundary.

💡 Research Summary

The paper develops a comprehensive functional‑analytic framework for Laplace operators equipped with Wentzell (also called Ventcel) boundary conditions on very general domains, including those with irregular or fractal boundaries. The authors combine the theory of sectorial sesquilinear forms with the concept of an approximative trace to define a natural realization of the Wentzell Laplacian in the product space (L^{2}(\Omega)\oplus L^{2}(\Gamma)), where (\Omega\subset\mathbb{R}^{d}) is a bounded open set, (\Gamma=\partial\Omega) its boundary, and (\sigma) a finite positive Borel measure on (\Gamma). The boundary condition is \