The Wentzell Laplacian via forms and the approximative trace

TL;DR

This paper develops a unified framework using form methods and the approximative trace to define and analyze the Wentzell Laplacian on irregular and fractal domains, accommodating complex boundary conditions and coefficients.

Contribution

It introduces a novel approach to defining Wentzell Laplacians via forms, extending applicability to irregular domains and complex boundary parameters.

Findings

Semigroup generation properties are established for the Wentzell Laplacian.

The approach handles irregular, fractal domains and complex coefficients.

A kernel continuous up to the boundary is obtained for Lipschitz domains.

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.