Elliptic fourth-order operators with Wentzell boundary conditions on Lipschitz domains

TL;DR

This paper develops a theoretical framework for elliptic fourth-order operators with Wentzell boundary conditions on Lipschitz domains, establishing well-posedness, regularity, and asymptotic properties of solutions.

Contribution

It introduces an abstract form-based approach for analyzing these operators, characterizes solution regularity, and extends results to a broad class of boundary conditions.

Findings

Generated an analytic semigroup on $L^2(\Omega) imes L^2(\Gamma)$.

Fully characterized the domain in terms of Sobolev regularity.

Proved Hölder regularity and discussed asymptotic behavior of solutions.

Abstract

For bounded domains $\Omega$ with Lipschitz boundary $\Gamma$, we investigate boundary value problems for elliptic operators with variable coefficients of fourth order subject to Wentzell (or dynamic) boundary conditions. Using form methods, we begin by showing general results for an even wider class of operators defined via two (intertwined) quadratic forms by defining very abstract concepts of weak traces. Even in this general setting, we prove generation of an analytic semigroup on the product space $L^2(\Omega) \times L^2(\Gamma)$. Using recent results concerning weak co-normal traces, we apply our abstract theory to the elliptic fourth-order case and are able to fully characterize the domain in terms of Sobolev regularity, also obtaining H\"older-regularity of solutions. Finally, we also discuss asymptotic behavior and (eventual) positivity.