The Bi-Laplacian with Wentzell boundary conditions on Lipschitz domains

TL;DR

This paper studies the mathematical properties of the Bi-Laplacian operator with Wentzell boundary conditions on Lipschitz domains, establishing spectral, regularity, and positivity results using advanced functional analysis techniques.

Contribution

It provides a comprehensive analysis of the Bi-Laplacian with Wentzell boundary conditions, including domain characterization, regularity, spectrum, and semigroup behavior, on Lipschitz domains.

Findings

Operator has compact resolvent and generates a holomorphic semigroup

Full Sobolev space characterization of the domain

Proved H"older regularity and spectral properties

Abstract

We investigate the Bi-Laplacian with Wentzell boundary conditions in a bounded domain $\Omega\subseteq\mathbb{R}^d$ with Lipschitz boundary $\Gamma$. More precisely, using form methods, we show that the associated operator on the ground space $L^2(\Omega)\times L^2(\Gamma)$ has compact resolvent and generates a holomorphic and strongly continuous real semigroup of self-adjoint operators. Furthermore, we give a full characterization of the domain in terms of Sobolev spaces, also proving H\"older regularity of solutions, allowing classical interpretation of the boundary condition. Finally, we investigate spectrum and asymptotic behavior of the semigroup, as well as eventual positivity.