The Bi-Laplacian with Wentzell boundary conditions on Lipschitz domains

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.