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 Hölder 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.

23 pages, no figures. Revision based on the referee's comments

Cited by in corpus (6)

• Spectrum and convergence of eventually positive operator semigroups
• Stability of (eventually) positive semigroups on spaces of continuous functions
• Spectral properties of locally eventually positive operator semigroups
• Boundary value problems with rough boundary data
• Local uniform convergence and eventual positivity of solutions to biharmonic heat equations
• Criteria for eventual domination of operator semigroups and resolvents