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.

arXiv comment23 pages, no figures. Revision based on the referee's comments

References in corpus (6)

• Regularity of Solutions of Linear Second Order Elliptic and Parabolic Boundary Value Problems on Lipschitz Domains
• Eventually Positive Semigroups of Linear Operators
• Eventually and asymptotically positive semigroups on Banach lattices
• Bi-Laplacians on graphs and networks
• A criterion for the uniform eventual positivity of operator semigroups
• Higher order operators on networks: hyperbolic and parabolic theory

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