Analytic semigroups generated by Dirichlet-to-Neumann operators on manifolds

We consider the Dirichlet-to-Neumann operator associated to a strictly elliptic operator on the space $$\mathrm {C}(\partial M)$$ C ( ∂ M ) of continuous functions on the boundary $$\partial M$$ ∂ M of a compact manifold $$\overline{M}$$ M ¯ with boundary. We prove that it generates an analytic semigroup of angle $$\frac{\pi }{2}$$ π 2 , generalizing and improving a result of Escher with a new proof. Combined with the abstract theory of operators with Wentzell boundary conditions developed by Engel and the author, this yields that the corresponding strictly elliptic operator with Wentzell boundary conditions generates a compact and analytic semigroups of angle $$\frac{\pi }{2}$$ π 2 on the space $$\mathrm {C}(\overline{M})$$ C ( M ¯ ) .

The Bi-Laplacian with Wentzell Boundary Conditions on Lipschitz Domains

We investigate the Bi-Laplacian with Wentzell boundary conditions in a bounded domain $$\Omega \subseteq \mathbb {R}^d$$ Ω ⊆ R d with Lipschitz boundary $$\Gamma $$ Γ . More precisely, using form methods, we show that the associated operator on the ground space $$L^2(\Omega )\times L^2(\Gamma )$$ L 2 ( Ω ) × L 2 ( Γ ) 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.