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

Abstract

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 ¯ ) .

Strictly elliptic operators with generalized Wentzell boundary conditions on continuous functions on manifolds with boundary

“…Our main theorem Theorem 4.6 generalizes these results to arbitrary strictly elliptic operators A m and C on smooth, compact, orientable Riemannian manifolds with smooth boundary. The situation q = 0 on bounded, smooth domains in R n was studied by Engel and Fragnelli [EF05] and, on smooth, compact, orientable Riemannian manifolds by [Bin18a]. The paper is organized as follows.…”