Dirichlet-to-Neumann operators on manifolds

TL;DR

This paper studies the Dirichlet-to-Neumann operator on manifolds, proving it generates an analytic semigroup, which implies related elliptic operators with Wentzell boundary conditions also generate compact, analytic semigroups.

Contribution

It establishes the generation of analytic semigroups by the Dirichlet-to-Neumann operator on manifolds, extending understanding of boundary value problems in this setting.

Findings

Dirichlet-to-Neumann operator generates an analytic semigroup of angle π/2.

Elliptic operators with Wentzell boundary conditions generate compact, analytic semigroups.

Results apply to the space of continuous functions on the boundary and the manifold.

Abstract

We consider the Dirichlet-to-Neumann operator associated to a strictly elliptic operator on the space $\mathrm{C}(\partial M)$ of continuous functions on the boundary $\partial M$ of a compact manifold $\overline{M}$ with boundary. We prove that it generates an analytic semigroup of angle $\frac{\pi}{2}$. This yields that the corresponding strictly elliptic operator with Wentzell boundary conditions generates a compact and analytic semigroups of angle $\frac{\pi}{2}$ on the space $\mathrm{C}(\overline{M})$.