Operators with Wentzell boundary conditions and the Dirichlet-to-Neumann operator

TL;DR

This paper establishes a connection between operators with Wentzell boundary conditions and the Dirichlet-to-Neumann operator, providing criteria for their generator properties and applications to differential operators.

Contribution

It introduces a novel approach using similarity transformations and perturbation arguments to relate boundary operators with generator properties, extending analysis to abstract Banach spaces.

Findings

Operator $A$ generates an analytic semigroup iff $N$ does on the boundary space.

The approach applies to second order differential operators.

Conditions are provided under which boundary operators influence generator properties.

Abstract

In this paper we relate the generator property of an operator $A$ with (abstract) generalized Wentzell boundary conditions on a Banach space $X$ and its associated (abstract) Dirichlet-to-Neumann operator $N$ acting on a "boundary" space $\partial X$. Our approach is based on similarity transformations and perturbation arguments and allows to split $A$ into an operator $A_{00}$ with Dirichlet-type boundary conditions on a space $X_0$ of states having "zero trace" and the operator $N$. If $A_{00}$ generates an analytic semigroup, we obtain under a weak Hille--Yosida type condition that $A$ generates an analytic semigroup on $X$ if and only if $N$ does so on $\partial X$. Here we assume that the (abstract) "trace" operator $L:X\to\partial X$ is bounded what is typically satisfied if $X$ is a space of continuous functions. Concrete applications are made to various second order differential operators.