Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator

TL;DR

This paper investigates the regularity properties of the Dirichlet-to-Neumann operator associated with fractional powers of sectorial operators, establishing its role as an isomorphism between interpolation spaces in Hilbert spaces.

Contribution

It provides a detailed analysis of the regularity of Dirichlet and Neumann problems and characterizes the Dirichlet-to-Neumann operator as an isomorphism linking fractional powers to interpolation spaces.

Findings

The Dirichlet-to-Neumann operator acts as an isomorphism between interpolation spaces.

The part of the operator in the Hilbert space corresponds exactly to the fractional power.

Regularity properties of boundary value problems are precisely characterized.

Abstract

In the very influential paper \cite{CS07} Caffarelli and Silvestre studied regularity of $(-\Delta)^s$, $0<s<1$, by identifying fractional powers with a certain Dirichlet-to-Neumann operator. Stinga and Torrea \cite{ST10} and Gal\'e, Miana and Stinga \cite{GMS13} gave several more abstract versions of this extension procedure. The purpose of this paper is to study precise regularity properties of the Dirichlet and the Neumann problem in Hilbert spaces. Then the Dirichlet-to-Neumann operator becomes an isomorphism between interpolation spaces and its part in the underlying Hilbert space is exactly the fractional power.