Regularity of spectral fractional Dirichlet and Neumann problems

TL;DR

This paper investigates the regularity properties of solutions to fractional Dirichlet and Neumann problems for elliptic operators, extending classical results to fractional powers and nonsmooth cases using advanced functional calculus.

Contribution

It extends classical regularity results to fractional elliptic boundary value problems, including nonsmooth domains, using complex interpolation and $H^ abla$-calculus techniques.

Findings

Regularity results in Sobolev and Hölder spaces for fractional elliptic problems

Extension of classical theory to nonsmooth domains and low regularity

Overview of boundary problems related to the fractional Laplacian

Abstract

Consider the fractional powers $(A_{\operatorname{Dir}})^a$ and $(A_{\operatorname{Neu}})^a$ of the Dirichlet and Neumann realizations of a second-order strongly elliptic differential operator $A$ on a smooth bounded subset $\Omega $ of ${\Bbb R}^n$. Recalling the results on complex powers and complex interpolation of domains of elliptic boundary value problems by Seeley in the 1970's, we demonstrate how they imply regularity properties in full scales of $H^s_p$-Sobolev spaces and H\"older spaces, for the solutions of the associated equations. Extensions to nonsmooth situations for low values of $s$ are derived by use of recent results on $H^\infty $-calculus. We also include an overview of the various Dirichlet- and Neumann-type boundary problems associated with the fractional Laplacian.