Fractional-order operators: Boundary problems, heat equations

TL;DR

This paper surveys fractional Laplacian operators and their boundary problems, and presents recent sharp regularity results for associated heat equations, including new higher regularity estimates and boundary regularity limitations.

Contribution

It provides a comprehensive survey of fractional Laplacian boundary problems and introduces new higher regularity estimates for the related heat equations.

Findings

Sharp interior regularity and $L_p$-estimates up to the boundary

New higher regularity estimates in $L_2$-spaces

Boundary $C^ abla$-regularity cannot generally be improved to $C^ abla$-smoothness

Abstract

The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential methods. The second half takes up the associated heat equation with homogeneous Dirichlet condition. Here we recall recently shown sharp results on interior regularity and on $L_p$-estimates up to the boundary, as well as recent H\"older estimates. This is supplied with new higher regularity estimates in $L_2$-spaces using a technique of Lions and Magenes, and higher $L_p$-regularity estimates (with arbitrarily high H\"older estimates in the time-parameter) based on a general result of Amann. Moreover, it is shown that an improvement to spatial $C^\infty $-regularity at the boundary is not in general possible.