An elliptic regularity theorem for fractional partial differential operators

TL;DR

This paper establishes an elliptic regularity theorem for fractional PDEs involving Riemann-Liouville derivatives, showing that solutions gain regularity in Sobolev spaces when the forcing term is regular.

Contribution

It extends classical elliptic regularity results to fractional derivatives, providing a new theoretical framework for fractional PDEs in Sobolev spaces.

Findings

Solutions gain regularity in Sobolev spaces when forcing terms are regular

The theorem applies to linear elliptic fractional PDEs with Riemann-Liouville derivatives

Potential applications include advanced analysis of fractional differential equations

Abstract

We present and prove a version of the elliptic regularity theorem for partial differential equations involving fractional Riemann-Liouville derivatives. In this case, regularity is defined in terms of Sobolev spaces $H^s(X)$: if the forcing of a linear elliptic fractional PDE is in one Sobolev space, then the solution is in the Sobolev space of increased order corresponding to the order of the derivatives. We also mention a few applications and potential extensions of this result.