Regularity of superposition operators of mixed fractional order

TL;DR

This paper extends regularity theory to superposition operators of mixed fractional order, establishing key inequalities and continuity properties for weak solutions, applicable even in classical linear cases.

Contribution

It introduces new regularity results for mixed fractional operators, including inequalities and continuity properties, broadening understanding beyond previous linear and nonlocal cases.

Findings

Established Caccioppoli inequality with tail for weak subsolutions

Proved local boundedness and Hölder continuity of weak solutions

Demonstrated lower semicontinuity and expansion of positivity for solutions

Abstract

We extend the De Giorgi--Nash--Moser theory to superposition operators of mixed fractional operators. In particular, we investigate several regularity properties for this class of operators. We establish the Caccioppoli-type inequality with tail for weak subsolutions, local boundedness of weak subsolutions, local H\"older continuity of weak solutions, the weak Harnack inequality for weak supersolutions, and the lower semicontinuity of weak supersolutions. Furthermore, we prove the expansion of positivity, a preliminary Harnack inequality, and the upper semicontinuity of weak subsolutions. Our results apply to both fixed-sign and sign-changing solutions involving mixed local--nonlocal superposition fractional operators. Notably, the results are new even in the classical linear case $p=2$, demonstrating the broader applicability of the techniques developed in this work.