Higher Sobolev regularity on the mixed local and nonlocal p-Laplace equations

TL;DR

This paper investigates the interior Sobolev regularity of solutions to mixed local and nonlocal p-Laplace equations, establishing new regularity results and fractional differentiability properties across the full range of p.

Contribution

It provides the first systematic analysis of higher Sobolev regularity for solutions to mixed local and nonlocal p-Laplace equations, including novel fractional differentiability results.

Findings

Solutions belong to W^{2,p}_loc and W^{2,2}_loc in subquadratic case

Gradient-related quantities are in W^{1,2}_loc in superquadratic case

Established higher fractional differentiability u in W^{1+β,q}_loc for all p in (1,∞)

Abstract

We develop a systematic study of the interior Sobolev regularity of weak solutions to the mixed local and nonlocal $p$-Laplace equations. To be precise, we show that the weak solution $u$ belongs to $W^{2, p}_\mathrm{loc}$ and even $W^{2, 2}_{\rm loc}$ Sobolev spaces in the subquadratic case, while $|\nabla u|^{\frac{p-2}{2}}\nabla u$ is of the class $W^{1, 2}_\mathrm{loc}$ in the superquadratic scenario, both of which coincide with that of the classical $p$-Laplace equations. Moreover, an improved higher fractional differentiability and integrability result $u\in W^{1+\beta, q}_\mathrm{loc}$ is proved in the full range $p\in (1, \infty)$ for any $q\in [\max\{p, 2\}, \infty)$ and $\beta\in(0, \frac 2q)$. The main analytical tools are the finite difference quotient technique, suitable energy method and tail estimates. As far as we know, our results are new within the context of such mixed problems.