On the regularity theory for mixed local and nonlocal quasilinear elliptic equations

TL;DR

This paper investigates the regularity properties of solutions to equations combining local and nonlocal p-Laplace operators, establishing key continuity and inequality results using analytic methods.

Contribution

It introduces a purely analytic approach to regularity theory for mixed local and nonlocal quasilinear elliptic equations, extending classical results to this combined setting.

Findings

Proved local boundedness of weak subsolutions

Established local Hölder continuity of solutions

Derived Harnack and weak Harnack inequalities

Abstract

We consider a combination of local and nonlocal $p$-Laplace equations and discuss several regularity properties of weak solutions. More precisely, we establish local boundedness of weak subsolutions, local H\"older continuity of weak solutions, Harnack inequality for weak solutions and weak Harnack inequality for weak supersolutions. We also discuss lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. Our approach is purely analytic and it is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. The main results apply to sign changing solutions and capture both local and nonlocal features of the equation.