Regularity of solutions to variable-exponent degenerate mixed fully nonlinear local and nonlocal equations

TL;DR

This paper studies the regularity of solutions to complex variable-exponent mixed local and nonlocal degenerate equations, establishing Lipschitz and interior $C^{1, ext{delta}}$ regularity under certain conditions.

Contribution

It introduces new regularity results for solutions to variable-exponent degenerate equations, including Lipschitz continuity and interior $C^{1, ext{delta}}$ regularity, using viscosity methods and flatness improvement.

Findings

Lipschitz regularity of solutions for $s ext{ in } (1/2,1)$

Existence of solutions via vanishing viscosity method

Interior $C^{1, ext{delta}}$ regularity near $s$ close to 1

Abstract

We consider a class of variable-exponent mixed fully nonlinear local and nonlocal degenerate elliptic equations, which degenerate along the set of critical points, $C:=\big\{x:\,Du(x)=0\big\}.$ Under general conditions, first, we establish the Lipschitz regularity of solutions using the Ishii-Lions viscosity method when the order of the fractional Laplacian, $s\in\big(\frac{1}{2},1\big).$ Due to inapplicability of comparison principle for the equations under consideration, one can not use the classical Perron's method for the existence of a solution. However, using the Lipschitz estimates established in theorem and vanishing viscosity method, we get the existence of solution. We further prove interior $C^{1,\delta}$ regularity of the viscosity solutions using an improvement of the flatness technique when $s$ is close enough to $1.$