Regularity of solutions to degenerate fully nonlinear elliptic equations with variable exponent

TL;DR

This paper establishes sharp local $C^{1,eta}$ regularity for viscosity solutions of a class of degenerate fully nonlinear elliptic equations with variable exponents, using geometric and compactness methods.

Contribution

It introduces a novel combination of geometric tangential methods with an improvement-of-flatness approach for variable exponent degenerate equations.

Findings

Proves sharp local $C^{1,eta}$ regularity of solutions.

Develops a refined method combining geometric tangential techniques with compactness.

Extends regularity results to equations with variable-exponent degeneracies.

Abstract

We consider the fully nonlinear equation with variable-exponent double phase type degeneracies $$ \big[|Du|^{p(x)}+a(x)|Du|^{q(x)}\big]F(D^2u)=f(x). $$ Under some appropriate assumptions, by making use of geometric tangential methods and combing a refined improvement-of-flatness approach with compactness and scaling techniques we obtain the sharp local $C^{1,\alpha}$ regularity of viscosity solutions to such equations.