A priori estimates for solutions of degenerate fully nonlinear elliptic equations with $L^p$ data

TL;DR

This paper derives optimal regularity estimates for solutions of degenerate fully nonlinear elliptic equations with integrable data, including interior $C^{1,eta}$ and log-Lipschitz estimates, using novel methods.

Contribution

It introduces new a priori regularity estimates for degenerate elliptic equations with $L^p$ data, including a Schauder-type estimate via an approximation lemma.

Findings

Proves interior $C^{1,eta}$ estimates for $p>n$

Establishes log-Lipschitz continuity in the critical case $f \,in \, L^{n,1}$

Develops uniform H"older estimates using sliding paraboloid and cusp methods

Abstract

We establish a priori regularity estimates for viscosity solutions of degenerate fully nonlinear elliptic equations with integrable right-hand sides. When the nonhomogeneous term belongs to $L^p$ with $p>n$, we prove optimal interior $C^{1,\alpha}$ estimates. In the critical case, we obtain a log-Lipschitz modulus of continuity under the Lorentz condition $f\in L^{n,1}$. We utilize sliding paraboloid or cusp methods to develop uniform H\"older estimates for equations that are elliptic only in suitable gradient regimes. Finally, we establish an approximation lemma for integrable right-hand sides via a corrector argument, which allows us to deduce the corresponding Schauder-type estimates.