Global regularity for a class of fully nonlinear PDEs with unbalanced variable degeneracy

TL;DR

This paper proves sharp regularity results for a class of fully nonlinear elliptic PDEs with unbalanced variable degeneracy, advancing understanding of their solutions' smoothness.

Contribution

It introduces new geometric tangential methods to establish global regularity for PDEs with switching degeneracy laws, extending previous results.

Findings

Established $C^{0, eta}$, $C^{0, 1}$, and $C^{1, eta}$ regularity estimates.

Applied results to nonlinear models and free boundary problems.

Utilized geometric tangential methods and compactness techniques.

Abstract

We establish the existence and sharp global regularity results ($C^{0, \gamma}$, $C^{0, 1}$ and $C^{1, \alpha}$ estimates) for a class of fully nonlinear elliptic PDEs with unbalanced variable degeneracy. In a precise way, the degeneracy law of the model switches between two different kinds of degenerate elliptic operators of variable order, according to the null set of a modulating function. Such sharp regularity estimates generalize and improve, to some extent, earlier ones via geometric treatments. Our results are consequences of geometric tangential methods and make use of compactness, localized oscillating and scaling techniques. In the end, our findings are applied in the study of a wide class of nonlinear models and free boundary problems.