Regularity estimates for fully nonlinear elliptic PDEs with general Hamiltonian terms and unbounded ingredients

TL;DR

This paper establishes optimal regularity estimates for solutions of fully nonlinear elliptic PDEs with unbounded Hamiltonian terms, covering various gradient regimes and providing new compatibility conditions.

Contribution

It introduces a unified approach to regularity for nonlinear elliptic PDEs with unbounded coefficients, extending classical results to broader settings.

Findings

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

Proved a priori BMO estimates and regularity under relaxed convexity.

Derived compatibility conditions linking regularity to source term integrability.

Abstract

We develop an optimal regularity theory for $L^p$-viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form whose gradient growth is described through a Hamiltonian function with measurable and possibly unbounded coefficients. Our approach treats both superlinear and sublinear gradient regimes in a unified way. We show $C^{0,\alpha}$, $C^{0,\textrm{Log-Lip}}$, $C^{1,\alpha}$, $ C^{1,\textrm{Log-Lip}}$ and $C^{2,\alpha}$ regularity estimates, by displaying the growth allowed to the Hamiltonian in order to deal with an unbounded nonlinear gradient coefficient, whose integrability in turn gets worse as we approach the quadratic regime. Moreover, we find proper compatibility conditions for which our regularity results depend intrinsically on the integrability of the underlying source term. As a byproduct of our methods, we prove a priori BMO estimates; sharp regularity to associated recession and flat profiles under relaxed convexity assumptions; improved regularity for a class of singular PDEs; and a Perron type result under unbounded ingredients.