$C^{1,\alpha}$ regularity for fully nonlinear elliptic equations with superlinear growth in the gradient

TL;DR

This paper extends $C^{1,eta}$ regularity results to solutions of fully nonlinear elliptic equations with superlinear gradient growth and unbounded coefficients, and proves existence of positive eigenvalues for related operators.

Contribution

It generalizes regularity estimates to broader classes of equations with unbounded coefficients and superlinear growth, and establishes eigenvalue existence results.

Findings

Extended $C^{1,eta}$ regularity to unbounded coefficient equations

Proved existence of positive eigenvalues for operators with unbounded weights

Established regularity and eigenvalue results for Pucci's operators with unbounded coefficients

Abstract

We extend the Caffarelli-\'Swiech-Winter $C^{1,\alpha}$ regularity estimates to $L^p$-viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form with superlinear growth in the gradient and unbounded coefficients. As an application, in addition to the usual $W^{2,p}$ results, we prove the existence of positive eigenvalues for proper operators with nonnegative unbounded weight, in particular for Pucci's operators with unbounded coefficients.