Polynomial decay in $W^{2,\varepsilon}$ estimates for viscosity supersolutions of fully nonlinear elliptic equations

TL;DR

This paper establishes polynomial decay rates for second-order derivatives of viscosity supersolutions in fully nonlinear elliptic equations, advancing understanding of regularity estimates in relation to ellipticity ratios.

Contribution

It proves polynomial decay in $W^{2, ext{ extepsilon}}$ estimates for viscosity supersolutions, addressing a conjecture about decay rates related to ellipticity ratios.

Findings

Proves polynomial decay of $W^{2, ext{ extepsilon}}$ estimates.

Relates decay rate to ellipticity ratio.

Advances regularity theory for nonlinear elliptic equations.

Abstract

We prove $W^{2,\varepsilon}$ estimates for viscosity supersolutions of fully nonlinear, uniformly elliptic equations where $\varepsilon$ decays polynomially with respect to the ellipticity ratio of the equations. Our result is related to a conjecture of Armstrong-Silvestre-Smart [Comm. Pure Appl. Math. 65 (2012), no. 8, 1169--1184] which predicts a linear decay for $\varepsilon$ with respect to the ellipticity ratio of the equations.