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

We prove $W^{2,\varepsilon}$ estimates for viscosity supersolutions of fully nonlinear, uniformly elliptic equations where ε 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.

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The research of the author was supported in part by the National Science Foundation under grant DMS-1764248.

Received 25 June 2018

Accepted 9 September 2018

Published 8 April 2020