New regularity estimates for fully nonlinear elliptic equations

Abstract

We establish new quantitative Hessian integrability estimates for viscosity supersolutions of fully nonlinear elliptic operators. As a corollary, we show that the optimal Hessian power integrability ε=ε(λ,Λ,n) in the celebrated W2,ε–regularity estimate satisfies(1+23(1−λΛ))n−1ln⁡n4⋅(λΛ)n−1≤ε≤nλ(n−1)Λ+λ, where n≥3 is the dimension and 0<λ<Λ are the ellipticity constants. In particular, (Λλ)n−1ε(λ,Λ,n) blows-up, as n→∞; previous results yielded fast decay of such a quantity. The upper estimate improves the one obtained by Armstrong, Silvestre, and Smart in [1].