New regularity estimates for fully nonlinear elliptic equations.

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 W 2 , ε –regularity estimate satisfies (1 + 2 3 (1 − λ Λ)) n − 1 ln ⁡ n 4 ⋅ (λ Λ) n − 1 ≤ ε ≤ n λ (n − 1) Λ + λ , where n ≥ 3 is the dimension and 0 < λ < Λ are the ellipticity constants. In particular, (Λ λ) n − 1 ε (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]. [ABSTRACT FROM AUTHOR]