On the sharp Hessian integrability conjecture in the plane

Abstract

We prove that if $u\in C^0(B_1)$ satisfies $F(x,D^{2}u)\le 0$ in $B_1\subset {\mathbb{R}}^2$, in the viscosity sense, for some fully nonlinear $(\lambda ,\Lambda )$-elliptic operator, then $u\in W^{2,\epsilon }(B_{1/2})$, with appropriate estimates, for a sharp exponent $\epsilon =\epsilon (\lambda ,\Lambda )$ verifying $$\frac{1.629}{\frac{\Lambda }{\lambda }+1}<\epsilon (\lambda ,\Lambda )\le \frac{2}{\frac{\Lambda }{\lambda }+1},$$ uniformly as $\frac{\lambda }{\Lambda }\to 0$. This is closely related to the Armstrong-Silvestre-Smart conjecture, raised in [Comm. Pure Appl. Math. 65 (2012), no. 8, 1169--1184], where the upper bound is postulated to be the optimal one.