On the sharp Hessian integrability conjecture in the plane

Abstract

We prove that if $u\in C^0(B_1)$ satisfies $F(x,D^2u) \le 0$ in $B_1\subset \mathbb{R}^2$, in the viscosity sense, for some fully nonlinear $(\lambda, \Lambda)$-elliptic operator, then $u \in W^{2,\varepsilon}(B_{1/2})$, with appropriate estimates, for a sharp exponent $ \varepsilon = \varepsilon(\lambda, \Lambda)$ verifying $$ \frac{1.629}{\frac{\Lambda}{\lambda} + 1} < \varepsilon(\lambda, \Lambda) \le \frac{2}{\frac{\Lambda}{\lambda} + 1}, $$ 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.