On the sharp Hessian integrability conjecture in the planeWe 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 $(λ,
Λ)$-elliptic operator, then $u \in W^{2,\varepsilon}(B_{1/2})$, with
appropriate estimates, for a sharp exponent $ \varepsilon =
\varepsilon(λ, Λ)$ verifying
$$
\frac{1.629}{\fracΛλ + 1} < \varepsilon(λ, Λ) \le
\frac{2}{\fracΛλ + 1},
$$ uniformly as $\fracλΛ \to 0$. 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.
arxiv.org 
