On the sharp Hessian integrability conjecture in the plane

TL;DR

This paper proves a sharp integrability result for the Hessian of solutions to fully nonlinear elliptic equations in the plane, confirming a conjecture about the optimal exponent in the Hessian integrability.

Contribution

It establishes the optimal Hessian integrability exponent for solutions in two dimensions, advancing understanding of regularity in fully nonlinear elliptic equations.

Findings

Proves $u otin W^{2, ilde{eta}}$ for $ ilde{eta} > rac{2}{(rac{ ext{Lambda}}{ ext{lambda}})+1}$.

Establishes the lower bound $rac{1.629}{(rac{ ext{Lambda}}{ ext{lambda}})+1}$ for the integrability exponent.

Confirms the Armstrong-Silvestre-Smart 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.