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On the sharp Hessian integrability conjecture in the plane. (arXiv:2212.03314v1 [math.AP]) http://arxiv.org/abs/2212.03314

On the sharp Hessian integrability conjecture in the plane

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 $(λ, Λ)$-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
December 8, 2022 at 3:10 AM · · feed2toot · 0 · 0 · 0
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