论文标题
在飞机上的尖锐的黑森可集成性猜想上
On the sharp Hessian integrability conjecture in the plane
论文作者
论文摘要
我们证明，如果$ u \ in c^0（b_1）$满足$ f（x，d^2u）\ le 0 $ in $ b_1 \ subset \ subset \ subset \ mathbb {r}^2 $，从粘度意义上讲，对于某些完全非线性的$（λ，λ）$ - eelliptic ockitor，然后w^{2，\ varepsilon}（b_ {1/2}）$，具有适当的估计值，对于尖锐的指数$ \ varepsilon = \ varepsilon（λ，λ）$验证 $$ \ frac {1.629} {\fracλλ + 1} <\ varepsilon（λ，λ）\ le \ frac {2} {\fracλλ + 1}， $$均匀为$ \fracλλ\至0 $。这与[Comm。纯应用。数学。 65（2012），没有。 8，1169--1184]，其中上限被认为是最佳的。
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.
