AI 中文总结
本文针对完全非线性一致椭圆方程的连续粘性解，证明在n≥5维时，其奇异集可具有任意介于0与n的豪斯多夫维数，甚至可具有正勒贝格测度，突破了此前需对F加额外C¹假设的限制。
AI 中文摘要
对于完全非线性一致椭圆方程F(D²u)=0的连续粘性解u，Nadirashvili、Tkachev与Vlăduţ的研究表明，在维数n≥5时，u未必是C²的。因此自然要问，由不存在u的C²邻域的点构成的奇异集Sing(u)能有多大。在F满足额外C¹假设的条件下，Armstrong、Silvestre与Smart证明，Sing(u)的豪斯多夫测度至多为n−ε，其中ε>0。本文证明，对于一般的一致椭圆F，模一个可数集后，Sing(u)可以是ℝ⁵中任意非空紧无处稠密集；特别地，在n≥5的维数下，若不对F作进一步假设，Sing(u)的豪斯多夫维数可介于0与n之间，且该奇异集甚至可以具有正的勒贝格测度。
英文摘要
For continuous viscosity solutions $u$ of fully nonlinear uniformly elliptic equations $F(D^2 u)=0$, the work of Nadirashvili--Tkachev--Vlăduţ shows that $u$ need not be $C^2$ in dimensions $n\ge 5$. It is therefore natural to ask how large the set $\operatorname{Sing}(u)$, consisting of points at which $u$ has no $C^2$ neighborhood, can be. Under an additional $C^1$ assumption on $F$, Armstrong--Silvestre--Smart proved that $\operatorname{Sing}(u)$ has Hausdorff dimension at most $n-\varepsilon$ for some $\varepsilon >0$. In this paper, we show that, in dimensions $n\ge 5$, the Hausdorff dimension of $\operatorname{Sing}(u)$ cannot be bounded away from $n$ under uniform ellipticity alone. In fact, we prove the stronger statement: $\operatorname{Sing}(u)$ can be any compact nowhere dense set in $\mathbb{R}^5$ modulo a countable set. Consequently, in every dimension $n\geq 5$, the Hausdorff dimension of $\operatorname{Sing}(u)$ can be any number in $[n-5, n]$; moreover, $\operatorname{Sing}(u)$ can even have positive Lebesgue measure. In contrast, for every dimension $n$ and every uniformly elliptic operator $F$, we prove that the set $Σ_{2,0}(u)$ of points at which $u$ is not twice differentiable has Hausdorff dimension at most $n-\varepsilon$ for some $\varepsilon>0$. In particular, this quantitatively strengthens Trudinger's theorem that $u$ is twice differentiable almost everywhere. More generally, we prove for every $α\in [0,1)$ that the set $Σ_{2,α}(u)$ of points at which $u$ has no $C^{2,α}$ expansion has Hausdorff dimension at most $n-\varepsilon(1-α)$.
CommentsAdded Hausdorff dimension estimates for pointwise singular sets (Theorem 1.7), and improved the exposition
