On Singular Sets of Fully Nonlinear Uniformly Elliptic Equations

Abstract

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 $\mathrm{Sing}(u)$, 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 $\mathrm{Sing}(u)$ has Hausdorff dimension at most $n-\epsilon$ for some $\epsilon >0$. In this paper, we show that, in dimensions $n\ge 5$, the Hausdorff dimension of $\mathrm{Sing}(u)$ cannot be bounded away from $n$ under uniform ellipticity alone. In fact, we prove the stronger statement: $\mathrm{Sing}(u)$ can be any compact nowhere dense set in ${\mathbb{R}}^5$ modulo a countable set. Consequently, in every dimension $n\ge 5$, the Hausdorff dimension of $\mathrm{Sing}(u)$ can be any number in $[n-5,n]$; moreover, $\mathrm{Sing}(u)$ can even have positive Lebesgue measure. In contrast, for every dimension $n$ and every uniformly elliptic operator $F$, we prove that the set ${\Sigma }_{2,0}(u)$ of points at which $u$ is not twice differentiable has Hausdorff dimension at most $n-\epsilon$ for some $\epsilon >0$. In particular, this quantitatively strengthens Trudinger's theorem that $u$ is twice differentiable almost everywhere. More generally, we prove for every $\alpha \in [0,1)$ that the set ${\Sigma }_{2,\alpha }(u)$ of points at which $u$ has no $C^{2,\alpha }$ expansion has Hausdorff dimension at most $n-\epsilon (1-\alpha )$.

Added Hausdorff dimension estimates for pointwise singular sets (Theorem 1.7), and improved the exposition