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--Vl\u{a}du\c{t} 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 $\Sigma_{2,0}(u)$ 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 $\alpha\in [0,1)$ that the set $\Sigma_{2,\alpha}(u)$ of points at which $u$ has no $C^{2,\alpha}$ expansion has Hausdorff dimension at most $n-\varepsilon(1-\alpha)$.