On Singular Sets of Fully Nonlinear Uniformly Elliptic Equations

For a continuous viscosity solution $u$ of a fully nonlinear uniformly elliptic equation $F(D^{2}u)=0$, the work of Nadirashvili–Tkachev–Vlăduţ shows that $u$ need not be $C^2$ in dimension $n\ge 5$. It is therefore natural to ask how large the singular set $\mathrm{Sing}(u)$, consisting of points having no $C^2$ neighborhood of $u$, can be. Under an additional $C^1$ assumption on $F$, Armstrong–Silvestre–Smart proved that $\mathrm{Sing}(u)$ has Hausdorff measure at most $n-\epsilon$ for some $\epsilon >0$. In this paper, we show that for general uniformly elliptic $F$, modulo a countable set, $\mathrm{Sing}(u)$ can be any nonempty compact nowhere-dense set in ${\mathbb{R}}^5$. In particular, without further assumptions on $F$, the Hausdorff dimension of $\mathrm{Sing}(u)$ can be any number between $0$ and $n$ in dimension $n\ge 5$; moreover, the singular set can even have positive Lebesgue measure.