Nirenberg's Theorem and Singular Solutions from Traceless Symmetric Matrices

Abstract

Nirenberg's theorem gives interior $C^{2,\alpha }$ regularity for continuous viscosity solutions of fully nonlinear uniformly elliptic equations $F(D^{2}u)=0$ in dimension two. In dimensions five and higher, the work of Nadirashvili--Tkachev--Vlăduţ shows that such solutions need not be $C^2$. We construct a family of $C^{1,1}\setminus C^2$ functions $\{w_m{\}}_{m\ge 3}$ and uniformly elliptic operators $F_m$ such that $F_m(D^2{w}_m)=0$ in ${\mathbb{R}}^{d_m}$, where $d_m=m(m+1)/2-1$. These solutions are given by $\mathrm{tr}(X^3)/\sqrt{\mathrm{tr}(X^2)}$ on the space of traceless real symmetric $m\times m$ matrices. When $m=3,4$, this family recovers the known singular solutions associated with the five-dimensional Cartan isoparametric cubic and the nine-dimensional Hsiang minimal cubic, respectively. Our construction treats all $m\ge 3$ in a unified way by direct analysis on traceless symmetric matrices. We also prove a geometric criterion for pointwise $C^{2,\beta }$ regularity of continuous viscosity solutions $u$ of uniformly elliptic equations $F(D^{2}u)=0$ in every dimension $n\ge 3$. Suppose that $n-2$ smooth hypersurfaces pass through a point $p$, each carrying a smooth vector field along which the directional derivative of $u$ is constant on that hypersurface. If these vector fields are linearly independent at $p$, then $u$ is twice differentiable at $p$ and admits a pointwise $C^{2,\beta }$ expansion for some $\beta \in (0,1)$. Our criterion is quantitative and yields interior $C^{2,\beta }$ regularity under uniform geometric hypotheses. As an application, we extend Nirenberg's theorem to a general class of solutions in dimensions three and higher. As a consequence, we establish interior $C^{2,\beta }$ regularity for solutions of Dirichlet problems with certain group symmetries. This extends a result of Nadirashvili--Vlăduţ for axially symmetric problems.