Nirenberg's Theorem and Singular Solutions from Traceless Symmetric Matrices
Abstract
Nirenberg's theorem gives interior C2,α regularity for continuous viscosity solutions of fully nonlinear uniformly elliptic equations F(D2u)=0 in dimension two. In dimensions five and higher, the work of Nadirashvili--Tkachev--Vlăduţ shows that such solutions need not be C2. We construct a family of C1,1∖ C2 functions {wm}m≥3 and uniformly elliptic operators Fm such that Fm(D2 wm)=0 in ℝdm, where dm=m(m+1)/2−1. These solutions are given by tr(X3)/√(tr(X2)) on the space of traceless real symmetric m× 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≥3 in a unified way by direct analysis on traceless symmetric matrices. We also prove a geometric criterion for pointwise C2,β regularity of continuous viscosity solutions u of uniformly elliptic equations F(D2u)=0 in every dimension n≥ 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 C2,β expansion for some β∈(0,1). Our criterion is quantitative and yields interior C2,β 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 C2,β regularity for solutions of Dirichlet problems with certain group symmetries. This extends a result of Nadirashvili--Vlăduţ for axially symmetric problems.
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BibTeX
@misc{chu2026nirenbergs,
title = {{Nirenberg's Theorem and Singular Solutions from Traceless Symmetric Matrices}},
author = {BaoZhi Chu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25416},
eprint = {2609.25416},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25416},
}
RIS
TY - UNPB
TI - Nirenberg's Theorem and Singular Solutions from Traceless Symmetric Matrices
AU - BaoZhi Chu
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25416
UR - https://arxiv.org/abs/2609.25416
AB - Nirenberg's theorem gives interior C^2,α regularity for continuous viscosity solutions of fully nonlinear uniformly elliptic equations F(D^2u)=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∖ C^2 functions {w_m}_m≥3 and uniformly elliptic operators F_m such that F_m(D^2 w_m)=0 in ℝ^d_m, where d_m=m(m+1)/2−1. These solutions are given by tr(X^3)/√(tr(X^2)) on the space of traceless real symmetric m× 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≥3 in a unified way by direct analysis on traceless symmetric matrices. We also prove a geometric criterion for pointwise C^2,β regularity of continuous viscosity solutions u of uniformly elliptic equations F(D^2u)=0 in every dimension n≥ 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,β expansion for some β∈(0,1). Our criterion is quantitative and yields interior C^2,β 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,β regularity for solutions of Dirichlet problems with certain group symmetries. This extends a result of Nadirashvili--Vlăduţ for axially symmetric problems.
ER -