发表机构
Northwestern University（西北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了高维一致椭圆方程的无迹对称矩阵奇异解族，并给出点态C^{2,\beta}正则性的几何判据，推广了Nirenberg定理至三维及以上。
AI 中文摘要
Nirenberg定理给出了二维空间中完全非线性一致椭圆方程F(D^2u)=0的连续粘性解的内部C^{2,\alpha}正则性。在五维及更高维空间中，Nadirashvili--Tkachev--Vlăduţ的工作表明此类解不必是C^2的。我们构造了一族C^{1,1}\setminus C^2函数\{w_m\}_{m\geq3}和一致椭圆算子F_m，使得F_m(D^2 w_m)=0在\mathbb{R}^{d_m}中成立，其中d_m=m(m+1)/2-1。这些解由无迹实对称m\times m矩阵空间上的\operatorname{tr}(X^3)/\sqrt{\operatorname{tr}(X^2)}给出。当m=3,4时，该族分别恢复了与五维Cartan等参三次型和九维Hsiang极小三次型相关的已知奇异解。我们的构造通过对无迹对称矩阵的直接分析，以统一方式处理所有m\geq3的情况。我们还证明了在任意维数n\geq3中，一致椭圆方程F(D^2u)=0的连续粘性解u的点态C^{2,\beta}正则性的几何判据。假设n-2个光滑超曲面经过点p，每个超曲面上携带一个光滑向量场，使得u沿该向量场的方向导数在该超曲面上为常数。如果这些向量场在p处线性无关，则u在p处二次可微，并允许对某个\beta\in(0,1)进行点态C^{2,\beta}展开。我们的判据是定量的，并在一致几何假设下产生内部C^{2,\beta}正则性。作为应用，我们将Nirenberg定理推广到三维及更高维空间中一类一般的解。作为推论，我们建立了具有某些群对称性的Dirichlet问题解的内部C^{2,\beta}正则性。这推广了Nadirashvili--Vlăduţ关于轴对称问题的结果。
英文摘要
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 $C^{1,1}\setminus C^2$ functions $\{w_m\}_{m\geq3}$ 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 $\operatorname{tr}(X^3)/\sqrt{\operatorname{tr}(X^2)}$ 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 $m\geq3$ 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 $n\geq 3$. 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 $β\in(0,1)$. 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.
