凸k-Hessian解内部奇异集的Sharp Hausdorff界
Sharp Hausdorff Bounds for the Interior Singular Set of Convex $k$-Hessian Solutions
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中文总结 AI 辅助
针对凸k-Hessian方程σ_k(D²u)=1的凸粘性解，证明其局部C²正则性缺失集的(n-1)维Hausdorff测度为0，得到3≤k<n时的余维1精细正则性结果，指数最优，还推导了相关正则性结论并刻画了平坦方向数。
中文摘要 AI 辅助
设2≤k≤n，Ω⊂ℝⁿ为开凸集，u是Ω上满足σ_k(D²u)=1的凸粘性解。我们证明u不具有局部C²正则性的集合的(n-1)维Hausdorff测度为0。在中间范围3≤k<n中，这给出了已知几乎处处部分正则性的余维1改进，且该指数是最优的。更一般地，对于满足σ_k(D²u)≥λ>0的凸粘性下解，我们得到由所有支撑接触集的仿射维数定义的层的Hausdorff界。证明结合了依赖支撑的Chou–Wang障碍论证、John椭球最小k个半轴乘积的估计以及Mooney的凸截面覆盖定理。作为直接分析结果，完整分布Hessian是绝对连续的，且u∈W²,¹_loc(Ω)，得到了奇异Monge–Ampère解已知的W²,¹正则性的k-Hessian对应结果。在逻辑上独立的结构部分，我们通过仿射截面上的渐近下确界均值公式刻画了特殊的平坦方向数n−k+1，并解释了该均值启发式如何导出证明中使用的支撑接触几何。
英文摘要
Let $2\le k\le n$, let $Ω\subset\mathbb{R}^n$ be open and convex, and let $u$ be a convex viscosity solution of $σ_k(D^2u)=1$ in $Ω$. We prove that the set on which $u$ fails to be locally $C^2$ has vanishing $(n-1)$-dimensional Hausdorff measure. In the intermediate range $3\le k<n$, this gives a codimension-one refinement of the known almost-everywhere partial regularity, and the exponent is sharp. More generally, for a convex viscosity subsolution of $σ_k(D^2u)\geλ>0$, we obtain Hausdorff bounds for strata defined by the affine dimension of all supporting contact sets. The proof combines a support-dependent Chou--Wang barrier argument, an estimate for the product of the smallest $k$ semiaxes of a John ellipsoid, and Mooney's convex section-covering theorem. As a direct analytical consequence, the full distributional Hessian is absolutely continuous and $u\in W^{2,1}_{\mathrm{loc}}(Ω)$, yielding a $k$-Hessian counterpart of the $W^{2,1}$ regularity known for singular Monge--Ampère solutions. In a logically separate structural part, we characterize the distinguished number of flat directions, $n-k+1$, by an asymptotic infimum mean-value formula over affine sections, and explain how this mean-value heuristic leads to the supporting-contact geometry used in the proof.
