arXiv 2309.080892023-09-15math.APmath.DG

具有 Hölder 系数的椭圆方程的奇异集与临界集的体积估计

Volume Estimates for Singular sets and Critical Sets of Elliptic Equations with Hölder Coefficients

Yiqi Huang, Wenshuai Jiang

AI总结：

本文研究了仅具 Hölder 系数的椭圆方程，在缺乏单调性公式的困难下，通过引入倍增指数的近似单调性公式与切向量图的定量唯一性，证明了奇异集与临界集的显式体积估计，并改进了定量分层的体积估计。

摘要：

考虑椭圆方程 $\mathcal{L}(u) = \partial_i(a^{ij}(x) \partial_j u) + b^i(x) \partial_i u + c(x) u= 0$ 的解 $u$，其中 $a^{ij}$ 仅假设为 Hölder 连续。本文证明了奇异集 $\mathcal{S}(u) = \{ x \in B_1 : u(x) = |\nabla u(x)| = 0\}$ 和临界集 $\mathcal{C}(u) \equiv \{ x\in B_{1} : |\nabla u(x)| = 0 \}$ 的 $(n-2)$ 维 Minkowski 估计的显式界，该界以倍增指数的界表示，取决于 $c \equiv 0$ 是否成立。这里的 Hölder 假设是尖锐的，因为它是根据椭圆估计定义 $u$ 的临界集的最弱条件。我们还能对每个定量分层 $\mathcal{S}^k_{\eta, r}$ 上的 Cheeger-Naber-Valtorta 体积估计获得最优改进。这种情况的主要困难在于缺乏对定量分层至关重要的单调性公式。在我们的证明中，一个关键要素是 Hölder 假设下倍增指数的新近似单调性公式。另一个关键要素是切向量图的定量唯一性。值得注意的是，我们的近似单调性足以解决微分方程分析中因缺乏单调性而产生的所有困难。我们相信这一思想可应用于其他相关研究。

英文摘要：

Consider the solutions $u$ to the elliptic equation $\mathcal{L}(u) = \partial_i(a^{ij}(x) \partial_j u) + b^i(x) \partial_i u + c(x) u= 0$ with $a^{ij}$ assumed only to be Hölder continuous. In this paper we prove an explicit bound for $(n-2)$-dimensional Minkowski estimates of singular set $\mathcal{S}(u) = \{ x \in B_1 : u(x) = |\nabla u(x)| = 0\}$ and critical set $\mathcal{C}(u) \equiv \{ x\in B_{1} : |\nabla u(x)| = 0 \}$ in terms of the bound on doubling index, depending on $c \equiv 0$ or not. Here the Hölder assumption is sharp as it is the weakest condition in order to define the critical set of $u$ according to elliptic estimates. We can also obtain an optimal improvement on Cheeger-Naber-Valtorta's volume estimates on each quantitative stratum $\mathcal{S}^k_{η, r}$. The main difficulty in this situation is the lack of monotonicity formula which is essential to the quantitative stratification. In our proof, one key ingredient is a new almost monotonicity formula for doubling index under the Hölder assumption. Another key ingredient is the quantitative uniqueness of tangent maps. It deserves to note that our almost monotonicity is sufficient to address all the difficulties arising from the absence of monotonicity in the analysis of differential equations. We believe the idea could be applied to other relevant study.

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