$C^1$ 裂缝域中调和函数的边界正则性
Boundary regularity of harmonic functions in $C^1$ slit domains
浏览论文内容
中文总结 AI 辅助
本文研究 $C^1$ 裂缝域中调和函数在裂缝上消失时的边界正则性，建立了带显式校正因子的 $\sqrt{d}$ 增长估计，校正因子由裂缝的连续模决定，并在 Dini 条件下保持有界，方法基于几何坐标变换，无需使用方程。
中文摘要 AI 辅助
我们在 $C^1$ 裂缝域中建立了在裂缝上消失的调和函数的精确上下界估计，且不假设裂缝位于超平面内。经典的在边缘附近的 $\sqrt{d}$ 增长（其中 $d$ 是到裂缝的距离）在此一般性下仍然成立，但需乘以一个由裂缝的 $C^1$ 连续模 $\omega$ 决定的显式因子 $\exp\big( \pm C \int_\rho^r \omega(s)\\, \frac{ds}{s} \big)$，其中 $\rho < r$ 是被比较的两个尺度。上界估计允许非零右端项和非零边界数据。当且仅当 $\omega$ 满足 Dini 条件时，校正因子在 $\rho\to0$ 时保持有界且上下界为正常数。超出 Dini 范围的连续模（例如相关自由边界问题中奇异集上出现的对数模）未被先前的 $C^{1,\alpha}$ 理论所覆盖。此前，$\sqrt{d}$ 增长仅对位于超平面内且具有 $C^{1,\alpha}$ 边缘的裂缝已知（De Silva, Savin）。对于 Lipschitz 裂缝，存在边界 Harnack 原理，但未精确确定增长率。主要技术工具是一个坐标变换，将 Lipschitz 裂缝域展平到模型半超平面裂缝上，并具有直到二阶的定量估计。该构造是几何性的，不依赖于方程，因此我们预期它可应用于其他边界正则性问题。
英文摘要
We establish precise upper and lower estimates for harmonic functions vanishing on the slit of a $C^1$ slit domain, with no assumption that the slit lies in a hyperplane. The classical $\sqrt{d}$ growth near the edge, $d$ being the distance to the slit, persists in this generality, up to an explicit factor \[\exp\Big( \pm C \int_ρ^r ω(s)\, \frac{ds}{s} \Big)\] determined by the $C^1$-modulus of continuity $ω$ of the slit, where $ρ< r$ are the two scales being compared. The upper estimates allow a right-hand side and non-zero boundary data. The correction factors remain bounded above and below by positive constants as $ρ\to0$ precisely when $ω$ satisfies the Dini condition. Moduli of continuity beyond the Dini regime, as is the case for the logarithmic moduli arising at singular sets in relevant free boundary problems, were not covered by the previous $C^{1,α}$ theory. Previously, the $\sqrt{d}$ growth was known for slits contained in a hyperplane, which additionally have a $C^{1,α}$ edge (De Silva, Savin). For Lipschitz slits, there are boundary Harnack principles, but no growth rate is identified precisely. The main technical ingredient is a change of coordinates flattening a Lipschitz slit domain onto the model half-hyperplane slit, with quantitative estimates up to second order. The construction is geometric and does not use the equation, so we expect it to be useful for other boundary regularity problems.
发表机构
- Universitat de Barcelona（巴塞罗那大学）
- Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas（西班牙高等科学研究所数学科学中心）
机构由 AI 辅助整理，请以论文原文为准。
