发表机构
School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究周期抛物方程奇异集，在非退化条件下获得Hausdorff测度界和余维二Minkowski估计，并证明质量比值控制加倍性质。
AI 中文摘要
我们研究了具有快速振荡系数的周期抛物型方程实值解的奇异集。在修正矩阵满足非退化条件的情况下，我们获得了局部抛物$n$维Hausdorff测度界和余维二Minkowski估计。该Minkowski估计在包括小于振荡尺度$\varepsilon$的每个尺度上都成立，且常数与$\varepsilon$无关。这些界依赖于时空$L^2$质量与终端切片$L^2$质量的粗尺度比值。独立于修正矩阵的非退化性，我们证明了该比值在所有更小尺度上一致地控制加倍性质，且一致于$\varepsilon$。奇异集论证结合了截断高斯近似中的可和误差、不同时间中心的调和多项式比较，以及具有微观奇异集估计的停止覆盖。
英文摘要
We study the singular sets of real-valued solutions to periodic parabolic equations with rapidly oscillating coefficients. Under a nondegeneracy condition on the corrector matrix, we obtain a local parabolic $n$-dimensional Hausdorff measure bound and a codimension-two Minkowski estimate. The Minkowski estimate holds at every scale, including scales below the oscillation scale $\varepsilon$, with a constant independent of $\varepsilon$. The bounds depend on a coarse-scale ratio of space-time $L^2$ mass to terminal-slice $L^2$ mass. Independently of corrector nondegeneracy, we prove that this ratio controls doubling at all smaller scales uniformly in $\varepsilon$. The singular-set argument combines summable errors in truncated Gaussian approximations, comparison of caloric polynomials at different time centers, and a stopping cover with microscopic singular-set estimates.
