Singular Sets and Doubling Properties in Parabolic Homogenization

Xiujin Chen

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arXiv · 2609.38952

Singular Sets and Doubling Properties in Parabolic Homogenization

Xiujin Chen

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Abstract

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.

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BibTeX RIS

Xiujin Chen. 2026-09-30. Singular Sets and Doubling Properties in Parabolic Homogenization. https://arxiv.org/abs/2609.38952

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