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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 ε, with a constant independent of ε. The bounds depend on a coarse-scale ratio of space-time L2 mass to terminal-slice L2 mass. Independently of corrector nondegeneracy, we prove that this ratio controls doubling at all smaller scales uniformly in ε. 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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Cite

BibTeXRIS
BibTeX
@misc{chen2026singular,
  title = {{Singular Sets and Doubling Properties in Parabolic Homogenization}},
  author = {Xiujin Chen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.38952},
  eprint = {2609.38952},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.38952},
}
RIS
TY  - UNPB
TI  - Singular Sets and Doubling Properties in Parabolic Homogenization
AU  - Xiujin Chen
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.38952
UR  - https://arxiv.org/abs/2609.38952
AB  - 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 ε, with a constant independent of ε. 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 ε. 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.
ER  -