The singular set in the thin Stefan problem
Abstract
We study the singular set of the free boundary for solutions of the thin Stefan problem in ℝn+1× ℝ, which models the melting of a thin sheet of ice over water. After a suitable transformation, we reduce to studying the parabolic Signorini (thin obstacle) problem with a time-monotonicity condition. First, we prove that the free boundary is locally the graph of a differentiable melting function t=τ(x) defined on the n-dimensional thin space, and that the singular set coincides with the critical set of τ. Next, we show that the singular set has parabolic Hausdorff dimension at most n. We then establish generic regularity up to dimension n+1=4, proving that the free boundary is regular for almost every time. Finally, under smoothness assumptions on the obstacle, we show that the singular set is contained in a C∞-hypersurface outside a set of parabolic dimension at most n−1. Our analysis combines ideas developed for the classical Stefan problem with several new ingredients specific to the thin setting, including frequency formulas for second blow-ups, epiperimetric inequalities, higher order expansions, and a fine stratification of the singular set.
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BibTeX
@misc{carducci2026singular,
title = {{The singular set in the thin Stefan problem}},
author = {Matteo Carducci and Roberto Colombo and Clara Torres-Latorre},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10751},
eprint = {2610.10751},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10751},
}
RIS
TY - UNPB
TI - The singular set in the thin Stefan problem
AU - Matteo Carducci
AU - Roberto Colombo
AU - Clara Torres-Latorre
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10751
UR - https://arxiv.org/abs/2610.10751
AB - We study the singular set of the free boundary for solutions of the thin Stefan problem in ℝ^n+1× ℝ, which models the melting of a thin sheet of ice over water. After a suitable transformation, we reduce to studying the parabolic Signorini (thin obstacle) problem with a time-monotonicity condition. First, we prove that the free boundary is locally the graph of a differentiable melting function t=τ(x) defined on the n-dimensional thin space, and that the singular set coincides with the critical set of τ. Next, we show that the singular set has parabolic Hausdorff dimension at most n. We then establish generic regularity up to dimension n+1=4, proving that the free boundary is regular for almost every time. Finally, under smoothness assumptions on the obstacle, we show that the singular set is contained in a C^∞-hypersurface outside a set of parabolic dimension at most n−1. Our analysis combines ideas developed for the classical Stefan problem with several new ingredients specific to the thin setting, including frequency formulas for second blow-ups, epiperimetric inequalities, higher order expansions, and a fine stratification of the singular set.
ER -