Boundary regularity of harmonic functions in C1 slit domains
Abstract
We establish precise upper and lower estimates for harmonic functions vanishing on the slit of a C1 slit domain, with no assumption that the slit lies in a hyperplane. The classical √d growth near the edge, d being the distance to the slit, persists in this generality, up to an explicit factor exp( ± C ∫ρr ω(s) ds/s ) determined by the C1-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→0 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 C1,α theory. Previously, the √d growth was known for slits contained in a hyperplane, which additionally have a C1,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.
Related
- Boundary regularity of harmonic map heat flows from RCD spaces into CAT(0) spaces
- Boundary pointwise regularity for the p-Laplace equation on C1,Dini domains
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BibTeX
@misc{domingopasarin2026boundary,
title = {{Boundary regularity of harmonic functions in $C^1$ slit domains}},
author = {Joan Domingo-Pasarin and Pablo Hidalgo-Palencia and Alejandro Martínez and Clara Torres-Latorre},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.15923},
eprint = {2609.15923},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15923},
}
RIS
TY - UNPB
TI - Boundary regularity of harmonic functions in C^1 slit domains
AU - Joan Domingo-Pasarin
AU - Pablo Hidalgo-Palencia
AU - Alejandro Martínez
AU - Clara Torres-Latorre
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2609.15923
UR - https://arxiv.org/abs/2609.15923
AB - 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 √d growth near the edge, d being the distance to the slit, persists in this generality, up to an explicit factor exp( ± C ∫_ρ^r ω(s) ds/s ) 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→0 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 √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.
ER -