Finiteness of singularities for higher-weight axially symmetric minimizers of the relaxed Dirichlet energy
Abstract
Let k≥2 be an integer. We prove that every minimizer of the Bethuel--Brezis--Coron relaxed Dirichlet energy in the signed meridional k-axial class, with continuous H1 degree-zero data on the boundary of a ball, is smooth outside a finite subset of the symmetry axis. This extends the weight-one finiteness theorem of Hardt--Lin--Poon and resolves the finite-singularity question posed by Martinazzi--Rivière as Open Problem~A, in the fixed-boundary Bethuel--Brezis--Coron setting with continuous degree-zero data. The interior argument uses a distributional graph-boundary identity, height-preserving half-turn replacements, and exact square completions retaining the current cost. Cap-valued crosscuts and a finite incidence count exclude singular accumulation at the boundary poles.
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BibTeX
@misc{dai2026finiteness,
title = {{Finiteness of singularities for higher-weight axially symmetric minimizers of the relaxed Dirichlet energy}},
author = {Zhiyuan Dai and Huaijie Wang and Wei Wang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07414},
eprint = {2611.07414},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07414},
}
RIS
TY - UNPB
TI - Finiteness of singularities for higher-weight axially symmetric minimizers of the relaxed Dirichlet energy
AU - Zhiyuan Dai
AU - Huaijie Wang
AU - Wei Wang
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07414
UR - https://arxiv.org/abs/2611.07414
AB - Let k≥2 be an integer. We prove that every minimizer of the Bethuel--Brezis--Coron relaxed Dirichlet energy in the signed meridional k-axial class, with continuous H^1 degree-zero data on the boundary of a ball, is smooth outside a finite subset of the symmetry axis. This extends the weight-one finiteness theorem of Hardt--Lin--Poon and resolves the finite-singularity question posed by Martinazzi--Rivière as Open Problem~A, in the fixed-boundary Bethuel--Brezis--Coron setting with continuous degree-zero data. The interior argument uses a distributional graph-boundary identity, height-preserving half-turn replacements, and exact square completions retaining the current cost. Cap-valued crosscuts and a finite incidence count exclude singular accumulation at the boundary poles.
ER -