PreprintCC BY 4.0arXiv
A remark on the global minimality of the cracktip
Abstract
We give a short proof that the planar cracktip is a global minimizer of the Mumford--Shah functional, based on interior and Dirichlet boundary regularity. We construct a Dirichlet minimizer with nonempty discontinuity set and show that this set contains no simple closed curves. The regularity results then imply the existence of an interior regular loose end, at which a blow-up yields the cracktip.
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BibTeX
@misc{deangelis2026remark,
title = {{A remark on the global minimality of the cracktip}},
author = {Francesco Deangelis and Camillo De Lellis and Matteo Focardi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33900},
eprint = {2610.33900},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33900},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A remark on the global minimality of the cracktip
AU - Francesco Deangelis
AU - Camillo De Lellis
AU - Matteo Focardi
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33900
UR - https://arxiv.org/abs/2610.33900
AB - We give a short proof that the planar cracktip is a global minimizer of the Mumford--Shah functional, based on interior and Dirichlet boundary regularity. We construct a Dirichlet minimizer with nonempty discontinuity set and show that this set contains no simple closed curves. The regularity results then imply the existence of an interior regular loose end, at which a blow-up yields the cracktip.
ER -