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The Sharp C1,1 Target Threshold for Minimizing Constraint Maps
Abstract
Let K⊂Rm be the closure of a bounded domain whose boundary is a compact embedded hypersurface, and let M̅=Rm∖intK be the allowed target. We prove that if ∂ K is of class C1,1, then every M̅-valued local minimizer of the Dirichlet energy E(u;B):=∫B|Du|2dd x is locally W2,∞, and hence locally C1,1, on its continuity set. For every 0<α<1, we also construct a convex body with C1,α boundary and an everywhere continuous global minimizer that is not C1,1. Thus C1,1 is the sharp target regularity threshold in the Hölder scale.
Related
BibTeX
@misc{liu2026sharp,
title = {{The Sharp $C^1,1$ Target Threshold for Minimizing Constraint Maps}},
author = {Yilu Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08952},
eprint = {2609.08952},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08952},
}
RIS
TY - UNPB
TI - The Sharp C^1,1 Target Threshold for Minimizing Constraint Maps
AU - Yilu Liu
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08952
UR - https://arxiv.org/abs/2609.08952
AB - Let K⊂R^m be the closure of a bounded domain whose boundary is a compact embedded hypersurface, and let M̅=R^m∖intK be the allowed target. We prove that if ∂ K is of class C^1,1, then every M̅-valued local minimizer of the Dirichlet energy E(u;B):=∫_B|Du|^2dd x is locally W^2,∞, and hence locally C^1,1, on its continuity set. For every 0<α<1, we also construct a convex body with C^1,α boundary and an everywhere continuous global minimizer that is not C^1,1. Thus C^1,1 is the sharp target regularity threshold in the Hölder scale.
ER -