Variational mean curvatures for general C1,α domains
Abstract
The notion of variational mean curvature (VMC) arises from Massari's variational approach to hypersurfaces of prescribed mean curvature and expresses a minimizing property for Massari's functional. Here, we complement the theory of VMCs with a new sharp relation between hypersurface regularity and VMC integrability. In fact, our main result asserts that every bounded open set in ℝn with C1,α boundary, 0<α<1, has a global VMC in Lp(ℝn) for every p<(1+α)/(1−α). In addition, we underpin sharpness of this conclusion by constructing examples of bounded open sets in ℝn with C1,α boundary which do not admit a VMC in Lp(ℝn) for any p>(1+α)/(1−α). Our proofs are based on Barozzi's level-set construction of global VMCs and determine integrability of these VMCs by geometric investigation of their superlevel sets and of related unions of balls.
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BibTeX
@misc{schmidt2026variational,
title = {{Variational mean curvatures for general $\mathrmC^1,α$ domains}},
author = {Thomas Schmidt and Jule Helena Schütt},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24224},
eprint = {2610.24224},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24224},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Variational mean curvatures for general C^1,α domains
AU - Thomas Schmidt
AU - Jule Helena Schütt
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24224
UR - https://arxiv.org/abs/2610.24224
AB - The notion of variational mean curvature (VMC) arises from Massari's variational approach to hypersurfaces of prescribed mean curvature and expresses a minimizing property for Massari's functional. Here, we complement the theory of VMCs with a new sharp relation between hypersurface regularity and VMC integrability. In fact, our main result asserts that every bounded open set in ℝ^n with C^1,α boundary, 0<α<1, has a global VMC in L^p(ℝ^n) for every p<(1+α)/(1−α). In addition, we underpin sharpness of this conclusion by constructing examples of bounded open sets in ℝ^n with C^1,α boundary which do not admit a VMC in L^p(ℝ^n) for any p>(1+α)/(1−α). Our proofs are based on Barozzi's level-set construction of global VMCs and determine integrability of these VMCs by geometric investigation of their superlevel sets and of related unions of balls.
ER -