Monotone Sobolev functions: approximation, critical points, and level sets
Abstract
We give an affirmative answer to the planar local smoothing problem in Question~1.7 of D.~Ntalampekos and positive and negative answers to the basic approximation and level-set parts of his higher-dimensional Question~1.8. In every dimension n≥2, each continuous Lebesgue monotone function in W1,p on a bounded open set admits uniform and strong W1,p approximation by monotone C1,αloc functions, with unchanged Sobolev boundary values and no increase of the p-Dirichlet energy, for 1<p<∞. The energy can be made strictly smaller unless the original function is p-harmonic. In the plane we obtain smooth local replacement at every isolated p-harmonic critical point, with arbitrarily small C1 and Sobolev error. A point of gradient index −m can be resolved into exactly m nondegenerate saddles. Together with Ntalampekos's planar theorem, this gives smooth monotone density with fixed boundary values; at a nonsmooth p-harmonic minimizer the energy increase is unavoidable but can tend to zero. In dimensions n≥3, an explicit Lipschitz monotone function has a nonmanifold point on every level in an interval, although it admits smooth monotone approximation. A product extension of a planar homogeneous $7$-harmonic function also rules out a general discrete exceptional set under the exact boundary and energy constraints. The answer to Question~1.8 thus distinguishes the five basic approximation properties from the stronger topological and exceptional-set conclusions. Applications include constrained integral functionals, strict BV convergence of superlevel sets, nonlinear flux convergence, and stability of persistence diagrams under tameness assumptions.
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BibTeX
@misc{zhong2026monotone,
title = {{Monotone Sobolev functions: approximation, critical points, and level sets}},
author = {Deguang Zhong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34295},
eprint = {2609.34295},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34295},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Monotone Sobolev functions: approximation, critical points, and level sets
AU - Deguang Zhong
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34295
UR - https://arxiv.org/abs/2609.34295
AB - We give an affirmative answer to the planar local smoothing problem in Question~1.7 of D.~Ntalampekos and positive and negative answers to the basic approximation and level-set parts of his higher-dimensional Question~1.8. In every dimension n≥2, each continuous Lebesgue monotone function in W^1,p on a bounded open set admits uniform and strong W^1,p approximation by monotone C^1,α_loc functions, with unchanged Sobolev boundary values and no increase of the p-Dirichlet energy, for 1<p<∞. The energy can be made strictly smaller unless the original function is p-harmonic. In the plane we obtain smooth local replacement at every isolated p-harmonic critical point, with arbitrarily small C^1 and Sobolev error. A point of gradient index −m can be resolved into exactly m nondegenerate saddles. Together with Ntalampekos's planar theorem, this gives smooth monotone density with fixed boundary values; at a nonsmooth p-harmonic minimizer the energy increase is unavoidable but can tend to zero. In dimensions n≥3, an explicit Lipschitz monotone function has a nonmanifold point on every level in an interval, although it admits smooth monotone approximation. A product extension of a planar homogeneous $7$-harmonic function also rules out a general discrete exceptional set under the exact boundary and energy constraints. The answer to Question~1.8 thus distinguishes the five basic approximation properties from the stronger topological and exceptional-set conclusions. Applications include constrained integral functionals, strict BV convergence of superlevel sets, nonlinear flux convergence, and stability of persistence diagrams under tameness assumptions.
ER -