Regularity for H-systems of bounded distortion
Abstract
We consider weak solutions u ∈ W1,n(U,Rn+1) of the H-system − div ( |D u|n−2 D u ) = (H ∘ u) J u, where n ≥ 2, U is the open unit ball in Rn, and H is bounded. We show that u is locally bounded if its distortion |D u|n / |J u| is bounded on the set where |u| is large. By a result of Fusco, Kristensen, Leone, and Verde, it follows that u is locally Hölder continuous if H is continuous, and continuously differentiable with locally Hölder continuous derivative if H is locally Lipschitz. The proof pushes forward |D u|n−2 D u ∘ D u* by u to a linear functional on continuous compactly supported maps of Rn+1 into Hom(Rn+1,Rn+1), whose values on derivatives are controlled by the equation. When the distortion is bounded, this functional satisfies the hypotheses of an inequality of Michael-Simon type for measures satisfying a first-order partial differential equation, due to De Philippis, Gennaioli, Pigati,and Rindler. The inequality yields lower bounds for the mass ratios of the push-forward of |D u|n ℒn by u, and these imply boundedness.
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BibTeX
@misc{kolasinski2026regularity,
title = {{Regularity for $H$-systems of bounded distortion}},
author = {Sławomir Kolasiński},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06441},
eprint = {2610.06441},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06441},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Regularity for H-systems of bounded distortion
AU - Sławomir Kolasiński
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.06441
UR - https://arxiv.org/abs/2610.06441
AB - We consider weak solutions u ∈ W^1,n(U,R^n+1) of the H-system − div ( |D u|^n−2 D u ) = (H ∘ u) J u, where n ≥ 2, U is the open unit ball in R^n, and H is bounded. We show that u is locally bounded if its distortion |D u|^n / |J u| is bounded on the set where |u| is large. By a result of Fusco, Kristensen, Leone, and Verde, it follows that u is locally Hölder continuous if H is continuous, and continuously differentiable with locally Hölder continuous derivative if H is locally Lipschitz. The proof pushes forward |D u|^n−2 D u ∘ D u^* by u to a linear functional on continuous compactly supported maps of R^n+1 into Hom(R^n+1,R^n+1), whose values on derivatives are controlled by the equation. When the distortion is bounded, this functional satisfies the hypotheses of an inequality of Michael-Simon type for measures satisfying a first-order partial differential equation, due to De Philippis, Gennaioli, Pigati,and Rindler. The inequality yields lower bounds for the mass ratios of the push-forward of |D u|^n ℒ^n by u, and these imply boundedness.
ER -