Regularity of elliptic systems with Uhlenbeck structure
Abstract
We study autonomous elliptic systems with Uhlenbeck structure of the form −div(a(|Du|)Du)=0 in Ω⊂ ℝn Under two-sided quantitative (p,q)-growth conditions on a, we prove gradient Hölder continuity with constants and exponent depending only on the dimensions, p and q. This improves the classical theory initiated by Uhlenbeck by removing the continuity requirement on a'. The proof uses nonlinear comparison systems whose regularity follows from a scalar uniformly elliptic equation. Under weaker (p,q)-growth conditions on the convex function φ(t):=∫0t sa(s) ds, we obtain local W1,s-regularity for every s>1, and hence Hölder continuity of solutions with every exponent less than one. An additional lower growth condition on a yields local Lipschitz continuity, even when a is discontinuous.
Related
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- Regularity Estimates for Quasilinear Elliptic Equations with Measure Data and Matrix-Valued Weight
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BibTeX
@misc{ok2026regularity,
title = {{Regularity of elliptic systems with Uhlenbeck structure}},
author = {Jihoon Ok},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08635},
eprint = {2611.08635},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08635},
}
RIS
TY - UNPB
TI - Regularity of elliptic systems with Uhlenbeck structure
AU - Jihoon Ok
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08635
UR - https://arxiv.org/abs/2611.08635
AB - We study autonomous elliptic systems with Uhlenbeck structure of the form −div(a(|Du|)Du)=0 in Ω⊂ ℝ^n Under two-sided quantitative (p,q)-growth conditions on a, we prove gradient Hölder continuity with constants and exponent depending only on the dimensions, p and q. This improves the classical theory initiated by Uhlenbeck by removing the continuity requirement on a'. The proof uses nonlinear comparison systems whose regularity follows from a scalar uniformly elliptic equation. Under weaker (p,q)-growth conditions on the convex function φ(t):=∫_0^t sa(s) ds, we obtain local W^1,s-regularity for every s>1, and hence Hölder continuity of solutions with every exponent less than one. An additional lower growth condition on a yields local Lipschitz continuity, even when a is discontinuous.
ER -