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Regularity and Rivière's GL(m)-Gauge Construction for Elliptic Systems with Antisymmetric Potentials in Arbitrary Dimensions
Abstract
Let 1 ≤ q ≤ 2 and denote by 2 ≤ q' its corresponding conjugate exponent. We prove the continuity of solutions u ∈ W1,(n/(n−1),q')(Bn, ℝm) to the critical elliptic system −Δu = Ω· ∇ u in dimension n ≥ 3, where the potential Ω∈ L(n,q)(Bn, so(m) ⊗ ∧1) is antisymmetric. First, we construct P ∈ W1,(n,q)(Bn, SO(m)) such that the PDE can be rewritten as −div(P−1du) = ∗ dξ· P−1du, which is nearly a Jacobian structure up to the rotation P. Second, we provide a Rivière's GL(m)-Gauge in order to establish a "full" (A,B)-conservation law, i.e. −div(Adu)=d∗ B · du. We show that the assumption on $Ω\in L^{(n,q)} (B^n, \mathfrak{so}(m) \otimes \wedge^1) $ for q≤ 2 is optimal.
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BibTeX
@misc{bayer2026regularity,
title = {{Regularity and Rivière's $GL(m)$-Gauge Construction for Elliptic Systems with Antisymmetric Potentials in Arbitrary Dimensions}},
author = {Carolin Bayer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00826},
eprint = {2609.00826},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00826},
}
RIS
TY - UNPB
TI - Regularity and Rivière's GL(m)-Gauge Construction for Elliptic Systems with Antisymmetric Potentials in Arbitrary Dimensions
AU - Carolin Bayer
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.00826
UR - https://arxiv.org/abs/2609.00826
AB - Let 1 ≤ q ≤ 2 and denote by 2 ≤ q' its corresponding conjugate exponent. We prove the continuity of solutions u ∈ W^1,(n/(n−1),q')(B^n, ℝ^m) to the critical elliptic system −Δu = Ω· ∇ u in dimension n ≥ 3, where the potential Ω∈ L^(n,q)(B^n, so(m) ⊗ ∧^1) is antisymmetric. First, we construct P ∈ W^1,(n,q)(B^n, SO(m)) such that the PDE can be rewritten as −div(P^−1du) = ∗ dξ· P^−1du, which is nearly a Jacobian structure up to the rotation P. Second, we provide a Rivière's GL(m)-Gauge in order to establish a "full" (A,B)-conservation law, i.e. −div(Adu)=d^∗ B · du. We show that the assumption on $Ω\in L^{(n,q)} (B^n, \mathfrak{so}(m) \otimes \wedge^1) $ for q≤ 2 is optimal.
ER -