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Rivière's regularity question for n-Laplace systems

For n≥2n\geq 2 and m≥1m\geq 1, let u∈W1,n(Bn,Rm)u\in W^{1,n}(\mathbb{B}^n,\mathbb{R}^m) be a weak solution of −div⁡(∣∇u∣n−2∇u)=Ω⋅(∣∇u∣n−2∇u)-\operatorname{div}(|\nabla u|^{n-2}\nabla u)=\Omega\mathbin{\cdot}(|\nabla u|^{n-2}\nabla u) in Bn\mathbb{B}^n, where Ω=(Ω1,…,Ωn)∈Ln(Bn,so(m)⊗Rn)\Omega=(\Omega_1,\ldots,\Omega_n)\in L^n(\mathbb{B}^n,\mathfrak{so}(m)\otimes\mathbb{R}^n) and Ωi(x)∈so(m)\Omega_i(x)\in\mathfrak{so}(m). Must uu be continuous?

References

Primary source

arXiv

Additional references

geometric-analysispartial-differential-equationssobolev-spaces
Posted

Progress summary

Refreshed
Claimed solved

A new unrefereed construction appears to show that the critical Lebesgue assumption is insufficient, but the claim has not been independently verified.

Rivière’s question asks whether weak solutions of critical nn-Laplace systems with antisymmetric potentials in LnL^n must be continuous. A May 2023 manuscript explicitly left the original question unanswered while proving continuity under stronger Lorentz and curl hypotheses.

Known results

  • Firoozye, 1995: constructed discontinuous maps in W1,nW^{1,n} with related critical regularity properties, but not for the exact antisymmetric-potential question.
  • May 2023: continuity was proved assuming f??f?? [sic] stronger Lorentz-space and curl conditions, including f??f?? [sic] f??f??; the original LnL^n case remained open.

August 24, 2026 claimed counterexample

The manuscript Non-Regularizing properties of nn-Laplace systems with antisymmetric potentials in critical Lebesgue spaces claims a construction showing that the Lorentz-space regularity theorem cannot extend to the corresponding critical Lebesgue space. If applicable to Rivière’s exact system, this gives a negative answer; the manuscript is unrefereed and the claim is unverified.

Current status (as of August 2026): Positive Lorentz-space results are established, while a 2026 manuscript claims but does not yet verify a negative answer to the original LnL^n regularity question.

Sources

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