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Rivière’s regularity question for critical nn-Laplace systems with antisymmetric potentials

Let n>2n>2. We construct a map U∈W1,n(Bn,Rn+2)U\in W^{1,n}(B^n,\mathbb{R}^{n+2}) that is discontinuous at the origin and smooth on the punctured ball Bn∖{0}B^n \setminus \{0\}, together with an antisymmetric potential Ω∈Ln(Bn,so(n+2)⊗Rn)\Omega\in L^n(B^n,so(n+2)\otimes\mathbb{R}^n) such that −Div(∣∇U∣n−2∇U)=Ω⋅∣∇U∣n−2∇U-\mathrm{Div}(|\nabla U|^{n-2}\nabla U)=\Omega\cdot |\nabla U|^{n-2}\nabla U in D′(Bn)D'(B^n). This gives a negative answer to a regularity question posed by Rivière.

Our potential admits the Lorentz-space regularity Ω∈⋂q>2L(n,q)∖L(n,2)\Omega \in \bigcap_{q>2}L^{(n,q)} \setminus L^{(n,2)}. In addition for given 1<p<∞1<p<\infty we can enforce ∇U∈L(n,p)\nabla U \in L^{(n,p)} but ∇U∉L(n,1)\nabla U \notin L^{(n,1)}. The construction does not give a counterexample to regularity for weakly nn-harmonic maps or for higher-dimensional HH-systems.

The example was generated by ChatGPT 5.6 Sol on August 5, 2026. The work itself was written by the author and thoroughly reviewed to ensure its correctness.

Result
Disproved(see note)
Status
Resolved
AI contribution
AI-discovered
Method
Construction
Field
Partial Differential Equations
Posed by
Tristan Rivière
Year posed
2011
Years open
15y
Solved
2026-08-05
Model
ChatGPT 5.6 Sol
Vendor
OpenAI
Collaborators
Dominik Schlagenhauf
Verification
Unreviewed
Publication
Preprint
Significance
30 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For every n>2n>2, the paper constructs a bounded map U∈W1,n(Bn,Rn+2)U\in W^{1,n}(B^n,\mathbb{R}^{n+2}), smooth on Bn∖{0}B^n\setminus\{0\} but discontinuous at the origin, together with an antisymmetric potential
Ω∈Ln(Bn,so(n+2)⊗Rn) \Omega\in L^n(B^n,so(n+2)\otimes\mathbb{R}^n)
such that
−Div(∣∇U∣n−2∇U)=Ω⋅∣∇U∣n−2∇Uin D′(Bn). -\mathrm{Div}\bigl(|\nabla U|^{n-2}\nabla U\bigr) = \Omega\cdot|\nabla U|^{n-2}\nabla U \qquad\text{in }D'(B^n).
Moreover, the potential satisfies the sharper Lorentz-space regularity
Ω∈⋂q>2L(n,q)∖L(n,2), \Omega\in\bigcap_{q>2}L^{(n,q)}\setminus L^{(n,2)},
and, for any prescribed 1<p<∞1<p<\infty, the construction can be arranged so that
∇U∈L(n,p)but∇U∉L(n,1). \nabla U\in L^{(n,p)} \qquad\text{but}\qquad \nabla U\notin L^{(n,1)}.
This gives a negative answer to Rivière’s general regularity question for critical nn-Laplace systems with antisymmetric LnL^n potentials: antisymmetry and critical LnL^n control alone do not imply continuity.

What the AI did

From the paper's own "AI Usage" section, which also appears in condensed form on the title page: "The example was provided by ChatGPT 5.6 Sol while the author was exploring possible counterexamples to the regularity problem of weakly nn-harmonic maps on August 5, 2026. The author identified it as a solution to the more general problem with the antisymmetric potential as in (1.1). Furthermore, the author simplified the paramters and notations for better readability. The proof has been checked by the author and is correct. The work was written by the author, however code snippets may occasionally come from LLMs including ChatGPT 5.6 Sol or Gemini 3.6 Thinking." For a disproof the counterexample is the entire result, and the model produced it; the author's contributions as he describes them are recognising what it settled, simplifying the parameters, checking the proof and writing the paper. Hence AI-discovered.

Verification

An arXiv preprint (v1, 25 August 2026, math.AP), unrefereed and with no independent endorsement, and no mathematics was checked here - there is no formalization and no computational certificate. The author states he has checked the proof himself, which is his own assurance rather than an independent one. Verified here on 26 August 2026: the paper exists at arXiv:2608.24393 with the title, sole author and construction this entry describes; its AI-usage section carries the disclosure quoted above word for word; and the question it answers is real and traceable - the abstract cites Rivière's 2011 chapter "The role of integrability by compensation in conformal geometric analysis" (Séminaires et Congrès 22) at Eq. (3.23), reformulated as open Problem 2.5 in Schikorra and Strzelecki's 2017 EMS survey on H-systems in higher dimensions.

Source

Submitted by VibeGene on

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