Continuity for critical n-Laplace systems with antisymmetric potentials
Follow the latest progress by email.
For dimensions n>2, does an antisymmetric Lⁿ potential in a critical n-Laplace system force bounded W¹,ⁿ weak solutions to be continuous?
Why it matters and what counts as progress
Why it matters: The question tests whether antisymmetry alone supplies the higher-dimensional regularity available in stronger Lorentz or curl-controlled settings.
What counts as progress: A continuity theorem under the exact critical hypotheses, or a bounded distributional solution discontinuous at a point with its potential meeting the claimed integrability.
How the problem developed
- 2007 · Partial result [1]
Tristan Rivière Established continuity regularity for a two-dimensional conformally invariant system, motivating the antisymmetric-potential framework.
- 2011 · Open problem [1]
Tristan Rivière Posed the higher-dimensional critical n-Laplace regularity question with antisymmetric potentials.
- 2024 · Partial result [1]
Dominik Martino and Armin Schikorra Proved continuity under the additional Lorentz condition Ω∈L^(n,2) for weakly n-harmonic maps.
- 25 Aug 2026 · Counterexample [1]
Dominik Schlagenhauf and ChatGPT 5.6 Sol Exhibited a discontinuous weak solution for the unrestricted critical system, with Ω in every L^(n,q), q>2, but not L^(n,2).
AI contribution
A Discontinuous Solution of the Critical n-Laplace System with Antisymmetric Potential
For n>2, Schlagenhauf constructs a bounded W¹,ⁿ map from the n-ball to Rⁿ⁺² that is discontinuous at the origin yet solves the critical n-Laplace system with an antisymmetric Lⁿ potential in distributions. The potential can lie in every L⁽ⁿ,ᑫ⁾ for q>2.
Sources
Website timeline
Publication dates and editorial changes.