Lions' Maximal Regularity Problem at the Half-Holder Endpoint
Lions asked whether the variational solution of a non-autonomous divergence-form problem has maximal L2-regularity under Holder continuity in time of the coefficients. Disproved at the half-Holder endpoint: a bounded, uniformly elliptic, real scalar coefficient, half-Holder in time and arbitrarily close to the heat equation, whose Lions solution has a time derivative that is not square integrable.
- Result
- Disproved(see note)
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Evolution equations
- Posed by
- Jacques-Louis Lions
- Year posed
- 1961
- Years open
- 65y
- Solved
- 2026-08-11
- Model
- GPT-5.5 Pro
- Vendor
- OpenAI
- Collaborators
- Lukas Niebel
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Tensorisation and parabolic rescaling carry the one-dimensional example to real symmetric isotropic counterexamples on R^d and on every bounded domain, in every dimension.
What the AI did
The Declaration of AI Use: "During an exploratory analysis, a first counterexample was found by OpenAI's GPT-5.5 Pro in two dimensions and on the full space. It was then verified and studied by the author, who simplified it and reduced it to the one-dimensional interval counterexample presented here." GPT-5.6 Sol was separately used for drafting and revision of the exposition.
Verification
A preprint days old, with no independent review.
Source
- PaperarXiv