Maximal parabolic regularity at half-Hölder time dependence
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Does one-half Hölder continuity in time suffice for L² maximal regularity of a uniformly elliptic parabolic equation? The endpoint question includes real scalar diffusion on a bounded interval, continuous L² forcing and zero initial data, even for coefficients close to the heat coefficient.
Why it matters and what counts as progress
Why it matters: The half-Hölder endpoint separates known positive regularity results from possible failure even for scalar diffusion close to the heat equation.
What counts as progress: Construct a solution whose time derivative is not in space-time L² or prove endpoint regularity under a sharper coefficient condition.
How the problem developed
- 11 Aug 2026 · Counterexample [1]
Lukas Niebel Constructed uniformly elliptic real scalar coefficients, arbitrarily close to the heat coefficient and 1/2-Hölder in time, for which a Lions solution has no L² time derivative.
AI contribution
Lions' Maximal Regularity Problem for Divergence-Form Differential Operators: Failure at the 1/2-Hölder Endpoint
Niebel constructs a real scalar diffusion coefficient on a bounded interval that is bounded, uniformly elliptic, one-half Hölder in time with spatial L∞ values, and arbitrarily close to the heat coefficient, yet its zero-data variational solution with continuous L² forcing has a time derivative outside space-time L². Extensions give real symmetric isotropic counterexamples in all dimensions and bounded domains.
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AI-origin example reaches the half-Hölder regularity endpoint