Establish higher logarithmic gradient integrability beyond μ = 3

Establish whether locally bounded BV-minimizers of μ-elliptic autonomous convex linear-growth functionals with μ > 3 satisfy a quantitative gradient-integrability estimate of the form ∇u ∈ L log^{α(μ)} L_loc, with α(μ) tending to zero as μ tends to infinity.

Background

The area integrand corresponds to the borderline ellipticity exponent μ = 3. At this threshold, locally bounded BV-minimizers are known to have W{1,1}-regularity together with a local L log² L gradient improvement.

The paper raises as a central conjecture the possibility that, beyond μ = 3, some weaker logarithmic gain in gradient integrability may still persist, even though the main theorem proves that qualitative W{1,1}-regularity can fail for μ > 3.

References

This leads to the central conjecture that, when going beyond $\mu=3$, higher gradient integrability might persist on a more fine-tuned scale, e.g., $\nabla u\inL\log{\alpha(\mu)}L_{loc}$ with $\alpha(\mu)\to 0$ as $\mu\to\infty$.

— The Sharp Ellipticity Threshold for $\mathrm{W}^{1,1}$-Regularity  (2609.29826 - Gmeineder, 24 Sep 2026) in Section 1, subsection “Non-uniform and μ-ellipticity”