Improve the unconditional ellipticity threshold

Determine whether the known condition μ < 1 + 2/n for W^{1,1}-regularity of BV-minimizers of μ-elliptic autonomous convex linear-growth functionals can be improved to μ < 1 + 2/(n−1).

Background

The paper reviews existing regularity theory for BV-minimizers of convex linear-growth functionals satisfying μ-ellipticity. The currently available unconditional result assumes μ < 1 + 2/n, while a stronger threshold 1 + 2/(n−1) is known in related (p,q)-growth settings under different hypotheses.

The authors explicitly identify whether the linear-growth threshold can be improved to the stronger value as unresolved. This question is relevant to determining the sharp range in which BV-minimizers necessarily belong to W{1,1}.

References

Moreover, as discussed in Remark 5.14, the first condition on $\mu$ from eq:muconditions is presently not known to be improvable to $\frac{2}{n-1}$ as in eq:pqbound1.

— 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”

For such integrands, De Filippis has recently obtained sharp criteria on $\mu$ for the underlying variational integrals to produce Non-Lipschitzian but Hölder regular $W{1,1}$-minimizers, thereby answering a question dating back to Ladyzhenskaya and Ural'tseva; see also Beck et al. and the references therein for positive results in the linear growth context. We point out that the construction in the proof of Theorem \ref{thm:main} does not work for radially symmetric integrands, and it is conceivable that radial symmetry leads to $W{1,1}$-regularity of $BV$-minimizers indeed.

— The Sharp Ellipticity Threshold for $\mathrm{W}^{1,1}$-Regularity  (2609.29826 - Gmeineder, 24 Sep 2026) in Remark 1.4