Construct SBV-regular singular minimizers beyond μ = 3

Determine whether BV-minimizers of autonomous μ-elliptic convex linear-growth functionals with μ > 3 can exhibit a Cantor singular part without relying solely on a jump across a hyperplane, in particular by establishing an SBV-regularity result for the singular minimizer construction.

Background

The main construction produces a singular BV-minimizer whose singularity is a jump across a hyperplane. Consequently, it demonstrates failure of W{1,1}-regularity but does not determine whether Cantor parts can occur.

The author explicitly reports being unable to obtain an SBV-regularity result even for μ > 3, leaving unresolved the broader structure of singularities that autonomous μ-elliptic functionals can generate.

References

The author has tried to obtain such a $\mathrm{SBV}$-regularity result even for $\mu>3$, but has not been successful in doing so.

— The Sharp Ellipticity Threshold for $\mathrm{W}^{1,1}$-Regularity  (2609.29826 - Gmeineder, 24 Sep 2026) in Section 1, immediately after Theorem 1.1

In particular, it is unclear how similar strategies can be made to work in the framework of Theorem \ref{thm:main}.

— The Sharp Ellipticity Threshold for $\mathrm{W}^{1,1}$-Regularity  (2609.29826 - Gmeineder, 24 Sep 2026) in Section 2.1, final paragraph of “Approaches to W^{1,1}-regularity and available thresholds”