Bouchitté’s vanishing mass conjecture

Let $V_n$ be a sequence of divergence-free matrix fields with $\sup_n\|V_n\|_{L^1}<\infty$ , each supported in an open set $A_n$ satisfying $|A_n|\to 0$ . The conjecture asserts a complete characterization of the limiting distributions of the directions carried by such sequences: every such limiting distribution is a superposition of microstructures whose barycenters are singular matrices, and conversely every superposition of microstructures with singular barycenters is generated by a sequence of this type.

References

Primary source

arXiv

Additional references

Divergence-free concentrations come from vanishing sequences — arXiv — Adolfo Arroyo-Rabasa, Francesco Nobili, Ivan Yuri Violo

calculus-of-variations partial-differential-equations

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A September 2026 paper claims to finish the conjecture for nonsymmetric objects, but it does not cover the symmetric case.

Bouchitté formulated the conjecture in 2001 as the missing ingredient in a lower-bound result for vanishing-mass limits. The problem concerns a complete characterization of the relevant concentration behavior, not merely necessary rigidity conditions.

Known results

A 2022 paper claimed rigorous vanishing-mass-limit results for optimal light structures and described the open problem as settled in dimensions two and three.

September 2026 converse construction

Arroyo-Rabasa, Nobili, and Violo report that the converse is established using support sets with smooth compact relative boundary. The result completes the characterization in the nonsymmetric setting, but this is a reported theorem rather than independently verified evidence in the scan.

Current status (as of September 2026): the conjecture is claimed complete in the nonsymmetric case, while the symmetric case remains outside the stated theorem.

Sources

No solutions have been posted yet.