De Giorgi conjecture on the regularity of minimizers of Cartesian area in one dimension and codimension one

For every pair of positive integers $n,m$ , every bounded open set $\Omega\subset\mathbb{R}^n$ , and admissible data $g:\Omega\to\mathbb{R}^m$ , consider minimizers $u\in BV(\Omega;\mathbb{R}^m)$ of the Cartesian-area functional $\mathcal{F}(u)=\int_{\Omega}\sqrt{\det\!\left(I_n+(Du)^{\mathsf T}Du\right)}\,dx+\int_{\Omega}|u-g|\,dx$ . The conjecture asserts that every such minimizer has the expected regularity, in particular is locally Lipschitz, in all dimensions and codimensions. The one-dimensional codimension-one case is known under a sufficiently smallness assumption on $\|g\|_{\infty}$ , where the minimizer is in fact $C^{1,1}$ ; the general conjecture remains open.

References

Primary source

Annali dell'Università di Ferrara

Additional references

A De Giorgi conjecture on the regularity of minimizers of Cartesian area in 1D — Annali dell'Università di Ferrara — Giovanni Bellettini, Shokhrukh Yu Kholmatov

calculus-of-variations geometric-measure-theory regularity-theory

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A 2025 paper proves a limited regularity result for one-dimensional minimizers with small data, but the broader conjecture remains open.

De Giorgi’s conjecture predicts regularity for minimizers of Cartesian-area functionals in all dimensions and codimensions. The new result addresses only the one-dimensional, codimension-one case under a smallness assumption.

2025 partial regularity result

Giovanni Bellettini and Shokhrukh Yu. Kholmatov prove that, for a bounded interval $I\subset\mathbb{R}$ and sufficiently small $\|g\|_\infty$ , every minimizer is Lipschitz; under ellipticity and $C^2$ anisotropy assumptions it is $C^{1,1}$ . The threshold depends on $|I|$ , and the paper also gives an anisotropic extension. The authors present this as only a partial result, not a solution of the general conjecture.

Current status (as of September 2026): The small-data one-dimensional, codimension-one case has a claimed $C^{1,1}$ regularity theorem, while the general conjecture remains open.

Sources

No solutions have been posted yet.