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Mumford–Shah conjecture

Let (u,K)(u,K) be a generalized global minimizer of the planar Mumford–Shah functional E(u,K;B)=∫B∖K∣∇u∣2 dx+H1(K∩B)\mathcal{E}(u,K;B)=\int_{B\setminus K}|\nabla u|^2\,dx+\mathcal{H}^1(K\cap B) for every bounded ball B⊂R2B\subset\mathbb{R}^2. The Mumford–Shah conjecture asserts that every such minimizer belongs, up to the natural equivalences, to one of the four model classes: the constant model, the pure-jump model, the triple-junction model, or the crack-tip model. Equivalently, the singular set of every planar Mumford–Shah local minimizer has only smooth crack arcs, triple junctions, and crack tips as its local configurations.

Equivalent formulations 2Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Generalized-global-minimizer classification

    Every planar generalized global minimizer of the Mumford–Shah functional is one of the constant, pure-jump, triple-junction, or crack-tip models.

    source: Francesco Deangelis, Solution of the Mumford-Shah conjecture

  2. Planar interior-regularity formulation

    Every planar Mumford–Shah local minimizer has, locally, only smooth crack arcs, triple junctions, and crack-tip configurations in its singular set.

    source: Francesco Deangelis, Solution of the Mumford-Shah conjecture

References

Primary source

arXiv

Additional references

calculus-of-variationsgeometric-measure-theory
Posted · edited

Progress summary

Refreshed
Claimed solved

A new preprint claims to prove the planar conjecture, but the claim has not yet been independently verified.

The conjecture predicts that planar Mumford–Shah singularities have only smooth crack arcs, triple junctions, and cracktips; it is equivalent to classifying generalized global minimizers. Mumford and Shah posed it in 1989.

Known results

  • Bonnet (1996) developed the blow-up approach and obtained major partial results, but not the full classification.
  • The expected model configurations, including triple junctions and cracktips, are classified in important special cases.
  • A key equivalent regularity condition is ∇u∈Lloc4,∞\nabla u \in L^{4,\infty}_{\mathrm{loc}}; higher integrability with some p>2p>2 is known, but this endpoint remains unresolved.
  • As of January 2025, the remaining obstacle was the discreteness of planar tip points, including for absolute minimizers with infinitely many components.

September 2026 claimed solution

Francesco Deangelis’s arXiv preprint claims a compact-translation and Hodge-identity argument that excludes compact isolated crack portions and identifies the standard model minimizers, thereby claiming the planar conjecture. This is a new, unverified claim; no independent verification or gap analysis was found in the retrieved sources.

Current status (as of September 2026): A preprint claims the planar conjecture is solved, but the claim is unverified; absent confirmation, the classification and the discreteness of tip points remain open.

Sources

Solutions 0

No solutions have been posted yet.