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Interior regularity of planar Mumford–Shah minimizers

What local shapes can occur in the interior discontinuity set of a reduced planar absolute Mumford–Shah minimizer? The regularity conjecture permits smooth crack arcs, endpoints and 120-degree triple junctions, with the exceptional points locally finite.

Why it matters and what counts as progress

Why it matters: A classification would complete the geometric description of interior cracks for this variational model.

What counts as progress: A proof covering every interior discontinuity point, including the remaining disconnected-crack case, or a minimizer with an excluded local structure.

How the problem developed

  1. 1989 · Conjecture [1][2]

    David Mumford and Jayant Shah Formulated the planar regularity problem for reduced minimizers; the conjecture asks for a precise local structure of the discontinuity set.

  2. 22 Sept 2026 · Proof claim [2]

    Francesco Deangelis Posted a classification of planar generalized global minimizer blow-ups and stated that the remaining noncompact, connected-complement case is ruled out, yielding the interior regularity conjecture.

  3. 5 Oct 2026 · Revision [2]

    Francesco Deangelis Revised the classification manuscript and its account of the compact-translation and second-variation argument.

AI contribution

Sources

  1. David Mumford and Jayant Shah, Optimal approximations by piecewise smooth functions and associated variational problems
  2. Francesco Deangelis, Solution of the Mumford–Shah conjecture
  3. Solution of the Mumford–Shah Conjecture, revised version
  4. Solution of the Mumford–Shah Conjecture, initial version
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