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The planar Mumford–Shah regularity conjecture and local weak-L $^{4}$ gradient bounds

Resolves the interior regularity conjecture for reduced absolute planar Mumford–Shah minimizers with bounded fidelity data. Locally, the closed discontinuity set is a

C^{1,\alpha}

arc, a regular crack tip, or three arcs meeting at

120^\circ

; only finitely many global connected components meet any compact interior region.

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Interior regularity of planar Mumford–Shah minimizers

Sep 24, 2026·Interior-regularity-of-planar-Mumford-Shah-minimizers-September-24-2026.pdf

We prove the interior regularity assertion of the planar Mumford–Shah conjecture for reduced absolute minimizers with bounded fidelity data. Every interior point of the closed discontinuity set has a neighborhood consisting of a

C^{1,\alpha}

arc, an arc ending at that point, or three such arcs meeting at 120 degrees. Only finitely many global connected components meet any relatively compact open set.

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The planar Mumford–Shah regularity conjecture and local weak-L4^{4}4 gradient bounds

Manuscripts in this family · 1

Interior regularity of planar Mumford–Shah minimizers

The planar Mumford–Shah regularity conjecture and local weak-L $^{4}$ gradient bounds