The planar Mumford–Shah regularity conjecture and local weak-L4 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.
released 2026-09-24 | 1 theorem · 5 lemmas · 9 proofs · 9,192 words |
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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.