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.

Manuscript

Interior regularity of planar Mumford–Shah minimizers

September 24, 2026 · source · cite as \cite{OAI:Interior-regularity-of-planar-Mumford-Shah-minimizers-September-24-2026}

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.

Result 366 · Partial differential equations. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.