The planar Mumford–Shah regularity conjecture and local weak-L⁴ gradient bounds

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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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Theorem 1.1

In Interior regularity of planar Mumford–Shah minimizers

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Let (u,K) be a reduced absolute minimizer of (1.1) in a bounded Lipschitz domain \Omega\subset\mathbb R^2, with g\in L^\infty(\Omega). Every x\in K has an interior neighborhood in which K has one of the following forms, with C^{1,\alpha} arcs for some \alpha>0:

(i)
a single embedded arc with x in its interior;
(ii)
a single embedded arc ending at x;
(iii)
three embedded arcs meeting only at x, with pairwise angles 120 degrees.

Moreover, for every open U\Subset\Omega,

\#\{C\in\pi_0(K):C\cap U\ne\varnothing\}<\infty,

where \pi_0(K) denotes the set of global connected components of K.

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Interior regularity of planar Mumford–Shah minimizers

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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.

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22 references in Interior regularity of planar Mumford–Shah minimizers

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