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On the singular sets of minimizers of the Mumford-Shah functional. (English) Zbl 0853.49010

Let \(\Omega\) be a bounded open subset in \(\mathbb{R}^2\) and let \(g \in L^\infty (\Omega)\). For \(K\) a closed subset of \(\Omega\) and \(u \in C^1 (\Omega \backslash K)\), define \[ J(u,K) = \int_{\Omega \backslash K} |\nabla u |^2 + \int_{\Omega \backslash K} |u - g |^2 + H^1(K), \] where \(H^1(K)\) is the “length” (i.e., the one-dimensional Hausdorff measure) of the set \(K\). Call a pair \((u,K)\) an irreducible minimizer if \(J(u,K)\) is minimum and also \(u\) cannot be extended to a function which is \(C^1\) on \(\Omega \backslash K'\) for some strictly smaller \(K'\). The main result of the paper is that, if \(\Omega\) is reasonably regular (piecewise \(C^1\) is more than enough) and if \((u,K)\) is an irreducible minimizer, then \(K\) is contained in a regular curve \(\Gamma\) with constant \(\leq C (\Omega, |g |_\infty)\). This means that there exists a parameterization \(z : \mathbb{R} \to \mathbb{R}^2\) of \(\Gamma\) such that \(|z(s) - z(t) |\leq |s - t |\) for all \(s,t\) and such that for all balls \(B\), \(|\{s \in \mathbb{R} : z(s) \in B\} |\leq C(\Omega, |g |_\infty) \text{diam} B \).
We refer to the paper for the statement of precise conditions on \(\Omega\) as well as more precise rectifiability properties satisfied by \(K\). Recall that Mumford and Shah have conjectured that \(K\) is a finite union of \(C^1\) curves; previous results in that direction have been obtained by Ambrosio; De Giorgi, Carriero, Leaci; Dal Maso, Morel, Solimini; Dibos, Koepfler,... . The proofs of this paper are based on lower estimates on \(H^1(K)\) of Dal Maso, Morel, Solimini, previous results of the authors on “uniform rectifiability”, and direct comparison arguments.
Reviewer: G.David (Orsay)

MSC:

49J40 Variational inequalities
49Q20 Variational problems in a geometric measure-theoretic setting
49J20 Existence theories for optimal control problems involving partial differential equations
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