## 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

### Keywords:

minimizers; Mumford-Shah functional