Rendiconti del Seminario Matematico della Università di Padova


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Volume 122, 2009, pp. 129–159
DOI: 10.4171/RSMUP/122-9

On the Homogeneity of Global Minimizers for the Mumford-Shah Functional when $K$ is a Smooth Cone

Antoine Lemenant[1]

(1) Université Paris XI Paris-Sud, Orsay, France

We show that if (u,K) is a global minimizer for the Mumford-Shah functional in RN, and if K is a smooth enough cone, then (modulo constants) u is a homogenous function of degree 1/2. We deduce some applications in R3 as for instance that an angular sector cannot be the singular set of a global minimizer, that if K is a half-plane then u is the corresponding cracktip function of two variables, or that if K is a cone that meets S2 with an union of C curvilinear convex polygones, then it is a P, Y or T.

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Lemenant Antoine: On the Homogeneity of Global Minimizers for the Mumford-Shah Functional when $K$ is a Smooth Cone. Rend. Sem. Mat. Univ. Padova 122 (2009), 129-159. doi: 10.4171/RSMUP/122-9