On the homogeneity of global minimizers for the Mumford-Shah functional when K is a smooth cone

2008/09/24 by Antoine Lemenant, Lemenant, Antoine
Mathematics · Medicine · #35J25 #35P15 #49Q05 #49Q20 #Analysis of PDEs (math.AP) #Bone and Joint Diseases #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.0809.4174

openalex publication_date 2008/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that if (u,K) is a global minimizer for the Mumford-Shah functional in R^N, 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 C¹ curvilinear convex polygones, then it is a P, Y or T.

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