Numerical study of a new global minimizer for the Mumford-Shah functional in R³

Abstract

Abstract.In [Progress Math. 233 (2005)], David suggested the existence of a new type of global minimizers for the Mumford-Shah functional in R 3 . The singular set of such a new minimizer belongs to a three parameters family of sets (0 < δ1, δ2, δ3 < π). We first derive necessary conditions satisfied by global minimizers of this family. Then we are led to study the first eigenvectors of the Laplace-Beltrami operator with Neumann boundary conditions on subdomains of S 2 with three reentrant corners. The necessary conditions are constraints on the eigenvalue and on the ratios between the three singular coefficients of the associated eigenvector. We use numerical methods (Singular Functions Method and Moussaoui's extraction formula) to compute the eigenvalues and the singular coefficients. We conclude that there is no (δ1, δ2, δ3) for which the necessary conditions are satisfied and this shows that the hypothesis was wrong.Mathematics Subject Classification. 35J25, 49R50, 65N38.

A selective review on Mumford–Shah minimizers

“…It was conjectured by David the existence of a new global minimizer that should look like a cone Y bevelled, whose intersection with the sphere would look like the union of three branches of arc of circles of length to be determined, meeting by 120 degree. But this possibility has been, surprisingly, excluded by Merlet in [Mer07] by use of a numerical method.…”

A rigidity result for global Mumford–Shah minimizers in dimension three

“…In addition, by [8, Proposition D.37. 18.] (which is just semicontinuity with respect to the weak convergence), we getˆB…”

“…For instance in [15] it is proved that an angular sector cannot be a global minimizer, unless it is a half-plane or a plane. In [18] a tentative construction for an extra global minimizer of particular type is proved to fail.…”

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

“…• Could one found an extra global minimizer in R 3 by blowing up the minimizer described in section 76.c. of [8] (see also [17])?…”