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Numerical study of a new global minimizer for the Mumford-Shah functional in \(\mathbb R^3\). (English) Zbl 1152.49045

Summary: In [Singular sets of minimizers for the Mumford-Shah functional. Basel: Birkhäuser (2005; Zbl 1086.49030)], G. David suggested the existence of a new type of global minimizers for the Mumford-Shah functional in \(\mathbb R^{3}\). The singular set of such a new minimizer belongs to a three parameters family of sets \((0<\delta_1,\delta_2,\delta_3<\pi)\). 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 eigenvalues 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 \((\delta_1,\delta_2,\delta_3)\) for which the necessary conditions are satisfied and this shows that the hypothesis is wrong.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
35J25 Boundary value problems for second-order elliptic equations
65N38 Boundary element methods for boundary value problems involving PDEs

Citations:

Zbl 1086.49030
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References:

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