  In [8], G. David suggested a new type of global minimizer for the
Mumford-Shah functional in $\R^3$, for which the singular sets belong 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 $\mathbf{S}^2$ with
three reentrant corners. The necessary conditions are constraints on the
eigenvalue and on the ratios between the 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 was wrong.
