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Non-local approximation of free-discontinuity problems with linear growth

Luca Lussardi, Enrico Vitali (2007)

ESAIM: Control, Optimisation and Calculus of Variations

We approximate, in the sense of Γ-convergence, free-discontinuity functionals with linear growth in the gradient by a sequence of non-local integral functionals depending on the average of the gradients on small balls. The result extends to higher dimension what we already proved in the one-dimensional case.

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

Benoît Merlet (2007)

ESAIM: Control, Optimisation and Calculus of Variations

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 < δ 1 , δ 2 , δ 3 < 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...

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