Numerical minimization of geometrical type problems related to calculus of variations

The minimization of the functional G(v)=H(S v)+∫ ∂Ω m·v-∫ Ω k·v is related to various geometrical type problems in calculus of variations, such as the minimal partition of a set, the segmentation of images, and the search for sets with prescribed curvature. The functional G is first regularized and next discretized by means of piecewise linear finite elements with numerical quadratures, thus allowing its actual minimization on a computer. The discrete functionals converge to G in the sense of Γ-convergence, which implies the convergence of the discrete minima to a minimum of G. Various numerical experiments illustrate the behaviour of the numerical algorithm.