Inserted: 17 nov 2009
Last Updated: 24 mar 2011
Journal: Calc. Var. Partial Diff. Eq.
According to the Weak-Membrane Model by Blake and Zisserman, a simple way to model free-discontinuity energies in a finite-difference scheme is by considering truncated quadratic energy densities (`defected springs' in a mass-spring model). Thanks to a scaling argument due to Chambolle this discrete model can be approximated by a continuous energy of fracture type defined on special functions with bounded variation. If not all springs are `defected', but a portion of them are simple quadratic linear springs then the problem is more complex, and a continuous description must take into account the location and `micro-geometry' of the two types of springs. In a probabilistic setting the location of the defected springs can be modeled in terms of realizations of i.i.d. random variables. In dimension two an analysis by Braides and Piatnitski shows that the $\Gamma$-limit is deterministic and depends almost surely on the probability $p$ of the weak springs. A deterministic study leads necessarily to a more complex statement. In this case we look at all possible $\Gamma$-limits of energies which mix arbitrarily weak and strong springs (a G-closure problem for free-discontinuity energies). We give a conjecture for the general form of the limit, and prove that it is sharp for homogeneous or isotropic energies.
Keywords: Homogenization, Discrete energies, Free-discontinuity problems, defects