AbstractWe analyze a finite-difference approximation of a functional of Ambrosio–Tortorelli type in brittle fracture, in the discrete-to-continuum limit. In a suitable regime between the competing scales, namely if the discretization step $$\delta $$
δ
is smaller than the ellipticity parameter $$\varepsilon $$
ε
, we show the $$\varGamma $$
Γ
-convergence of the model to the Griffith functional, containing only a term enforcing Dirichlet boundary conditions and no $$L^p$$
L
p
fidelity term. Restricting to two dimensions, we also address the case in which a (linearized) constraint of non-interpenetration of matter is added in the limit functional, in the spirit of a recent work by Chambolle, Conti and Francfort.
Exact solutions for the total variation denoising problem of piecewise constant images in dimension one