We approximate, in the sense of Γ-convergence, the Mumford-Shah functional by means of a sequence of non-local integral functionals depending on the average of the absolute value of the gradient.

forse per le mie origini contadine, mi sconsigliarono vivamente di intraprendere studi scientifici. 4Preface A number of variational problems recently under consideration involves integral functionals with “free discontinuities ” (according to a terminology introduced in [22]): the variable function u is required to be smooth only outside a surface K, depending on u, and both u and K enter the structure of the functional. Hence, a typical form is: F(u, K) = φ(|∇u(x)|)dx + f(|u

Abstract. Usually smeared crack techniques are based on the following features: the fracture is represented by means of a band of finite elements and by a softening constitutive law of damage type. Often these methods are implemented with nonlocal operators which control the localization effects and reduce the mesh bias. We consider a non-local smeared crack energy defined for a finite element space on a structured grid. We characterize the limit energy as the mesh size h tends to zero and we establish a precise link between the discrete and continuum formulations of the fracture energies, showing the correct scaling and the explicit form of the mesh bias. 1