Non-local approximation of free-discontinuity functionals with linear growth: the one-dimensional case

Abstract

We approximate, in the sense of -convergence, one-dimensional freediscontinuity functionals with linear growth in the gradient by means of a sequence of non-local integral functionals depending on the averages of the gradient on small intervals.

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

“…If n = 1 and Ω = (a, b), this result follows as a particular case of Theorem 3.3 in [21]; the corresponding form of the jump energy density is…”

“…4.7) when applying the so-called slicing method. Actually, we will need the stronger form of the lower Γ-limit estimate contained in Remark 3.4 of [21]: if n = 1 and Ω = (a, b), then…”

“…Unlike the one-dimensional case we dealt with in [21], here we restrict the study to a fixed integrand function f , independent of ε. In the limit (see Th.…”

An approximation result for free discontinuity functionals by means of non‐local energies

“…In this paper, in Section 4, we first extend result [10] to a family of functions f ε , depending on ε, instead of f , in view of obtaining a convergence result (already proved in the one-dimensional case in [11]) to a functional (1) with a more general growth in the gradient with respect to the case = Id considered in [10]. Then, in Section 5, we will investigate whether it is possible to construct a family (F ε ) ε>0 , which -converges to a given functional F of type (1) with linear growth in the gradient.…”

Γ-limits of convolution functionals

“…A variant of the method proposed in [10] has been used in [22] to deal with the approximation of a functional F of the form (1.1), with φ having linear growth and θ independent on the normal ν u (see also [20,21]). More precisely, in [22] the Γ -limit of the family…”