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

Abstract

Abstract. We approximate, in the sense of Γ-convergence, free-discontinuity functionals with linear growth in the gradient by a sequence of non-local integral functionals depending on the average of the gradients on small balls. The result extends to higher dimension what we already proved in the one-dimensional case.Mathematics Subject Classification. 49Q20, 49J45, 49M30.

Γ-limits of convolution functionals

“…Denote by X the space of all functions v ∈ W 1,1 loc (R × P ⊥ C ) which are non-decreasing in the first variable and such that there exist ξ 0 < ξ 1 with v(x) = 0 if x 1 < ξ 0 and v(x) = s if x 1 > ξ 1 . Then, exploiting the same argument as in [22], we have θ(s, e 1 ) ≥ inf X G, where…”

“…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…”

“…3.1 for details). In Section 7 we have been able to show that the method used in [22] to write θ in a more explicit form works only if n = 1. In the case n > 1 such an argument does not work.…”

“…, n. In the case C = B 1 and ρ = 1 ωn χ B1 , treated in [22], one is able to prove that inf X G = inf Y G computing directly inf X G by a discretization argument (see Prop. 5.7 in [22]). In general, inf X G = inf Y G does not hold.…”

“…In general, inf X G = inf Y G does not hold. Indeed proceeding at first as in the proof of Proposition 5.6 in [22], one is able to show that for any C ⊂ R 2 open, bounded, convex and symmetrical set (i.e. C = −C) and for ρ =…”