Approximation by Γ-convergence of a total variation with discontinuous coefficients

Abstract

We determine the Γ-limit as ε→0 of the sequence ∫_Ω[εϕ²(x,∇u)+ε⁻¹u²(1−u²)]dx of functionals defined on H¹(Ω). The free energy density ϕ has linear growth, is convex and positively homogeneous of degree one in the second variable and is upper semi-continuous in the first variable. The Γ-limit can be interpreted as a generalized total variation with discontinuous coefficients on the class of the characteristic functions of sets of finite perimeter. The proof is based on some variational properties of the generalized total variation strictly related to the upper semi-continuity of ϕ and on constructing a suitable approximation of ϕ, from above.