Affiliations: Dipartimento di Matematica, Università di Pavia, Via Abbiategrasso 209, 27100 Pavia, Italy | International School for Advanced Studies SISSA/ISAS, via Beirut 4, 34014 Trieste, Italy | Dipartimento di Matematica, Università di Bologna, Piazza Porta S. Donato 5, 40127 Bologna, Italy | International School for Advanced Studies SISSA/ISAS, via Beirut 4, 34014 Trieste, Italy
Note: [] Correspondence to: M. Amar, Dipartimento di Matematica, Università di Pavia, Via Abbiategrasso 209, 27100 Pavia, Italy.
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.
