We study the [Gamma]-convergence of a family of vectorial integral functionals, which are the sum of a vanishing anisotropic quadratic form in the gradients and a penalizing double-well potential depending only on a linear combination of the components of their argument. This particular feature arises from the study of the so-called `bidomain model' for the cardiac electric field; one of its consequences is that the L1-norm of a minimizing sequence can be unbounded and therefore a lack of coercivity occurs. We characterize the [Gamma]-limit as a surface integral functional, whose integrand is a convex function of the normal and can be computed by solving a localized minimization problem.
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Luigi Ambrosio, Giuseppe Savaré, Piero Colli Franzone, On the asymptotic behaviour of anisotropic energies arising in the cardiac bidomain model. Interfaces Free Bound. 2 (2000), no. 3, pp. 213–266DOI 10.4171/IFB/19