On the asymptotic behaviour of anisotropic energies arising in the cardiac bidomain model

Abstract

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 $L^1$-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.