Variational models for phase separation. (Q1396046)

The paper is concerned with the asymptotic behaviour of a family of integral functionals connected with classical models for phase separation. Let \(W\) be a nonnegative, continuous, double well type energy density with minima located at \(u=0\) and \(u=1\), and with linear growth at infinity. Let \(g\) be nonnegative, even, Lipschitz continuous, positively 1-homogeneous, and positive for \(z\not=0\). Let \(\Omega\) be an open subset of \({\mathbb R}^n\). In the paper the asymptotic behaviour as \(\varepsilon\to 0\) of the family of functionals \[ F_\varepsilon(u,v)={1\over\varepsilon}\int_\Omega W(u)dx+{\alpha\over\varepsilon} \int_\Omega(u-v)^2\,dx+ \varepsilon\int_\Omega g(\nabla v)^2\,dx \] is studied, where \(\alpha\) is a positive parameter. In the one-dimensional setting, such functionals were proposed by Rogers and Truskinovsky with \(g(z)=| z| \) as a model for the longitudinal deformation of an elastic bar which could take into account an elastic energy of Ericksen's type together with an internal scalar variable which measures the deviation from one-dimensional deformation. It is proved that the limit functional, in the sense of \(\Gamma(L^1(\Omega)) \)-convergence, \(F\) is finite at \((u,v)\) if and only if \(u\) is a function of bounded variation and \(u=v\in\{0,1\}\) a.e., and that in this case \[ F(u,v)=c_W(\alpha)\int_{S(u)}g^{**}(\nu_u)d{\mathcal H}^{n-1}, \] where \(g^{**}\) denotes the convex envelope of \(g\), \[ c_W(\alpha)=\sqrt\alpha\inf\bigg\{\int_{\mathbb R}W(\varphi)dx+{\alpha^2\over4}\iint_{{\mathbb R}^2}e^{-\alpha | x-y| }(\varphi(x)-\varphi(y))^2 \,dx\,dy : \] \[ \varphi \colon{\mathbb R}\to[0,1]\text{ measurable such that }\lim_{x\to-\infty}\varphi(x)=0,\;\lim_{x\to+\infty}\varphi(x)=1\bigg\}, \] \(S(u)\) is the approximate discontinuity set of \(u\), and \(\nu_u\) is a normal unit vector field on \(S(u)\). Additional representation and qualitative results for \(c_W\) are also obtained. It is also proved that the limit \(\lim_{\alpha\to+\infty}c_W(\alpha)\) agrees with the well-known Modica-Mortola constant \(\min\{\int_{\mathbb R}W(\varphi)dx+\int_{\mathbb R}\varphi'(x)^2dx :\varphi\in H^1({\mathbb R}),\;\lim_{x\to-\infty}\varphi(x)=0,\;\lim_{x\to+\infty}\varphi(x)=1\}\).