A priori estimates for finite-energy sequences of one-dimensional Cahn-Hilliard functional with non-standard multi-well potential

Abstract

In this paper we provide some results pertaining to asymptotic behaviour as $\varepsilon\str 0$ of the finite-energy  sequences of the one-dimensional  Cahn-Hilliard functional  
$$I^{\varepsilon}_0(u)=\int_{0}^{1}\Big({\varepsilon}^2u'^2(s)+W(u(s))\Big)ds,$$
where $u\in {\rm H}^{1}\oi{0}{1}$ and where $W$ is a multi-well potential endowed with a non-standard integrability condition. We introduce a new class of finite-energy sequences, we recover its underlying geometric properties  as $\vrepsilon\str 0$, and obtain the related a priori estimates.