Some results in asymptotic analysis of finite-energy sequences of one-dimensional Cahn–Hilliard functional with non-standard two-well potential

Abstract

In this paper we extend the consideration of G. Leoni pertaining to the finite-energy sequences of the one-dimensional Cahn-Hilliard functional \[ I^{\varepsilon}_0(u)=\int_{0}^{1}\Big({\varepsilon}^2 u'^2(s)+W(u(s))\Big)ds, \] where \(u\in {\rm H}^{1}(0,1)\) and where \(W\) is a two-well potential with symmetrically placed wells endowed with a non-standard integrability condition. We introduce several new classes of finite-energy sequences, we recover their underlying geometric properties as \(\varepsilon\longrightarrow 0\), and we prove the related compactness result.