A diffused interface whose chemical potential lies in a Sobolev space

Tonegawa, Yoshihiro

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 3, pp. 487-510.

We study a singular perturbation problem arising in the scalar two-phase field model. Given a sequence of functions with a uniform bound on the surface energy, assume the Sobolev norms $W^{1,p}$ of the associated chemical potential fields are bounded uniformly, where $p>\frac{n}{2}$ and $n$ is the dimension of the domain. We show that the limit interface as $\epsilon$ tends to zero is an integral varifold with a sharp integrability condition on the mean curvature.

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