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MiS Preprint
38/2005

Gamma-convergence of the Allen-Cahn energy with an oscillating forcing term

Marcello Lucia, Nicolas Dirr and Matteo Novaga

Abstract

We consider a standard functional in the mesoscopic theory of phase transitions, consisting of a gradient term with a double-well potential, and we add to it a bulk term modeling the interaction with a periodic mean zero external field. This field is amplified and dilated with a power of the transition layer thickness $\epsilon$ leading to a nontrivial interaction of forcing and concentration when $\epsilon \to 0$.

We show that the functionals $\Gamma$--converge after additive renormalization to an anisotropic surface energy, if the period of the oscillation is larger than the interface thickness. Difficulties arise from the fact that the functionals have non constant absolute minimizers and are not uniformly bounded from below.

Received:
Apr 19, 2005
Published:
Apr 19, 2005
Keywords:
gamma-convergence, allen-cahn, homogenization, phase transition

Related publications

inJournal
2006 Repository Open Access
Marcello Lucia, Nicolas Dirr and Matteo Novaga

Gamma-convergence of the Allen-Cahn energy with an oscillating forcing term

In: Interfaces and free boundaries, 8 (2006) 1, pp. 57-78