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

Marcello Lucia, Nicolas Dirr, and Matteo Novaga

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Submission date: 19. Apr. 2005 (revised version: November 2005)
Pages: 37
published in: Interfaces and free boundaries, 8 (2006) 1, p. 57-78 
DOI number (of the published article): 10.4171/IFB/135
Bibtex
Keywords and phrases: gamma-convergence, allen-cahn, homogenization, phase transition
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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 leading to a nontrivial interaction of forcing and concentration when . We show that the functionals -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.