A homogenization result in the gradient theory of phase transitions

Cristoferi, Riccardo; Fonseca, Irene; Hagerty, Adrian; Popovici, Cristina

doi:10.48550/arxiv.1808.01972

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2018

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“…This allows for a reduction to a problem already studied by Fonseca and Tartar [6]. In particular, we see an isotropic transition potential, in contrast with the case where the scales are roughly comensurate [4]. This note is intended as a prelude to a more complete analysis of the full range of scales.…”

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A Note on Homogenization Effects on Phase Transition Problems

Hagerty

2018

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In fluid-fluid phase transitions problems featuring small scale heterogeneity, we see that when the scale heterogeneity is sufficiently small, the periodic potential function W (x, p) can be replaced with a homogenized potential function WH(p). This allows for a reduction to a problem already studied by Fonseca and Tartar [6]. In particular, we see an isotropic transition potential, in contrast with the case where the scales are roughly comensurate [4]. This note is intended as a prelude to a more complete analysis of the full range of scales.

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“…In [4], this problem was studied in the case where ε and δ are commensurate. This was incorporated by assuming that δ = ε, that is to say…”

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confidence: 99%