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An example in the gradient theory of phase transitions


De Lellis, C (2002). An example in the gradient theory of phase transitions. ESAIM: Control, Optimisation and Calculus of Variations, 7:285-289 (electronic).

Abstract

We prove by giving an example that when n ≥ 3 the asymptotic behavior of functionals ∫Ω[ε|∇2u|2+(1−|∇u|2)2/ε] is quite different with respect to the planar case. In particular we show that the one-dimensional ansatz due to Aviles and Giga in the planar case (see Aviles and Giga 1987) is no longer true in higher dimensions.

Abstract

We prove by giving an example that when n ≥ 3 the asymptotic behavior of functionals ∫Ω[ε|∇2u|2+(1−|∇u|2)2/ε] is quite different with respect to the planar case. In particular we show that the one-dimensional ansatz due to Aviles and Giga in the planar case (see Aviles and Giga 1987) is no longer true in higher dimensions.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:phase transitions; Γ-convergence; asymptotic analysis; singular perturbation; Ginzburg-Landau energy
Language:English
Date:2002
Deposited On:29 Nov 2010 16:27
Last Modified:21 Nov 2017 14:22
Publisher:EDP Sciences
ISSN:1262-3377
Publisher DOI:https://doi.org/10.1051/cocv:2002012
Related URLs:http://www.ams.org/mathscinet-getitem?mr=1925030

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