An example in the gradient theory of phase transitions

Lellis, Camillo De

doi:10.1051/cocv:2002012

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An example in the gradient theory of phase transitions

Lellis, Camillo De

ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 285-289

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Résumé

We prove by giving an example that when $n \geq 3$ the asymptotic behavior of functionals $\int_{Ω} ε | \nabla^{2} {u |}^{2} {+ (1 - | \nabla u |}^{2})^{2} / ε$ 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 [2]) is no longer true in higher dimensions.

MR Zbl

DOI : 10.1051/cocv:2002012

Classification : 49J45, 74G65, 76M30
Keywords: phase transitions, $\Gamma $-convergence, asymptotic analysis, singular perturbation, Ginzburg-Landau

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     title = {An example in the gradient theory of phase transitions},
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     publisher = {EDP-Sciences},
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     year = {2002},
     doi = {10.1051/cocv:2002012},
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     zbl = {1037.49010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2002012/}
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Lellis, Camillo De. An example in the gradient theory of phase transitions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 285-289. doi: 10.1051/cocv:2002012

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