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An example in the gradient theory of phase transitions

Published online by Cambridge University Press:  15 September 2002

Camillo De Lellis
Camillo De Lellis*
Affiliation:
Scuola Normale Superiore, P.zza dei Cavalieri 7, 56100 Pisa, Italy; [email protected].
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Abstract

We prove by giving an example that when n ≥ 3 the asymptotic behavior of functionals $\int_\Omega \varepsilon|\nabla^2 u|^2+(1-|\nabla u|^2)^2/\varepsilon$ 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.

Keywords

Phase transitionsΓ-convergenceasymptotic analysissingular perturbationGinzburg–Landau.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 7 , 2002 , pp. 285 - 289
Copyright
© EDP Sciences, SMAI, 2002

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References

Ambrosio, L., De Lellis, C. and Mantegazza, C., Line energies for gradient vector fields in the plane. Calc. Var. Partial Differential Equations 9 (1999) 327-355. CrossRef
Aviles, P. and Giga, Y., A mathematical problem related to the physical theory of liquid crystal configurations. Proc. Centre Math. Anal. Austral. Nat. Univ. 12 (1987) 1-16.
Aviles, P. and Giga, Y., On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg-Landau type energy for gradient fields. Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 1-17. CrossRef
C. De Lellis, Energie di linea per campi di gradienti, Ba. D. Thesis. University of Pisa (1999).
De Simone, A., Kohn, R.W., Müller, S. and Otto, F., A compactness result in the gradient theory of phase transition. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 833-844. CrossRef
Jabin, P.-E. and Perthame, B., Compactness in Ginzburg-Landau energy by kinetic averaging. Comm. Pure Appl. Math. 54 (2001) 1096-1109. CrossRef
W. Jin, Singular perturbation and the energy of folds, Ph.D. Thesis. Courant Insitute, New York (1999).
Jin, W. and Kohn, R.V., Singular perturbation and the energy of folds. J. Nonlinear Sci. 10 (2000) 355-390. CrossRef
Ortiz, M. and Gioia, G., The morphology and folding patterns of buckling driven thin-film blisters. J. Mech. Phys. Solids 42 (1994) 531-559. CrossRef