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

We prove by giving an example that when $n\ge 3$ the asymptotic behavior of functionals ${\int }_{\Omega }\epsilon |{\nabla }^{2}{u|}^{2}{+\left(1-|\nabla u|}^{2}{\right)}^{2}/\epsilon$ 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.

DOI: 10.1051/cocv:2002012
Classification: 49J45,  74G65,  76M30
Keywords: phase transitions, $\Gamma$-convergence, asymptotic analysis, singular perturbation, Ginzburg-Landau
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Lellis, Camillo De. An example in the gradient theory of phase transitions. ESAIM: Control, Optimisation and Calculus of Variations, Volume 7 (2002), pp. 285-289. doi : 10.1051/cocv:2002012. http://www.numdam.org/articles/10.1051/cocv:2002012/

[1] L. Ambrosio, C. De Lellis and C. Mantegazza, Line energies for gradient vector fields in the plane. Calc. Var. Partial Differential Equations 9 (1999) 327-355. | MR | Zbl

[2] P. Aviles and Y. Giga, A mathematical problem related to the physical theory of liquid crystal configurations. Proc. Centre Math. Anal. Austral. Nat. Univ. 12 (1987) 1-16. | MR

[3] P. Aviles and Y. Giga, 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. | Zbl

[4] C. De Lellis, Energie di linea per campi di gradienti, Ba. D. Thesis. University of Pisa (1999).

[5] A. De Simone, R.W. Kohn, S. Müller and F. Otto, A compactness result in the gradient theory of phase transition. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 833-844. | MR | Zbl

[6] P.-E. Jabin and B. Perthame, Compactness in Ginzburg-Landau energy by kinetic averaging. Comm. Pure Appl. Math. 54 (2001) 1096-1109. | Zbl

[7] W. Jin, Singular perturbation and the energy of folds, Ph.D. Thesis. Courant Insitute, New York (1999).

[8] W. Jin and R.V. Kohn, Singular perturbation and the energy of folds. J. Nonlinear Sci. 10 (2000) 355-390. | MR | Zbl

[9] M. Ortiz and G. Gioia, The morphology and folding patterns of buckling driven thin-film blisters. J. Mech. Phys. Solids 42 (1994) 531-559. | MR | Zbl

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