# An example in the gradient theory of phase transitions

ESAIM: Control, Optimisation and Calculus of Variations (2002)

- Volume: 7, page 285-289
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topLellis, Camillo De. "An example in the gradient theory of phase transitions." ESAIM: Control, Optimisation and Calculus of Variations 7 (2002): 285-289. <http://eudml.org/doc/245679>.

@article{Lellis2002,

abstract = {We prove by giving an example that when $n\ge 3$ 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.},

author = {Lellis, Camillo De},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {phase transitions; $\Gamma $-convergence; asymptotic analysis; singular perturbation; Ginzburg–Landau; -convergence; Ginzburg-Landau energy},

language = {eng},

pages = {285-289},

publisher = {EDP-Sciences},

title = {An example in the gradient theory of phase transitions},

url = {http://eudml.org/doc/245679},

volume = {7},

year = {2002},

}

TY - JOUR

AU - Lellis, Camillo De

TI - An example in the gradient theory of phase transitions

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2002

PB - EDP-Sciences

VL - 7

SP - 285

EP - 289

AB - We prove by giving an example that when $n\ge 3$ 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.

LA - eng

KW - phase transitions; $\Gamma $-convergence; asymptotic analysis; singular perturbation; Ginzburg–Landau; -convergence; Ginzburg-Landau energy

UR - http://eudml.org/doc/245679

ER -

## References

top- [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. Zbl0960.49013MR1731470
- [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. MR924423
- [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. Zbl0923.49008
- [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. Zbl0986.49009MR1854999
- [6] P.-E. Jabin and B. Perthame, Compactness in Ginzburg–Landau energy by kinetic averaging. Comm. Pure Appl. Math. 54 (2001) 1096-1109. Zbl1124.35312
- [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. Zbl0973.49009MR1752602
- [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. Zbl0832.73051MR1264947

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.