An example in the gradient theory of phase transitions

Camillo De Lellis

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

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

Abstract

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We prove by giving an example that when n ≥ 3 the asymptotic behavior of functionals ∫ Ω ε | ∇ 2 u | 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 [2]) is no longer true in higher dimensions.

How to cite

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Lellis, 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

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  1. [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. [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. [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. [4] C. De Lellis, Energie di linea per campi di gradienti, Ba. D. Thesis. University of Pisa (1999). 
  5. [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. [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. [7] W. Jin, Singular perturbation and the energy of folds, Ph.D. Thesis. Courant Insitute, New York (1999). 
  8. [8] W. Jin and R.V. Kohn, Singular perturbation and the energy of folds. J. Nonlinear Sci. 10 (2000) 355-390. Zbl0973.49009MR1752602
  9. [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

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