An example in the gradient theory of phase transitions

Camillo De Lellis

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • 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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De Lellis, Camillo. "An example in the gradient theory of phase transitions." ESAIM: Control, Optimisation and Calculus of Variations 7 (2010): 285-289. <http://eudml.org/doc/90623>.

@article{DeLellis2010,
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. },
author = {De Lellis, Camillo},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Phase transitions; Γ-convergence; asymptotic analysis; singular perturbation; Ginzburg–Landau. ; phase transitions; -convergence; asymptotic analysis; Ginzburg-Landau energy},
language = {eng},
month = {3},
pages = {285-289},
publisher = {EDP Sciences},
title = {An example in the gradient theory of phase transitions},
url = {http://eudml.org/doc/90623},
volume = {7},
year = {2010},
}

TY - JOUR
AU - De Lellis, Camillo
TI - An example in the gradient theory of phase transitions
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 7
SP - 285
EP - 289
AB - 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.
LA - eng
KW - Phase transitions; Γ-convergence; asymptotic analysis; singular perturbation; Ginzburg–Landau. ; phase transitions; -convergence; asymptotic analysis; Ginzburg-Landau energy
UR - http://eudml.org/doc/90623
ER -

References

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  1. L. Ambrosio, C. De Lellis and C. Mantegazza, Line energies for gradient vector fields in the plane. Calc. Var. Partial Differential Equations9 (1999) 327-355.  Zbl0960.49013
  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.  
  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. A129 (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. A131 (2001) 833-844.  Zbl0986.49009
  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.  Zbl0973.49009
  9. M. Ortiz and G. Gioia, The morphology and folding patterns of buckling driven thin-film blisters. J. Mech. Phys. Solids42 (1994) 531-559.  Zbl0832.73051

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