An example in the gradient theory of phase transitions
ESAIM: Control, Optimisation and Calculus of Variations (2010)
- Volume: 7, page 285-289
- ISSN: 1292-8119
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topDe 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
top- 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.
- 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.
- 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.
- C. De Lellis, Energie di linea per campi di gradienti, Ba. D. Thesis. University of Pisa (1999).
- 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.
- P.-E. Jabin and B. Perthame, Compactness in Ginzburg-Landau energy by kinetic averaging. Comm. Pure Appl. Math.54 (2001) 1096-1109.
- W. Jin, Singular perturbation and the energy of folds, Ph.D. Thesis. Courant Insitute, New York (1999).
- W. Jin and R.V. Kohn, Singular perturbation and the energy of folds. J. Nonlinear Sci.10 (2000) 355-390.
- M. Ortiz and G. Gioia, The morphology and folding patterns of buckling driven thin-film blisters. J. Mech. Phys. Solids42 (1994) 531-559.
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