# 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.
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- 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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