Regularity of flat level sets in phase transitions
Annals of mathematics, Tome 169 (2009) no. 1, pp. 41-78
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We consider local minimizers of the Ginzburg-Landau energy functional \[\int \frac{1}{2}|\nabla u|^2 + \frac{1}{4}(1-u^2)^2dx\] and prove that, if the $0$ level set is included in a flat cylinder then, in the interior, it is included in a flatter cylinder. As a consequence we prove a conjecture of De Giorgi which states that level sets of global solutions of \[\triangle u=u^3-u\] such that \[\quad |u|\le 1, \quad \partial_n u>0, \quad \lim_{x_n \to \pm \infty}u(x’,x_n)=\pm 1\] are hyperplanes in dimension $n \le 8$.
@article{10_4007_annals_2009_169_41,
author = {Ovidiu Savin},
title = {Regularity of flat level sets in phase transitions},
journal = {Annals of mathematics},
pages = {41--78},
year = {2009},
volume = {169},
number = {1},
doi = {10.4007/annals.2009.169.41},
mrnumber = {2480601},
zbl = {1180.35499},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.169.41/}
}
Ovidiu Savin. Regularity of flat level sets in phase transitions. Annals of mathematics, Tome 169 (2009) no. 1, pp. 41-78. doi: 10.4007/annals.2009.169.41
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