We consider a class of two-dimensional Ginzburg-Landau problems which are characterized by energy density concentrations on a one-dimensional set. In this paper, we investigate the states of vanishing energy. We classify these zero-energy states in the whole space: They are either constant or a vortex. A bounded domain can sustain a zero-energy state only if the domain is a disk and the state a vortex. Our proof is based on specific entropies which lead to a kinetic formulation, and on a careful analysis of the corresponding weak solutions by the method of characteristics.
@article{ASNSP_2002_5_1_1_187_0,
author = {Jabin, Pierre-Emmanuel and Otto, Felix and Perthame, Beno\^It},
title = {Line-energy {Ginzburg-Landau} models : zero-energy states},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {187--202},
publisher = {Scuola normale superiore},
volume = {Ser. 5, 1},
number = {1},
year = {2002},
mrnumber = {1994807},
zbl = {1072.35051},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2002_5_1_1_187_0/}
}
TY - JOUR AU - Jabin, Pierre-Emmanuel AU - Otto, Felix AU - Perthame, BenoÎt TI - Line-energy Ginzburg-Landau models : zero-energy states JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2002 SP - 187 EP - 202 VL - 1 IS - 1 PB - Scuola normale superiore UR - http://www.numdam.org/item/ASNSP_2002_5_1_1_187_0/ LA - en ID - ASNSP_2002_5_1_1_187_0 ER -
%0 Journal Article %A Jabin, Pierre-Emmanuel %A Otto, Felix %A Perthame, BenoÎt %T Line-energy Ginzburg-Landau models : zero-energy states %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2002 %P 187-202 %V 1 %N 1 %I Scuola normale superiore %U http://www.numdam.org/item/ASNSP_2002_5_1_1_187_0/ %G en %F ASNSP_2002_5_1_1_187_0
Jabin, Pierre-Emmanuel; Otto, Felix; Perthame, BenoÎt. Line-energy Ginzburg-Landau models : zero-energy states. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 1, pp. 187-202. http://www.numdam.org/item/ASNSP_2002_5_1_1_187_0/
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