Local minimizers of the Ginzburg–Landau functional with prescribed degrees

• Published in: Journal of functional analysis Vol. 257; no. 4; pp. 1053 - 1091
• Main Author: Dos Santos, Mickaël
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.08.2009; Elsevier
• Subjects: Analysis of PDEs; Ginzburg–Landau functional; Local minimizers; Mathematics; Prescribed degrees; Ginzburg–Landau functional; Prescribed degrees; Local minimizers; Ginzburg-Landau functional; local minimizers; prescribed degrees
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2009.02.023

We consider, in a smooth bounded multiply connected domain D ⊂ R 2 , the Ginzburg–Landau energy E ε ( u ) = 1 2 ∫ D | ∇ u | 2 + 1 4 ε 2 ∫ D ( 1 − | u | 2 ) 2 subject to prescribed degree conditions on each component of ∂ D . In general, minimal energy maps do not exist [L. Berlyand, P. Mironescu, Ginzburg–Landau minimizers in perforated domains with prescribed degrees, preprint, 2004]. When D has a single hole, Berlyand and Rybalko [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html] proved that for small ε local minimizers do exist. We extend the result in [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html]: E ε ( u ) has, in domains D with 2 , 3 , … holes and for small ε, local minimizers. Our approach is very similar to the one in [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html]; the main difference stems in the construction of test functions with energy control.