Local Minimizers of the Ginzburg-landau Functional with Prescribed Degrees

We consider, in a smooth bounded multiply connected domain D ⊂ R, the Ginzburg-Landau energy Eε(u) = 1 2 ∫ D { |∇u|2 + 1 2ε2 (1− |u|2)2 } subject to prescribed degree conditions on each component of ∂D. In general, minimal energy maps do not exist [4]. When D has a single hole, Berlyand and Rybalko [5] proved that for small ε local minimizers do exist. We extend the result in [5]: Eε(u) has, in domains D with 2, 3, ... holes and for small ε, local minimizers. Our approach is very similar to the one in [5]; the main difference stems in the construction of test functions with energy control.