Minimizing properties of arbitrary solutions to the Ginzburg-Landau equation Myriam COMTE and Petru MIRONESCU Abstract We consider the Ginzburg-Landau type equation −Δu ε = 1 ε 2 u ε (1 −|u ε | 2 ) in G, u ε = g on ∂G where G is a smooth bounded domain in 2 , g ∈ C ∞ (∂G ; 2 \{0}), and ε> 0 is a small parameter. We prove the uniqueness of solutions to this equation under some non-vanishing assumptions on u ε , or under conditions on the boundary function g . 1. Introduction We consider the Ginzburg-Landau type equation (1)    − Δu ε = 1 ε 2 u ε (1 −|u ε | 2 ) in G u ε = g on ∂G, where G is a smooth bounded domain in 2 , g ∈ C ∞ (∂G ; 2 \{0}), and ε> 0 is a small parameter. This problem has an associated (Ginzburg-Landau) energy (2) E ε (u)= E ε (u, G)= 1 2  G |∇u| 2 + 1 4ε 2  G (1 −|u| 2 ) 2 and the natural class of testing functions is (3) H 1 g (G, ) := { u ∈ H 1 (G ; ); u = g on ∂G } (we identify 2 and ). In general, (1) admits non-minimizing solutions. For example, if G = B 1 = the unit disc in 2 , g (z)= e 2iθ =  z |z|  2 , there is a (unique) radial solution to (1), that is a solution of the form (4) u ε (z)= f ε (|z|)e 2iθ 1