278 NAW 5/1 nr. 3 september 2000 Minimization of the renormalized energy in the unit ball of R 2 L. Ignat, C. Lefter, V.D. Radulescu L. Ignat Department of Mathematics, University of Craiova 1100 Craiova, Romania C. Lefter Department of Mathematics, University of Iasi 6600 Iasi, Romania lefter@uaic.ro V.D. Radulescu Department of Mathematics, University of Craiova 1100 Craiova, Romania radules@ann.jussieu.fr (corresponding author) Minimization of the renormalized energy in the unit ball of R 2 We establish an explicit formula for the renormalized energy cor- responding to the Ginzburg-Landau functional. Then we find the location of vortices in the case of the unit ball in R 2 , provided that the topological Brouwer degree of the boundary data equals to 2 or 3. Our proofs use techniques related to linear partial differential equa- tions (Green’s formula for the Neumann problem), convex functions, elementary identities or inequalities in the complex plane. Superconductivity was discovered in 1911 by the Dutch physicist Kamerlingh-Onnes. Superconducting materials exhibit two main properties: i. Their electric resistance is virtually zero. ii. They have peculiar magnetic behavior. From this point of view, superconductors can be classified into two types. In type I, magnetic fields are excluded from the mate- rial (except for a very thin layer near the surface). Type II super- conductors, on the other hand, do allow penetration of magnetic fields, but these fields concentrate in narrow regions or points, called vortices. In fact, type II superconductors can sustain very high magnetic fields. The first successful theory for superconductivity was the phe- nomenological macroscopic model proposed in 1935 by London. His theory accounted for the expulsion of magnetic fields and predicted the quantization of magnetic fluxoids. Then, in 1950 Ginzburg and Landau [3] proposed a more involved theory which allowed for spatial variations of both the magnetic field and the superconductivity order parameter. In addition to the model’s success in explaining the experimental observations of the day, it was by Abrikosov in 1957 to predict in [1] the existence of type II superconductors, and the formation of large array of magnetic vortices for such materials. In 1994, Bethuel, Brezis and Hélein proposed a mathematical model of the Ginzburg-Landau theory which relates the number of vortices to a topological invariant of the boundary condition. A fundamental role in their analysis is played by the notion of renormalized energy. We give in what follows a partial answer to a problem raised by Bethuel, Brezis and Hélein in [2]. Let B 1 = {x =(x 1 , x 2 ) ∈ R 2 ; x 2 1 + x 2 2 = | x| 2 < 1}. Fix d a positive integer and consider a configuration a =(a 1 ,..., a d ) of distinct points in B 1 . Let ρ > 0 be sufficiently small such that the balls B(a i , ρ) are mutually disjoint and contained in B 1 and set Ω ρ = B 1 \  d i=1 B(a i , ρ). Consider the boundary data g : S 1 → S 1 defined by g( θ)= e idθ . We observe that the Brouwer degree deg ( g, S 1 ) is equal to d. We recall that if G ⊂ R 2 is a smooth, bounded and simply connected domain and h =(h 1 , h 2 ) ∈ C 1 (∂G, S 1 ) then the topological Brouwer de- gree (i.e., the winding number of h considered as a map from ∂G into S 1 ) is defined by deg (h, ∂G)= 1 2π  ∂G  h 1 ∂h 2 ∂τ − h 2 ∂h 1 ∂τ  , where τ denotes the unit tangent vector to ∂G. In [2], F. Bethuel, H. Brezis and F. Hélein have studied the be- havior as ρ → 0 of solutions of the minimization problem (1) E ρ, g = min v∈E ρ,g  Ω ρ |∇v | 2 , where E ρ, g = {v ∈ H 1 (Ω ρ ; S 1 ); v = g on ∂G and deg(v, ∂B(a i , ρ)) = +1, for i = 1, ..., d} . We have denoted by H 1 (Ω ρ ; S 1 ) the space of all measurable func- tions u : Ω ρ → R 2 such that u ∈ H 1 (Ω ρ ) and |u| = 1 for a.e. x ∈ Ω ρ . We also point out that all the derivatives appearing in this paper are taken in distributional sense. It is proved in [2] that problem (1) has a unique solution, say u ρ . By analyzing the behavior of u ρ as ρ → 0 the following asymptotic estimate is obtained as well: (2) 1 2  Ω ρ |∇u ρ | 2 = π d log 1 ρ + W(a)+ O(ρ) , as ρ → 0.