Abrikosov Lattices in Superconductivity

Abrikosov Lattices in Superconductivity ˘dulescu1 (Center of Nonlinear Analysis and Applications, Problem 06-006, by Vicent ¸ iu Ra University of Craiova, Craiova, Romania). Let B1 = {x = (x1 , x2 ) ∈ R2 ; x21 + x22 = |x|2 < 1}. Fix d a positive integer, and consider a configuration S = (a1 , . . . , ad ) of distinct points in B1 . (i) Solve the linear Neumann problem ΔΦ = 2π

d 

δai

in B1 ,

i=1

∂Φ = d on S 1 , ∂ν 

S1

Φ = 0,

where ν is the outward formal to S 1 and δb is the Dirac mass concentrated at b ∈ B1 . (ii) Define d   d log |ai − aj | + Φ dσ − π R(ai ), W (S) = −π 2 S1 i=1 i=j where R(x) = Φ(x) −

d 

log |x − ai |.

i=1

Prove that W (S) = −π



log | ai − aj |2 −π

1≤i