Sobolev maps with integer degree and applications to Skyrme’s problem

Abstract

Let ɸ : R 3 → S 3 ⊂ R 4 , ∣ A ( ɸ )∣ 2 ═ Ʃ 3 α,β═1 │∂ ɸ /∂ x α ∧ ∂ ɸ /∂ x β ∣ 2 and let k ϵ Z . Skyrme's problem consists in minimizing the energy ε( ɸ ) : ═ ∫ R 3 ∣∇ ɸ ∣ 2 + ∣ A ( ɸ )∣ 2 d x among maps with degree k ═ d ( ɸ ) : ═ 1/2π 2 ∫ R 3 det ( ɸ , ∇ ɸ ) d x . We show that for all ɸ with finite energy d ( ɸ ) is an integer and then obtain existence of a minimizer of ε in the natural class of maps with finite energy.