Compactness and bubble analysis for 1/2-harmonic maps

In this paper we study compactness and quantization properties of sequences of 1/2-harmonic maps uk:R→Sm−1uk:R→Sm−1 such that ‖uk‖H˙1/2(R,Sm−1)⩽C. More precisely we show that there exist a weak 1/2-harmonic map u∞:R→Sm−1u∞:R→Sm−1, a finite and possible empty set {a1,…,aℓ}⊂R{a1,…,aℓ}⊂R such that up to subsequences|(−Δ)1/4uk|2dx⇀|(−Δ)1/4u∞|2dx+∑i=1ℓλiδai,in Radon measure, as k→+∞k→+∞, with λi⩾0λi⩾0.The convergence of ukuk to u∞u∞ is strong in W˙loc1/2,p(R∖{a1,…,aℓ}), for every p⩾1p⩾1. We quantify the loss of energy in the weak convergence and we show that in the case of non-constant 1/2-harmonic maps with values in S1S1 one has λi=2πniλi=2πni, with nini a positive integer.