2012/10/09 by Francesca Da Lio, Da Lio, Francesca · 1 citation
Mathematics · #35B65 #35J20 #35J60 #35S99 #58E20 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP #msc:35B65 #msc:35J20 #msc:35J60 #msc:35S99 #msc:58E20
paper · pdf · doi:10.48550/arxiv.1210.2653
31 pages
arxiv created 2012/10/09 · openalex publication_date 2012/10/09 · arxiv updated 2012/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study compactness and quantization properties of sequences of 1/2-harmonic maps uk\colon\R→ \calSm-1 such that |uk|_ H1/2(\R,\calSm-1)≤ C. More precisely we show that there exist a weak 1/2-harmonic map u_∞\colon\R→ \calSm-1, a possible empty set a1,...,a_ℓ in \R such that up to subsequences (|(-Δ)1/4uk|2 \rightharpoonup |(-Δ)1/4u∞|2)dx+∑i=1ℓλi δai, in Radon measure, as k→ +∞, with λi≥ 0. The convergence of uk to u_∞ is strong in W1/2,ploc(\R∖a1,...,a_ℓ), for every p≥ 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 \calS2 one has λi=2 πni, with ni a positive integer.