{"paper":{"title":"Compactness and Bubbles Analysis for 1/2-harmonic Maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Francesca Da Lio","submitted_at":"2012-10-09T16:09:32Z","abstract_excerpt":"In this paper we study compactness and quantization properties of sequences of 1/2-harmonic maps $u_k\\colon\\R\\to {\\cal{S}}^{m-1}$ such that $|u_k|_{\\dot H^{1/2}(\\R,{\\cal{S}}^{m-1})}\\le C.$ More precisely we show that there exist a weak 1/2-harmonic map $u_\\infty\\colon\\R\\to {\\cal{S}}^{m-1}$, a possible empty set ${a_1,...,a_\\ell}$ in $\\R$ such that up to subsequences $$(|(-\\Delta)^{1/4}u_k|^2 \\rightharpoonup |(-\\Delta)^{1/4}u_{\\infty}|^2)dx+\\sum_{i=1}^{\\ell}\\lambda_i \\delta_{a_i}, in Radon measure,$$ as $k\\to +\\infty$, with $\\lambda_i\\ge 0.$\n  The convergence of $u_k$ to $u_\\infty$ is strong in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.2653","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}