{"paper":{"title":"The nonlocal Liouville-type equation in $\\mathbb{R}$ and conformal immersions of the disk with boundary singularities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CV"],"primary_cat":"math.DG","authors_text":"Francesca Da Lio, Luca Martinazzi","submitted_at":"2016-07-12T21:42:30Z","abstract_excerpt":"In this paper we perform a blow-up and quantization analysis of the fractional Liouville equation in dimension $1$. More precisely, given a sequence $u_k :\\mathbb{R} \\to \\mathbb{R}$ of solutions to \\begin{equation}\n  (-\\Delta)^\\frac{1}{2} u_k =K_ke^{u_k}\\quad \\text{in }\\mathbb{R}, \\end{equation} with $K_k$ bounded in $L^\\infty$ and $e^{u_k}$ bounded in $L^1$ uniformly with respect to $k$, we show that up to extracting a subsequence $u_k$ can blow-up at (at most) finitely many points $B=\\{a_1,\\dots, a_N\\}$ and either (i) $u_k\\to u_\\infty$ in $W^{1,p}_{loc}(\\mathbb{R}\\setminus B)$ and $K_ke^{u_k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.03525","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"}