{"paper":{"title":"Blow-up analysis of a nonlocal Liouville-type equation","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, Tristan Rivi\\`ere","submitted_at":"2015-03-30T15:08:42Z","abstract_excerpt":"In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \\begin{equation}(-\\Delta)^\\frac12 u= \\kappa e^u-1~\\mbox{in $S^1$,} \\end{equation} where $(-\\Delta)^\\frac{1}{2}$ stands for the fractional Laplacian and $\\kappa$ is a bounded function. We interpret the above equation as the prescribed curvature equation to a curve in conformal parametrization. We also establish a relation between this equation and the analogous equation in $\\mathbb{R}$ \\begin{equation}\n  (-\\Delta)^\\frac{1}{2} u =Ke^u \\quad \\text{in }\\mathbb{R}, \\end{equation} with $K$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.08701","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"}