The nonlocal Liouville-type equation in R and conformal immersions of the disk

Da Lio, Francesca and Martinazzi, Luca. (2017) The nonlocal Liouville-type equation in R and conformal immersions of the disk. Calculus of Variations and Partial Differential Equations, 56 (5). p. 152.

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In this paper we perform a blow-up and quantization analysis of the fractional Liouville equation in dimension 1. More precisely, given a sequence uk:R→R of solutions to (−Δ)12uk=KkeukinR,(1) with Kk bounded in L∞ and euk bounded in L1 uniformly with respect to k, we show that up to extracting a subsequence uk can blow-up at (at most) finitely many points B={a1,…,aN} and that either (i) uk→u∞ in W1,ploc(R∖B) and Kkeuk⇀∗K∞eu∞+∑Nj=1πδaj, or (ii) uk→−∞ uniformly locally in R∖B and Kkeuk⇀∗∑Nj=1αjδaj with αj≥π for every j. This result, resting on the geometric interpretation and analysis of (1) provided in a recent collaboration of the authors with T. Rivière and on a classical work of Blank about immersions of the disk into the plane, is a fractional counterpart of the celebrated works of Brézis–Merle and Li–Shafrir on the 2-dimensional Liouville equation, but providing sharp quantization estimates (αj=π and αj≥π) which are not known in dimension 2 under the weak assumption that (Kk) be bounded in L∞ and is allowed to change sign.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Analysis (Martinazzi)
UniBasel Contributors:Martinazzi, Luca
Item Type:Article, refereed
Article Subtype:Research Article
Note:Publication type according to Uni Basel Research Database: Journal article
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Last Modified:09 Sep 2020 08:33
Deposited On:09 Sep 2020 08:25

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