Vol. 8, No. 7, 2015

Download this article
Download this article For screen
For printing
Recent Issues

Volume 10
Issue 2, 253–512
Issue 1, 1–252

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
Cover
About the Cover
Editorial Board
Editors’ Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Subscriptions
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Blow-up analysis of a nonlocal Liouville-type equation

Francesca Da Lio, Luca Martinazzi and Tristan Rivière

Vol. 8 (2015), No. 7, 1757–1805
Abstract

We establish an equivalence between the Nirenberg problem on the circle and the boundary of holomorphic immersions of the disk into the plane. More precisely we study the nonlocal Liouville-type equation

(Δ)1 2 u = κeu 1 in S1, (1)

where (Δ)1 2 stands for the fractional Laplacian and κ is a bounded function. The equation (1) can actually be interpreted as the prescribed curvature equation for a curve in conformal parametrization. Thanks to this geometric interpretation we perform a subtle blow-up and quantization analysis of (1). We also show a relation between (1) and the analogous equation in ,

(Δ)1 2 u = Keu in  (2)

with K bounded on .

Keywords
nonlocal Liouville equation, Nirenberg problem, fractional harmonic maps, blow-up analysis of solutions, regularity of solutions, conformal variational problems, quasiconformal mappings in the plane
Mathematical Subject Classification 2010
Primary: 30C20, 35B65, 58E20, 35B44, 35R11
Secondary: 30C62
Milestones
Received: 30 April 2015
Revised: 2 July 2015
Accepted: 29 July 2015
Published: 18 September 2015
Authors
Francesca Da Lio
Departement Mathematik
ETH Zürich
CH-8092 Zürich
Switzerland
Luca Martinazzi
Departement Mathematik und Informatik
Universität Basel
CH-4051 Basel
Switzerland
Tristan Rivière
Departement Mathematik
ETH Zürich
CH-8092 Zürich
Switzerland