Vol. 8, No. 7, 2015

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Blow-up analysis of a nonlocal Liouville-type equation

Francesca Da Lio, Luca Martinazzi and Tristan Rivière

Vol. 8 (2015), No. 7, 1757–1805

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 .

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
Received: 30 April 2015
Revised: 2 July 2015
Accepted: 29 July 2015
Published: 18 September 2015
Francesca Da Lio
Departement Mathematik
ETH Zürich
CH-8092 Zürich
Luca Martinazzi
Departement Mathematik und Informatik
Universität Basel
CH-4051 Basel
Tristan Rivière
Departement Mathematik
ETH Zürich
CH-8092 Zürich