Blow-up analysis of a nonlocal Liouville-type equation

Da Lio, Francesca; Martinazzi, Luca; Rivière, Tristan

doi:10.2140/apde.2015.8.1757

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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

DOI: 10.2140/apde.2015.8.1757

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)}^{\frac{1}{2}} u = κ e^{u} - 1 in S^{1},

where ${(- Δ)}^{\frac{1}{2}}$ 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 $ℝ$ ,

{(- Δ)}^{\frac{1}{2}} u = K e^{u} in ℝ

with $K$ bounded on $ℝ$ .

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

Vol. 8, No. 7, 2015