#### 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
##### 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

 (1)

where ${\left(-\Delta \right)}^{\frac{1}{2}}$ stands for the fractional Laplacian and $\kappa$ 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 $ℝ$,

 (2)

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