Calculus of Variations and Geometric Measure Theory
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F. Da Lio - L. Martinazzi - T. Rivière

Blow-up analysis of a nonlocal Liouville-type equation

created by paolini on 17 Jul 2018

[BibTeX]

preprint

Inserted: 17 jul 2018

Year: 2015

ArXiv: 1503.08701 PDF

Abstract:

In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-\Delta)\frac12 u= \kappa eu-1~\mbox{in $S^1$,} \end{equation} where $(-\Delta)^\frac{1}{2}$ stands for the fractional Laplacian and $\kappa$ is a bounded function. We interpret the above equation as the prescribed curvature equation to a curve in conformal parametrization. We also establish a relation between this equation and the analogous equation in $\mathbb{R}$ \begin{equation} (-\Delta)\frac{1}{2} u =Keu \quad \text{in }\mathbb{R}, \end{equation} with $K$ bounded on $\mathbb{R}$.

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