Blow-up analysis of a nonlocal Liouville-type equation

By Francesca Da Lio, L. Martinazzi, T. Rivière. 2015

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In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-\Delta)^\frac12 u= \kappa e^u-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 =Ke^u \quad \text{in }\mathbb{R}, \end{equation} with $K$ bounded on $\mathbb{R}$.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1c249e36dc1a447eab7a2b41ae5e146ae1493894