Calculus of Variations and Geometric Measure Theory

A. Jevnikar - R. Lopez-Soriano - M. Medina - D. Ruiz

Blow-up analysis of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures

created by jevnikar on 01 May 2020
modified on 05 May 2021


Accepted Paper

Inserted: 1 may 2020
Last Updated: 5 may 2021

Journal: Anal. PDE
Year: 2021


This paper is concerned with the compactness of metrics of the disk with prescribed Gaussian and geodesic curvatures. We consider a blowing-up sequence of metrics and give a precise description of its asymptotic behavior. In particular, the metrics blow-up at a unique point on the boundary and we are able to give necessary conditions on its location. It turns out that such conditions depend locally on the Gaussian curvatures but they depend on the geodesic curvatures in a nonlocal way. This is a novelty with respect to the classical Nirenberg problem where the blow-up conditions are local, and this new aspect is driven by the boundary condition.

Keywords: blow-up analysis, Prescribed curvature problem, conformal metric, Pohozaev-type identity