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

Aleks Jevnikar, Rafael López-Soriano, María Medina and David Ruiz

Vol. 15 (2022), No. 8, 1897–1931

Here we are concerned with the compactness of metrics on 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.

prescribed curvature problem, conformal metric, blow-up analysis, Pohozaev-type identity
Mathematical Subject Classification
Primary: 35B44, 35J20, 58J32
Received: 30 April 2020
Accepted: 4 May 2021
Published: 10 February 2023
Aleks Jevnikar
Department of Mathematics, Computer Science and Physics
University of Udine
Rafael López-Soriano
Universidad de Granada
Campus de Fuentenueva
María Medina
Departamento de Matemáticas
Universidad Autónoma de Madrid
Ciudad Universitaria de Cantoblanco
David Ruiz
Universidad de Granada
Campus de Fuentenueva