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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 |
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Vol. 15 (2022), No. 8, 1897–1931
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Abstract |
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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. |
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Keywords
prescribed curvature problem, conformal metric, blow-up
analysis, Pohozaev-type identity
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Mathematical Subject Classification
Primary: 35B44, 35J20, 58J32
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Milestones
Received: 30 April 2020
Accepted: 4 May 2021
Published: 10 February 2023
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Authors |
| Aleks Jevnikar | |
| Department of Mathematics, Computer
Science and Physics University of Udine Udine Italy |
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| Rafael López-Soriano | |
| Universidad de Granada Campus de Fuentenueva Granada Spain |
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| María Medina | |
| Departamento de Matemáticas Universidad Autónoma de Madrid Ciudad Universitaria de Cantoblanco Madrid Spain |
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| David Ruiz | |
| Universidad de Granada Campus de Fuentenueva Granada Spain |
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