We consider the problem of prescribing the Gaussian and the geodesic curvatures of a compact surface with boundary by a confor- mal deformation of the metric. We derive some existence results using a variational approach, either by minimization of the Euler-Lagrange en- ergy or via min-max methods. One of the main tools in our approach is a blow-up analysis of solutions, which in the present setting can have di- verging volume. To our knowledge, this is the first time in which such an aspect is treated. Key ingredients in our arguments are: a blow-up anal- ysis around a sequence of points different from local maxima; the use of holomorphic domain-variations; and Morse-index estimates.

Conformal metrics with prescribed gaussian and geodesic curvatures

Malchiodi, Andrea;
2022

Abstract

We consider the problem of prescribing the Gaussian and the geodesic curvatures of a compact surface with boundary by a confor- mal deformation of the metric. We derive some existence results using a variational approach, either by minimization of the Euler-Lagrange en- ergy or via min-max methods. One of the main tools in our approach is a blow-up analysis of solutions, which in the present setting can have di- verging volume. To our knowledge, this is the first time in which such an aspect is treated. Key ingredients in our arguments are: a blow-up anal- ysis around a sequence of points different from local maxima; the use of holomorphic domain-variations; and Morse-index estimates.
2022
Settore MAT/05 - Analisi Matematica
Constant mean-curvature; blow-up analysis; palais-smale sequences; scalar curvature; yamabe problem; delta-u; boundary; existence; compactness; surfaces; problème de courbure prescrite; métrique conforme; méthodes variationnelles; analyse de explosion
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/76385
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