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Prescribing nearly constant curvatures on balls

Part of: Elliptic equations and systems Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  27 October 2023

Luca Battaglia Open the ORCID record for Luca Battaglia [Opens in a new window] ,
Sergio Cruz-Blázquez  and
Angela Pistoia
Luca Battaglia
Affiliation:
Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy (luca.battaglia@uniroma3.it)
Sergio Cruz-Blázquez
Affiliation:
Departamento de Análisis Matemático, Universidad de Granada, Avenida de Fuente Nueva s/n, 18071 Granada, Spain (sergiocruz@ugr.es)
Angela Pistoia
Affiliation:
Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via Antonio Scarpa 10, 00161 Roma, Italy (angela.pistoia@uniroma1.it)
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Abstract

In this paper, we address two boundary cases of the classical Kazdan–Warner problem. More precisely, we consider the problem of prescribing the Gaussian and boundary geodesic curvature on a disk of $\mathbb {R}^2$, and the scalar and mean curvature on a ball in higher dimensions, via a conformal change of the metric. We deal with the case of negative interior curvature and positive boundary curvature. Using a Ljapunov–Schmidt procedure, we obtain new existence results when the prescribed functions are close to constants.

Keywords

prescribed curvatureconformal metricsLjapunov–Schmidt construction

MSC classification

Primary: 35J25: Boundary value problems for second-order elliptic equations
Secondary: 58J32: Boundary value problems on manifolds
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 36
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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