Multipolar Hardy inequalities on Riemannian manifolds

Dedicated to Professor Enrique Zuazua on the occasion of his 55th birthday

Francesca Faraci, Csaba Farkas, A. Kristály

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

We prove multipolar Hardy inequalities on complete Riemannian manifolds, providing various curved counterparts of some Euclidean multipolar inequalities due to Cazacu and Zuazua [Improved multipolar Hardy inequalities, 2013]. We notice that our inequalities deeply depend on the curvature, providing (quantitative) information about the deflection from the flat case. By using these inequalities together with variational methods and group-theoretical arguments, we also establish non-existence, existence and multiplicity results for certain Schrödinger-type problems involving the Laplace-Beltrami operator and bipolar potentials on Cartan-Hadamard manifolds and on the open upper hemisphere, respectively.

Original languageEnglish
Pages (from-to)551-567
Number of pages17
JournalESAIM - Control, Optimisation and Calculus of Variations
Volume24
Issue number2
DOIs
Publication statusPublished - Apr 1 2018

Fingerprint

Hardy Inequality
Riemannian Manifold
Hadamard Manifolds
Laplace-Beltrami Operator
Hemisphere
Multiplicity Results
Variational Methods
Deflection
Nonexistence
Existence Results
Euclidean
Curvature

Keywords

  • Hardy inequality
  • Multipolar
  • Riemannian manifolds

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Control and Optimization
  • Computational Mathematics

Cite this

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abstract = "We prove multipolar Hardy inequalities on complete Riemannian manifolds, providing various curved counterparts of some Euclidean multipolar inequalities due to Cazacu and Zuazua [Improved multipolar Hardy inequalities, 2013]. We notice that our inequalities deeply depend on the curvature, providing (quantitative) information about the deflection from the flat case. By using these inequalities together with variational methods and group-theoretical arguments, we also establish non-existence, existence and multiplicity results for certain Schr{\"o}dinger-type problems involving the Laplace-Beltrami operator and bipolar potentials on Cartan-Hadamard manifolds and on the open upper hemisphere, respectively.",
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AU - Kristály, A.

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AB - We prove multipolar Hardy inequalities on complete Riemannian manifolds, providing various curved counterparts of some Euclidean multipolar inequalities due to Cazacu and Zuazua [Improved multipolar Hardy inequalities, 2013]. We notice that our inequalities deeply depend on the curvature, providing (quantitative) information about the deflection from the flat case. By using these inequalities together with variational methods and group-theoretical arguments, we also establish non-existence, existence and multiplicity results for certain Schrödinger-type problems involving the Laplace-Beltrami operator and bipolar potentials on Cartan-Hadamard manifolds and on the open upper hemisphere, respectively.

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