Concentration phenomenon in the critical exponent problems on hyperbolic space
TL;DR: In this article, the authors studied the critical exponent problem (Pλ,μ)−ΔHnu+(μg(x)−λ)u = up−1 ǫ in Hn,u>0ǫ ∈ H1(Hn) where Hn is the n-dimensional hyperbolic space, n≥4, p=2n/(n−2) is...
Abstract: This article deals with the study of the following critical exponent problem (Pλ,μ)−ΔHnu+(μg(x)−λ)u=up−1 in Hn,u>0 in Hn,u∈H1(Hn) where Hn is the n-dimensional hyperbolic space, n≥4, p=2n/(n−2) is ...
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26 Nov 2022
TL;DR: In this article , the authors propose a solution to solve the problem of the problem: this article ] of "uniformity" and "uncertainty" of the solution.
Abstract: ,
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TL;DR: In this article, the existence of a fonction u satisfaisant l'equation elliptique non lineaire is investigated, i.e., a domaine borne in R n avec n ≥ 3.
Abstract: Soit Ω un domaine borne dans R n avec n≥3 On etudie l'existence d'une fonction u satisfaisant l'equation elliptique non lineaire -Δu=u P +f(x,u) sur Ω, u>0 sur Ω, u=0 sur ∂Ω, ou p=(n+2)/(n−2), f(x,0)=0 et f(x,u) est une perturbation de u P de bas ordre au sens ou lim u→+α f(x,u)/u P =0
2,676 citations
Book•
01 Jan 1984
TL;DR: The Dirichlet Heat Kernel for Regular Domains as mentioned in this paper is a heat kernel for non-compact manifolds that is based on the Laplacian on forms (LFP).
Abstract: Preface. The Laplacian. The Basic Examples. Curvature. Isoperimetric Inequalities. Eigenvalues and Kinematic Measure. The Heat Kernel for Compact Manifolds. The Dirichlet Heat Kernel for Regular Domains. The Heat Kernel for Noncompact Manifolds. Topological Perturbations with Negligible Effect. Surfaces of Constant Negative Curvature. The Selberg Trace Formula. Miscellanea. Laplacian on Forms. Bibliography. Index.
2,059 citations
Additional excerpts
...λ1(Q) = inf 0 ≡u∈H1 0(Q) ∫ Q |∇Hnu|(2) dvHn ∫ Q |u|2 dvHn where H1 0(Q) = the completion of {u ∈ C∞ c (Q) : ‖u‖0 < +∞} under ‖ · ‖0 (refer [18])....
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TL;DR: The best constant for the simplest Sobolev inequality was proved in this paper by symmetrizations (rearrangements in the sense of Hardy-Littlewood) and one-dimensional calculus of variations.
Abstract: The best constant for the simplest Sobolev inequality is exhibited. The proof is accomplished by symmetrizations (rearrangements in the sense of Hardy-Littlewood) and one-dimensional calculus of variations.
2,011 citations
Book•
01 Jan 1994
TL;DR: In this paper, an exposition of the theoretical foundations of hyperbolic manifolds is presented, which is intended to be used both as a textbook and as a reference for algebra and topology courses.
Abstract: This book is an exposition of the theoretical foundations of hyperbolic manifolds. It is intended to be used both as a textbook and as a reference. The reader is assumed to have a basic knowledge of algebra and topology at the first year graduate level of an American university. The book is divided into three parts. The first part, Chapters 1-7, is concerned with hyperbolic geometry and discrete groups. The second part, Chapters 8-12, is devoted to the theory of hyperbolic manifolds. The third part, Chapter 13, integrates the first two parts in a development of the theory of hyperbolic orbifolds. There are over 500 exercises in this book and more than 180 illustrations.
1,527 citations
Book•
02 Oct 1996
TL;DR: Sobolev spaces in the presence of symmetries were studied in this paper, where Sobolev embeddings and best constants problems were also considered in the context of symmetric spaces.
Abstract: Geometric preliminaries.- Sobolev spaces.- Sobolev embeddings.- The best constants problems.- Sobolev spaces in the presence of symmetries.
495 citations