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Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries
Author(s) -
Li Yanyan,
Zhu Meijun
Publication year - 1997
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199705)50:5<449::aid-cpa2>3.0.co;2-9
Subject(s) - mathematics , trace (psycholinguistics) , riemannian manifold , boundary (topology) , sobolev space , manifold (fluid mechanics) , pure mathematics , sobolev inequality , constant (computer programming) , inequality , mathematical analysis , mechanical engineering , computer science , engineering , programming language , philosophy , linguistics
In this paper, we establish some sharp Sobolev trace inequalities on n ‐dimensional, compact Riemannian manifolds with smooth boundaries. More specifically, let q = 2( n ‐ 1)/( n ‐ 2), 1/ S = inf {∫ R + n|∇ u | 2 : ∇ u ∈ L 2 (R + n ), ∫ dR + n| u | q = 1}. We establish for any Riemannian manifold with a smooth boundary, denoted as ( M, g ), that there exists some constant A = A ( M, g ) > 0, (∫ dM | u | q ds g ) 2/ q < or = to S ∫ M |∇ g u | 2 dv g + A ∫ dM u 2 ds g , for all u ∈ H 1 ( M ). The inequality is sharp in the sense that the inequality is false when S is replaced by any smaller number. © 1997 John Wiley & Sons, Inc.