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On the sharp constant in the Bianchi–Egnell stability inequality: On the sharp constant in the Bianchi-Egnell stability inequality

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Title: On the sharp constant in the Bianchi–Egnell stability inequality: On the sharp constant in the Bianchi-Egnell stability inequality
Authors: König, Tobias
Source: Bulletin of the London Mathematical Society. 55:2070-2075
Publication Status: Preprint
Publisher Information: Wiley, 2023.
Publication Year: 2023
Subject Terms: Mathematics - Analysis of PDEs, Bianchi-Egnell inequality, Sobolev inequality, FOS: Mathematics, Inequalities involving derivatives and differential and integral operators, 0101 mathematics, Sobolev spaces and other spaces of ''smooth' functions, embedding theorems, trace theorems, 01 natural sciences, fractional exponents, Analysis of PDEs (math.AP)
Description: This note is concerned with the Bianchi–Egnell inequality, which quantifies the stability of the Sobolev inequality, and its generalization to fractional exponents . We prove that in dimension the best constant is strictly smaller than the spectral gap constant associated to sequences that converge to the manifold of Sobolev optimizers. In particular, cannot be asymptotically attained by such sequences. Our proof relies on a precise expansion of the Bianchi–Egnell quotient along a well‐chosen sequence of test functions converging to .
Document Type: Article
File Description: application/xml
Language: English
ISSN: 1469-2120
0024-6093
DOI: 10.1112/blms.12837
DOI: 10.48550/arxiv.2210.08482
Access URL: http://arxiv.org/abs/2210.08482
Rights: CC BY NC
arXiv Non-Exclusive Distribution
Accession Number: edsair.doi.dedup.....546b3d0e5cda692045d6c4beb47f92cb
Database: OpenAIRE
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  Data: <i>Bulletin of the London Mathematical Society</i>. 55:2070-2075
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  Data: <searchLink fieldCode="DE" term="%22Mathematics+-+Analysis+of+PDEs%22">Mathematics - Analysis of PDEs</searchLink><br /><searchLink fieldCode="DE" term="%22Bianchi-Egnell+inequality%22">Bianchi-Egnell inequality</searchLink><br /><searchLink fieldCode="DE" term="%22Sobolev+inequality%22">Sobolev inequality</searchLink><br /><searchLink fieldCode="DE" term="%22FOS%3A+Mathematics%22">FOS: Mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Inequalities+involving+derivatives+and+differential+and+integral+operators%22">Inequalities involving derivatives and differential and integral operators</searchLink><br /><searchLink fieldCode="DE" term="%220101+mathematics%22">0101 mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Sobolev+spaces+and+other+spaces+of+''smooth'+functions%2C+embedding+theorems%2C+trace+theorems%22">Sobolev spaces and other spaces of ''smooth' functions, embedding theorems, trace theorems</searchLink><br /><searchLink fieldCode="DE" term="%2201+natural+sciences%22">01 natural sciences</searchLink><br /><searchLink fieldCode="DE" term="%22fractional+exponents%22">fractional exponents</searchLink><br /><searchLink fieldCode="DE" term="%22Analysis+of+PDEs+%28math%2EAP%29%22">Analysis of PDEs (math.AP)</searchLink>
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  Data: This note is concerned with the Bianchi–Egnell inequality, which quantifies the stability of the Sobolev inequality, and its generalization to fractional exponents . We prove that in dimension the best constant is strictly smaller than the spectral gap constant associated to sequences that converge to the manifold of Sobolev optimizers. In particular, cannot be asymptotically attained by such sequences. Our proof relies on a precise expansion of the Bianchi–Egnell quotient along a well‐chosen sequence of test functions converging to .
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      – SubjectFull: 01 natural sciences
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