Estimation of Sobolev embedding constant on a domain dividable into bounded convex domains

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5 Citations (Scopus)

Abstract

This paper is concerned with an explicit value of the embedding constant from W1 , q(Ω) to Lp(Ω) for a domain Ω ⊂ RN (N∈ N), where 1 ≤ q≤ p≤ ∞. We previously proposed a formula for estimating the embedding constant on bounded and unbounded Lipschitz domains by estimating the norm of Stein’s extension operator. Although this formula can be applied to a domain Ω that can be divided into a finite number of Lipschitz domains, there was room for improvement in terms of accuracy. In this paper, we report that the accuracy of the embedding constant is significantly improved by restricting Ω to a domain dividable into bounded convex domains.

Original languageEnglish
Article number299
JournalJournal of Inequalities and Applications
Volume2017
DOIs
Publication statusPublished - 2017 Jan 1

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Sobolev Embedding
Convex Domain
Bounded Domain
Lipschitz Domains
Extension Operator
Unbounded Domain
Norm

Keywords

  • Hardy-Littlewood-Sobolev inequality
  • Sobolev embedding constant
  • Young inequality

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

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abstract = "This paper is concerned with an explicit value of the embedding constant from W1 , q(Ω) to Lp(Ω) for a domain Ω ⊂ RN (N∈ N), where 1 ≤ q≤ p≤ ∞. We previously proposed a formula for estimating the embedding constant on bounded and unbounded Lipschitz domains by estimating the norm of Stein’s extension operator. Although this formula can be applied to a domain Ω that can be divided into a finite number of Lipschitz domains, there was room for improvement in terms of accuracy. In this paper, we report that the accuracy of the embedding constant is significantly improved by restricting Ω to a domain dividable into bounded convex domains.",
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AU - Tanaka, Kazuaki

AU - Sekine, Kouta

AU - Oishi, Shinichi

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N2 - This paper is concerned with an explicit value of the embedding constant from W1 , q(Ω) to Lp(Ω) for a domain Ω ⊂ RN (N∈ N), where 1 ≤ q≤ p≤ ∞. We previously proposed a formula for estimating the embedding constant on bounded and unbounded Lipschitz domains by estimating the norm of Stein’s extension operator. Although this formula can be applied to a domain Ω that can be divided into a finite number of Lipschitz domains, there was room for improvement in terms of accuracy. In this paper, we report that the accuracy of the embedding constant is significantly improved by restricting Ω to a domain dividable into bounded convex domains.

AB - This paper is concerned with an explicit value of the embedding constant from W1 , q(Ω) to Lp(Ω) for a domain Ω ⊂ RN (N∈ N), where 1 ≤ q≤ p≤ ∞. We previously proposed a formula for estimating the embedding constant on bounded and unbounded Lipschitz domains by estimating the norm of Stein’s extension operator. Although this formula can be applied to a domain Ω that can be divided into a finite number of Lipschitz domains, there was room for improvement in terms of accuracy. In this paper, we report that the accuracy of the embedding constant is significantly improved by restricting Ω to a domain dividable into bounded convex domains.

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