Evaluation of the One-Dimensional Lp Sobolev Type Inequality

Takemura, Kazuo; Kametaka, Yoshinori

doi:10.3390/math8020296

Open AccessArticle

Evaluation of the One-Dimensional L^p Sobolev Type Inequality

by

Kazuo Takemura

^1,*,†,‡ and

Yoshinori Kametaka

^2,‡

¹

General Education College of Science and Technology, Nihon University, Funabashi 274-0812, Japan

²

Faculty of Engineering Science, Osaka University, Toyonaka 560-8531, Japan

^*

Author to whom correspondence should be addressed.

^†

Current address: 7-24-1, Narashinodai, Funabashi, Chiba 274-8501, Japan.

^‡

These authors contributed equally to this work.

Mathematics 2020, 8(2), 296; https://doi.org/10.3390/math8020296

Submission received: 30 December 2019 / Revised: 17 February 2020 / Accepted: 17 February 2020 / Published: 21 February 2020

(This article belongs to the Section C1: Difference and Differential Equations)

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Abstract

This study applies the extended

L^{2}

Sobolev type inequality to the

L^{p}

Sobolev type inequality using Hölder’s inequality. The sharp constant and best function of the

L^{p}

Sobolev type inequality are found using a Green function for the nth order ordinary differential equation. The sharp constant is shown to be equal to the

L^{p}

norm of the Green function and to the pth root of the value of the origin of the best function.

Keywords: sobolev inequality; sharp constant; green function

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MDPI and ACS Style

Takemura, K.; Kametaka, Y. Evaluation of the One-Dimensional L^p Sobolev Type Inequality. Mathematics 2020, 8, 296. https://doi.org/10.3390/math8020296

AMA Style

Takemura K, Kametaka Y. Evaluation of the One-Dimensional L^p Sobolev Type Inequality. Mathematics. 2020; 8(2):296. https://doi.org/10.3390/math8020296

Chicago/Turabian Style

Takemura, Kazuo, and Yoshinori Kametaka. 2020. "Evaluation of the One-Dimensional L^p Sobolev Type Inequality" Mathematics 8, no. 2: 296. https://doi.org/10.3390/math8020296

APA Style

Takemura, K., & Kametaka, Y. (2020). Evaluation of the One-Dimensional L^p Sobolev Type Inequality. Mathematics, 8(2), 296. https://doi.org/10.3390/math8020296

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Evaluation of the One-Dimensional L^p Sobolev Type Inequality

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