Some discrete Sobolev's inequalities in three-dimensional spherical and cylindrical coordinates
type: Item
instance of: scholarly article
title: Some discrete Sobolev's inequalities in three-dimensional spherical and cylindrical coordinates
zbMATH Open document ID: 0716.46035
DOI: 10.1007/BF02006748
published in: Acta Mathematicae Applicatae Sinica. English Series
publication date: 1990-01-01T00:00:00Z
review text: Soit u une fonction définie dans un ensemble de \({\mathbb{R}}^ 3\) muni des coordonnées sphèriques r, \(\theta\), \(\phi\), si on considère les pas \(h_ 1\) (resp. \(h_ 2\), \(h_ 3)\) et si on localise les points par \(r_ i=(i-)h_ 1\) (resp. \(\theta_ j\), \(\phi_ k)\), on désigne la norme discrète de u dans \(L^ p\) par: \[ | u|^ p_{L^ p}=h_ 1h_ 2h_ 3\sum_{ijk}(u_{ijk})^ pr^ 2_ i \sin \theta_ j. \] Pour \(1\leq p<3\), q: \(1/q=1/p-1/3\), u dont le support est compact l'auteur obtient un majoration \[ | u|_{L^ q}\leq C(p)[| \delta_ ru|_{L^ p}+| \delta_{\theta}u|_{L^ p}+| \delta_{\phi}u|_{L^ p}] \] (où \(\delta_ ru_{ijk}=(1/h_ 1)(u_{i+1,j,k}-u_{ijk})\) resp. \(\delta_{\theta},\delta_{\phi})\) (Theorem I). Il a un résultat analogue en coordonnées cylindriques.
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46E39
zbMATH DE Number: 4180106
zbMATH Keywords: difference method
zbMATH Keywords: discrete Sobolev's inequalities in three-dimensional spherical and cylindrical coordinates
author: Qianshun Chang
reviewed by: Marie-Therese Lacroix
MaRDI profile type: Publication
cites work: The existence of unbounded solutions of the korteweg‐de vries equation
full work available at URL: https://doi.org/10.1007/bf02006748
OpenAlex ID: W2313118235
description: scientific article; zbMATH DE number 4180106