{"title":"On the discrete Sobolev inequalities","authors":"Sédrick Kameni Ngwamou, Michael Ndjinga","doi":"10.1515/jnma-2023-0086","DOIUrl":null,"url":null,"abstract":"Abstract We prove a discrete version of the famous Sobolev inequalities [1] in R d for d ∈ N ∗ , p ∈ [ 1 , + ∞ [ $\\mathbb{R}^{d} \\text { for } d \\in \\mathbb{N}^{*}, p \\in[1,+\\infty[$ for general non orthogonal meshes with possibly non convex cells. We follow closely the proof of the continuous Sobolev inequality based on the embedding of B V R d into L d d − 1 $B V\\left(\\mathbb{R}^{d}\\right) \\text { into } \\mathrm{L}^{\\frac{d}{d-1}}$ [1, theorem 9.9],[12, theorem 1.1] by introducing discrete analogs of the directional total variations. In the case p > d (Gagliardo-Nirenberg inequality), we adapt the proof of the continuous case ( [1, theorem 9.9], [9, theorem 4.8]) and use techniques from [3, 5]. In the case p > d (Morrey’s inequality), we simplify and extend the proof of [12, theorem 1.1] to more general meshes.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"66 1","pages":""},"PeriodicalIF":3.8000,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Numerical Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/jnma-2023-0086","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract We prove a discrete version of the famous Sobolev inequalities [1] in R d for d ∈ N ∗ , p ∈ [ 1 , + ∞ [ $\mathbb{R}^{d} \text { for } d \in \mathbb{N}^{*}, p \in[1,+\infty[$ for general non orthogonal meshes with possibly non convex cells. We follow closely the proof of the continuous Sobolev inequality based on the embedding of B V R d into L d d − 1 $B V\left(\mathbb{R}^{d}\right) \text { into } \mathrm{L}^{\frac{d}{d-1}}$ [1, theorem 9.9],[12, theorem 1.1] by introducing discrete analogs of the directional total variations. In the case p > d (Gagliardo-Nirenberg inequality), we adapt the proof of the continuous case ( [1, theorem 9.9], [9, theorem 4.8]) and use techniques from [3, 5]. In the case p > d (Morrey’s inequality), we simplify and extend the proof of [12, theorem 1.1] to more general meshes.
摘要本文证明了著名的Sobolev不等式[1]在rd中的离散形式,对于可能具有非凸单元的一般非正交网格,对于d∈N∗,p∈[1,+∞[$\mathbb{R}^{d} \text { for } d \in \mathbb{N}^{*}, p \in[1,+\infty[$。我们通过引入定向总变分的离散类比,密切关注基于bv R d嵌入到ld d−1 $B V\left(\mathbb{R}^{d}\right) \text { into } \mathrm{L}^{\frac{d}{d-1}}$[1,定理9.9],[12,定理1.1]的连续Sobolev不等式的证明。在p b> d (Gagliardo-Nirenberg不等式)的情况下,我们采用连续情况([1,定理9.9],[9,定理4.8])的证明,并使用[3,5]中的技术。在p b> d (Morrey’s不等式)的情况下,我们将[12,定理1.1]的证明简化并推广到更一般的网格。
期刊介绍:
The Journal of Numerical Mathematics (formerly East-West Journal of Numerical Mathematics) contains high-quality papers featuring contemporary research in all areas of Numerical Mathematics. This includes the development, analysis, and implementation of new and innovative methods in Numerical Linear Algebra, Numerical Analysis, Optimal Control/Optimization, and Scientific Computing. The journal will also publish applications-oriented papers with significant mathematical content in computational fluid dynamics and other areas of computational engineering, finance, and life sciences.