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Licensed Unlicensed Requires Authentication Published by De Gruyter September 5, 2023

On the discrete Sobolev inequalities

  • Sedrick Kameni Ngwamou EMAIL logo and Michael Ndjinga

Abstract

We prove a discrete version of the famous Sobolev inequalities [1] in R d  for  d N , p [ 1 , + [ 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 [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.

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Published Online: 2023-09-05

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