On the discrete Sobolev inequalities

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.