Abstract
We prove a discrete version of the famous Sobolev inequalities in ℝd 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 BV(ℝd) into Ld/(d–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 and use techniques from prior works. In the case p > d (Morrey’s inequality), we simplify and extend the proof of Porreta to more general meshes.
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