On the discrete Sobolev inequalities
Sédrick Kameni Ngwamou, Michaël Ndjinga · Journal of Numerical Mathematics · 2023
Abstract We prove a discrete version of the famous Sobolev inequalities 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 BV(ℝ d ) into L d/(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.