A Note on the Sobolev and Gagliardo--Nirenberg Inequality when 𝑝 > 𝑁

Abstract

It is known that the Sobolev space W1,p⁢(ℝN){W^{1,p}(\mathbb{R}^{N})} is embedded into LN⁢p/(N-p)⁢(ℝN){L^{Np/(N-p)}(\mathbb{R}^{N})} if pN{p>N}. There is usually a discontinuity in the proof of those two different embeddings since, for p>N{p>N}, the estimate ∥u∥∞≤C⁢∥D⁢u∥pN/p⁢∥u∥p1-N/p{\lVert u\rVert_{\infty}\leq C\lVert Du\rVert_{p}^{N/p}\lVert u\rVert_{p}^{1-N% /p}} is commonly obtained together with an estimate of the Hölder norm. In this note, we give a proof of the L∞{L^{\infty}}-embedding which only follows by an iteration of the Sobolev–Gagliardo–Nirenberg estimate ∥u∥N/(N-1)≤C⁢∥D⁢u∥1{\lVert u\rVert_{N/(N-1)}\leq C\lVert Du\rVert_{1}}. This kind of proof has the advantage to be easily extended to anisotropic cases and immediately exported to the case of discrete Lebesgue and Sobolev spaces; we give sample results in case of finite differences and finite volumes schemes.

Keywords

• sobolev spaces
• gagliardo–nirenberg inequality
• discrete sobolev inequalities
• 35a23