A Note on the Sobolev and Gagliardo--Nirenberg Inequality when 𝑝 > 𝑁
| Autor: | Porretta Alessio |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | Advanced Nonlinear Studies, Vol 20, Iss 2, Pp 361-371 (2020) |
| Druh dokumentu: | article |
| ISSN: | 1536-1365 2169-0375 |
| DOI: | 10.1515/ans-2020-2086 |
| Popis: | It is known that the Sobolev space W1,p(ℝN){W^{1,p}(\mathbb{R}^{N})} is embedded into LNp/(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∥Du∥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∥Du∥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. |
| Databáze: | Directory of Open Access Journals |
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