A Note on the Sobolev and Gagliardo--Nirenberg Inequality when

It is known that the Sobolev space is embedded into if and into if . There is usually a discontinuity in the proof of those two different embeddings since, for , the estimate is commonly obtained together with an estimate of the Hölder norm. In this note, we give a proof of the -embedding which only...

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Bibliographic Details
Published in:Advanced nonlinear studies 2020-05, Vol.20 (2), p.361-371
Main Author: Porretta, Alessio
Format: Article
Language:English
Subjects:
35A23
Discrete Sobolev Inequalities
Gagliardo–Nirenberg Inequality
Sobolev spaces
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Summary:It is known that the Sobolev space is embedded into if and into if . There is usually a discontinuity in the proof of those two different embeddings since, for , the estimate is commonly obtained together with an estimate of the Hölder norm. In this note, we give a proof of the -embedding which only follows by an iteration of the Sobolev–Gagliardo–Nirenberg estimate . 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.
ISSN:1536-1365
2169-0375
DOI:10.1515/ans-2020-2086