| Title:
|
Concentration-Compactness Principle for embedding into multiple exponential spaces on unbounded domains (English) |
| Author:
|
Černý, Robert |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
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1572-9141 (online) |
| Volume:
|
65 |
| Issue:
|
2 |
| Year:
|
2015 |
| Pages:
|
493-516 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\Omega \subset \mathbb R^n$ be a domain and let $\alpha <n-1$. We prove the Concentration-Compactness Principle for the embedding of the space $W_0^1L^n\log ^{\alpha }L(\Omega )$ into an Orlicz space corresponding to a Young function which behaves like $\exp (t^{{n}/{(n-1-\alpha )}})$ for large $t$. We also give the result for the embedding into multiple exponential spaces. \endgraf Our main result is Theorem \ref {lions4} where we show that if one passes to unbounded domains, then, after the usual modification of the integrand in the Moser functional, the statement of the Concentration-Compactnes Principle is very similar to the statement in the case of a bounded domain. In particular, in the case of a nontrivial weak limit the borderline exponent is still given by the formula $$ P:=(1-\|\Phi (|\nabla u|)\|_{L^1(\mathbb R^n)})^{-{1}/{(n-1)}}. $$ (English) |
| Keyword:
|
Sobolev space |
| Keyword:
|
Orlicz-Sobolev space |
| Keyword:
|
Moser-Trudinger inequality |
| Keyword:
|
sharp constant |
| Keyword:
|
concentration-compactness principle |
| MSC:
|
26D10 |
| MSC:
|
46E30 |
| MSC:
|
46E35 |
| idZBL:
|
Zbl 06486960 |
| idMR:
|
MR3360440 |
| DOI:
|
10.1007/s10587-015-0189-y |
| . |
| Date available:
|
2015-06-16T18:01:32Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144283 |
| . |
| Reference:
|
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