Concentration-compactness principle for embedding into multiple exponential spaces on unbounded domains

Title:

Concentration-compactness principle for embedding into multiple exponential spaces on unbounded domains

Creator:

Černý, Robert

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:03800cfc-7b7d-47ae-aad2-ac74343a6d46
uuid:03800cfc-7b7d-47ae-aad2-ac74343a6d46
issn:0011-4642

Subject:

matematika, mathematics, Sobolev space, Orlicz-Sobolev space, Moser-Trudinger inequality, sharp constant, concentration-compactness principle, 13, and 51

Type:

model:article and TEXT

Format:

print, bez média, and svazek

Description:

Let Ω \subset \mathbb{R}^{n} be a domain and let α < n − 1. We prove the Concentration-Compactness Principle for the embedding of the space W_{0}^{1}L^{n} log^{\alpha } L(Ω) into an Orlicz space corresponding to a Young function which behaves like (t^{n/n-1-\alpha }) for large t. We also give the result for the embedding into multiple exponential spaces. Our main result is Theorem 1.6 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: = \left( {1 - \left\| {\Phi (|\nabla u|)} \right\|_{L^1 (\mathbb{R}^n )} } \right)^{ - 1/(n - 1)} ., Robert Černý., and Obsahuje seznam literatury

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

493-516

Source:

Czechoslovak Mathematical Journal | 2015 Volume:65 | Number:2

Harvested from:

CDK

Metadata only:

false