Concentration-Compactness Principle for embedding into multiple exponential spaces on unbounded domains

Robert Černý

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 2, page 493-516
  • ISSN: 0011-4642

Abstract

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Let Ω ⊂ 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 α L ( Ω ) into an Orlicz space corresponding to a Young function which behaves like exp ( t n / ( n - 1 - α ) ) for large t . We also give the result for the embedding into multiple exponential spaces. Our main result is Theorem 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 - ∥ Φ ( | ∇ u | ) ∥ L 1 ( ℝ n ) ) - 1 / ( n - 1 ) .

How to cite

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Černý, Robert. "Concentration-Compactness Principle for embedding into multiple exponential spaces on unbounded domains." Czechoslovak Mathematical Journal 65.2 (2015): 493-516. <http://eudml.org/doc/270126>.

@article{Černý2015,
abstract = {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. Our main result is Theorem 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 \[ P:=(1-\Vert \Phi (|\nabla u|)\Vert \_\{L^1(\mathbb \{R\}^n)\})^\{-\{1\}/\{(n-1)\}\}. \]},
author = {Černý, Robert},
journal = {Czechoslovak Mathematical Journal},
keywords = {Sobolev space; Orlicz-Sobolev space; Moser-Trudinger inequality; sharp constant; concentration-compactness principle},
language = {eng},
number = {2},
pages = {493-516},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Concentration-Compactness Principle for embedding into multiple exponential spaces on unbounded domains},
url = {http://eudml.org/doc/270126},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Černý, Robert
TI - Concentration-Compactness Principle for embedding into multiple exponential spaces on unbounded domains
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 2
SP - 493
EP - 516
AB - 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. Our main result is Theorem 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 \[ P:=(1-\Vert \Phi (|\nabla u|)\Vert _{L^1(\mathbb {R}^n)})^{-{1}/{(n-1)}}. \]
LA - eng
KW - Sobolev space; Orlicz-Sobolev space; Moser-Trudinger inequality; sharp constant; concentration-compactness principle
UR - http://eudml.org/doc/270126
ER -

References

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