Note on the concentration-compactness principle for generalized Moser-Trudinger inequalities

Robert Černý

Open Mathematics (2012)

  • Volume: 10, Issue: 2, page 590-602
  • ISSN: 2391-5455

Abstract

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Let Ω ⊂ ℝn, n ≥ 2, be a bounded domain and let α < n − 1. Motivated by Theorem I.6 and Remark I.18 of [Lions P.-L., The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1985, 1(1), 145–201] and by the results of [Černý R., Cianchi A., Hencl S., Concentration-Compactness Principle for Moser-Trudinger inequalities: new results and proofs, Ann. Mat. Pura Appl. (in press), DOI: 10.1007/s10231-011-0220-3], we give a sharp estimate of the exponent concerning the Concentration-Compactness Principle for the embedding of the Orlicz-Sobolev space W 01 L n logα L(Ω) into the Orlicz space corresponding to a Young function that behaves like exp t n/(n−1−α) for large t. We also give the result for the case of the embedding into double and other multiple exponential spaces.

How to cite

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Robert Černý. "Note on the concentration-compactness principle for generalized Moser-Trudinger inequalities." Open Mathematics 10.2 (2012): 590-602. <http://eudml.org/doc/269628>.

@article{RobertČerný2012,
abstract = {Let Ω ⊂ ℝn, n ≥ 2, be a bounded domain and let α < n − 1. Motivated by Theorem I.6 and Remark I.18 of [Lions P.-L., The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1985, 1(1), 145–201] and by the results of [Černý R., Cianchi A., Hencl S., Concentration-Compactness Principle for Moser-Trudinger inequalities: new results and proofs, Ann. Mat. Pura Appl. (in press), DOI: 10.1007/s10231-011-0220-3], we give a sharp estimate of the exponent concerning the Concentration-Compactness Principle for the embedding of the Orlicz-Sobolev space W 01 L n logα L(Ω) into the Orlicz space corresponding to a Young function that behaves like exp t n/(n−1−α) for large t. We also give the result for the case of the embedding into double and other multiple exponential spaces.},
author = {Robert Černý},
journal = {Open Mathematics},
keywords = {Orlicz spaces; Orlicz-Sobolev spaces; Embedding theorems; Sharp constants; Moser-Trudinger inequality; Concentration-Compactness Principle; concentration-compactness principle; sharp constants},
language = {eng},
number = {2},
pages = {590-602},
title = {Note on the concentration-compactness principle for generalized Moser-Trudinger inequalities},
url = {http://eudml.org/doc/269628},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Robert Černý
TI - Note on the concentration-compactness principle for generalized Moser-Trudinger inequalities
JO - Open Mathematics
PY - 2012
VL - 10
IS - 2
SP - 590
EP - 602
AB - Let Ω ⊂ ℝn, n ≥ 2, be a bounded domain and let α < n − 1. Motivated by Theorem I.6 and Remark I.18 of [Lions P.-L., The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1985, 1(1), 145–201] and by the results of [Černý R., Cianchi A., Hencl S., Concentration-Compactness Principle for Moser-Trudinger inequalities: new results and proofs, Ann. Mat. Pura Appl. (in press), DOI: 10.1007/s10231-011-0220-3], we give a sharp estimate of the exponent concerning the Concentration-Compactness Principle for the embedding of the Orlicz-Sobolev space W 01 L n logα L(Ω) into the Orlicz space corresponding to a Young function that behaves like exp t n/(n−1−α) for large t. We also give the result for the case of the embedding into double and other multiple exponential spaces.
LA - eng
KW - Orlicz spaces; Orlicz-Sobolev spaces; Embedding theorems; Sharp constants; Moser-Trudinger inequality; Concentration-Compactness Principle; concentration-compactness principle; sharp constants
UR - http://eudml.org/doc/269628
ER -

References

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