The concentration-compactness principle in the calculus of variations. The limit case, Part I.

Pierre-Louis Lions

Revista Matemática Iberoamericana (1985)

  • Volume: 1, Issue: 1, page 145-201
  • ISSN: 0213-2230

Abstract

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After the study made in the locally compact case for variational problems with some translation invariance, we investigate here the variational problems (with constraints) for example in RN where the invariance of RN by the group of dilatations creates some possible loss of compactness. This is for example the case for all the problems associated with the determination of extremal functions in functional inequalities (like for example the Sobolev inequalities). We show here how the concentration-compactness principle has to be modified in order to be able to treat this class of problems and we present applications to Functional Analysis, Mathematical Physics, Differential Geometry and Harmonic Analysis.

How to cite

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Lions, Pierre-Louis. "The concentration-compactness principle in the calculus of variations. The limit case, Part I.." Revista Matemática Iberoamericana 1.1 (1985): 145-201. <http://eudml.org/doc/39314>.

@article{Lions1985,
abstract = {After the study made in the locally compact case for variational problems with some translation invariance, we investigate here the variational problems (with constraints) for example in RN where the invariance of RN by the group of dilatations creates some possible loss of compactness. This is for example the case for all the problems associated with the determination of extremal functions in functional inequalities (like for example the Sobolev inequalities). We show here how the concentration-compactness principle has to be modified in order to be able to treat this class of problems and we present applications to Functional Analysis, Mathematical Physics, Differential Geometry and Harmonic Analysis.},
author = {Lions, Pierre-Louis},
journal = {Revista Matemática Iberoamericana},
keywords = {Cálculo de variaciones; Principio de concentración-compacidad; Teoría de Morse; Dominios no acotados; Masas de Dirac; Grupo de invariancias; Minimización; functional inequalities; Sobolev inequalities; concentration-compactness principle},
language = {eng},
number = {1},
pages = {145-201},
title = {The concentration-compactness principle in the calculus of variations. The limit case, Part I.},
url = {http://eudml.org/doc/39314},
volume = {1},
year = {1985},
}

TY - JOUR
AU - Lions, Pierre-Louis
TI - The concentration-compactness principle in the calculus of variations. The limit case, Part I.
JO - Revista Matemática Iberoamericana
PY - 1985
VL - 1
IS - 1
SP - 145
EP - 201
AB - After the study made in the locally compact case for variational problems with some translation invariance, we investigate here the variational problems (with constraints) for example in RN where the invariance of RN by the group of dilatations creates some possible loss of compactness. This is for example the case for all the problems associated with the determination of extremal functions in functional inequalities (like for example the Sobolev inequalities). We show here how the concentration-compactness principle has to be modified in order to be able to treat this class of problems and we present applications to Functional Analysis, Mathematical Physics, Differential Geometry and Harmonic Analysis.
LA - eng
KW - Cálculo de variaciones; Principio de concentración-compacidad; Teoría de Morse; Dominios no acotados; Masas de Dirac; Grupo de invariancias; Minimización; functional inequalities; Sobolev inequalities; concentration-compactness principle
UR - http://eudml.org/doc/39314
ER -

Citations in EuDML Documents

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  1. O. Rey, Le rôle de la frontière dans un problème elliptique avec non-linéarité critique et conditions au bord de Neumann
  2. A. Carpio Rodriguez, M. Comte, R. Lewandowski, A nonexistence result for a nonlinear equation involving critical Sobolev exponent
  3. Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n -laplacian
  4. Anass Ourraoui, On a class of nonlocal problem involving a critical exponent
  5. Bruno Nazaret, Stability results for some nonlinear elliptic equations involving the p -laplacian with critical Sobolev growth
  6. Ge Yuxin, Estimations of the best constant involving the L 2 norm in Wente’s inequality and compact H -surfaces in euclidean space
  7. A. Bahri, Critical points at infinity in the variational calculus
  8. Yuxin Ge, Inégalité de Wente et ses applications aux H -surfaces
  9. Michel Willem, Minimization problems with lack of compactness
  10. A. El Hamidi, J. M. Rakotoson, Extremal functions for the anisotropic Sobolev inequalities

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