A min-max theorem for multiple integrals of the Calculus of Variations and applications

David Arcoya; Lucio Boccardo

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1995)

  • Volume: 6, Issue: 1, page 29-35
  • ISSN: 1120-6330

Abstract

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In this paper we deal with the existence of critical points for functionals defined on the Sobolev space W 0 1 , 2 ⁢ Ω by J ⁢ v = ∫ Ω I ⁢ x , v , D ⁢ v ⁢ d ⁢ x , v ∈ W 0 1 , 2 ⁢ Ω , where Ω is a bounded, open subset of R N . Since the differentiability can fail even for very simple examples of functionals defined through multiple integrals of Calculus of Variations, we give a suitable version of the Ambrosetti-Rabinowitz Mountain Pass Theorem, which enables us to the study of critical points for functionals which are not differentiable in all directions. Then we present some applications of this theorem to the study of the existence and multiplicity of nonnegative critical points for multiple integrals of the Calculus of Variations.

How to cite

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Arcoya, David, and Boccardo, Lucio. "A min-max theorem for multiple integrals of the Calculus of Variations and applications." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 6.1 (1995): 29-35. <http://eudml.org/doc/244334>.

@article{Arcoya1995,
abstract = {In this paper we deal with the existence of critical points for functionals defined on the Sobolev space \( W\_\{0\}^\{1,2\} (\Omega) \) by \( J(v) = \int\_\{\Omega\} \mathfrak\{I\} (x,v,Dv) \, dx \), \( v \in W\_\{0\}^\{1,2\} (\Omega) \), where \( \Omega \) is a bounded, open subset of \( \mathbb\{R\}^\{N\} \). Since the differentiability can fail even for very simple examples of functionals defined through multiple integrals of Calculus of Variations, we give a suitable version of the Ambrosetti-Rabinowitz Mountain Pass Theorem, which enables us to the study of critical points for functionals which are not differentiable in all directions. Then we present some applications of this theorem to the study of the existence and multiplicity of nonnegative critical points for multiple integrals of the Calculus of Variations.},
author = {Arcoya, David, Boccardo, Lucio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Critical points; Multiple integrals of Calculus of Variations; Quasilinear equations; Ambrosetti-Rabinowitz mountain pass theorem; existence; critical points; functionals; Sobolev space; multiple integrals},
language = {eng},
month = {3},
number = {1},
pages = {29-35},
publisher = {Accademia Nazionale dei Lincei},
title = {A min-max theorem for multiple integrals of the Calculus of Variations and applications},
url = {http://eudml.org/doc/244334},
volume = {6},
year = {1995},
}

TY - JOUR
AU - Arcoya, David
AU - Boccardo, Lucio
TI - A min-max theorem for multiple integrals of the Calculus of Variations and applications
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1995/3//
PB - Accademia Nazionale dei Lincei
VL - 6
IS - 1
SP - 29
EP - 35
AB - In this paper we deal with the existence of critical points for functionals defined on the Sobolev space \( W_{0}^{1,2} (\Omega) \) by \( J(v) = \int_{\Omega} \mathfrak{I} (x,v,Dv) \, dx \), \( v \in W_{0}^{1,2} (\Omega) \), where \( \Omega \) is a bounded, open subset of \( \mathbb{R}^{N} \). Since the differentiability can fail even for very simple examples of functionals defined through multiple integrals of Calculus of Variations, we give a suitable version of the Ambrosetti-Rabinowitz Mountain Pass Theorem, which enables us to the study of critical points for functionals which are not differentiable in all directions. Then we present some applications of this theorem to the study of the existence and multiplicity of nonnegative critical points for multiple integrals of the Calculus of Variations.
LA - eng
KW - Critical points; Multiple integrals of Calculus of Variations; Quasilinear equations; Ambrosetti-Rabinowitz mountain pass theorem; existence; critical points; functionals; Sobolev space; multiple integrals
UR - http://eudml.org/doc/244334
ER -

References

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  4. ARCOYA, D. - BOCCARDO, L., Critical points for multiple integrals of Calculus of Variations. To appear. Zbl0884.58023MR1412429DOI10.1007/BF00379536
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  8. EKELAND, I., Nonconvex minimization problems. Bull. Amer. Math. Soc. (NS), 1, 1979, 443-474. Zbl0441.49011MR526967DOI10.1090/S0273-0979-1979-14595-6
  9. DACOROGNA, B., Direct Methods in the Calculus of Variations. Springer-Verlag, 1989. Zbl1140.49001MR990890
  10. DE FIGUEIREDO, D. G., The Ekeland Variational Principle with Applications and Detours. Springer-Verlag, 1989. MR1019559
  11. LADYZENSKAYA, O. A. - URALCEVA, N. N., Equations aux dérivées partielles de type elliptique. Dunod, Paris1968. Zbl0164.13001
  12. MAWHIN, I. - WILLEM, M., Critical Point Theory and Hamiltonian Systems. Springer-Verlag, 1989. Zbl0676.58017MR982267
  13. MORREY, C. B., Multiple Integrals in the Calculus of Variations. Springer-Verlag, 1966. Zbl1213.49002MR202511
  14. RABINOWTTZ, P. H., Minimax methods in critical point theory with applications to differential equations. CBMS Regional Conference Series Math., 65, Amer. Math. Soc., Providence1986. Zbl0609.58002MR845785

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