Abstract

We prove the existence of radially symmetric minimizers, in the class of Sobolev vector-valued functions vanishing on the boundary of a ball, for convex non-coercive integral functionals. We associate to the functional a system of differential inclusions of Euler-Lagrange type, and we prove that the solvability of these inclusions is a necessary and sufficient condition for the existence of a radially symmetric minimizer.

Author information

Contact details are reproduced from the original publication and may be historical.

Graziano Crasta

Dip. di Matematica Pura ed Applicata, Università di Modena, 41100 Modena, Italy

Annalisa Malusa

Dip. di Matematica, Università di Roma 1, Piazzale A. Moro 2, 00185 Roma, Italy

Keywords

  • Calculus of variations
  • existence
  • Euler-Lagrange inclusions
  • radially symmetric solutions
  • non-coercive problems

Mathematics Subject Classification

49J10, 45K05; 49J30

References

  1. A. Cellina, S. Perrotta: On minima of radially symmetric functionals of the gradient, Nonlinear Anal. 23 (1994) 239-249.https://doi.org/10.1016/0362-546X(94)90045-0Google Scholar
  2. A. Cellina, G. Treu, S. Zagatti: On the minimum problem for a class of non-coercive functionals, J. Differential Equations 127 (1996) 225-262.https://doi.org/10.1006/jdeq.1996.0069Google Scholar
  3. L. Cesari: Optimization - Theory and Applications, Springer-Verlag, New York, 1983.https://doi.org/10.1007/978-1-4613-8165-5Google Scholar
  4. F. H. Clarke: An indirect method in the calculus of variations, Trans. Amer. Math. Soc. 336 (1993) 655-673.https://doi.org/10.1090/S0002-9947-1993-1118823-3Google Scholar
  5. G. Crasta: On the minimum problem for a class of non-coercive non-convex variational problems, SIAM J. Control Optim. 38 (1999) 237-253.https://doi.org/10.1137/S0363012997330701Google Scholar
  6. G. Crasta: Existence, uniqueness and qualitative properties of minima to radially symmetric non-coercive non-convex variational problems, Math. Z., to appear.Google Scholar
  7. G. Crasta, A. Malusa: Existence results for non-convex variational problems, SIAM J. Control Optim. 34 (1996) 2064-2076.https://doi.org/10.1137/S0363012994278201Google Scholar
  8. G. Crasta, A. Malusa: Non-convex minimization problems for functionals defined on vector-valued functions, in preparation.Google Scholar
  9. B. Dacorogna: Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin, Heidelberg, 1989.https://doi.org/10.1007/978-3-642-51440-1Google Scholar
  10. K. Deimling: Multivalued Differential Equations, De Gruyter, Berlin, 1992.https://doi.org/10.1515/9783110874228Google Scholar
  11. I. Ekeland, R. Temam: Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1977.Google Scholar
  12. R. T. Rockafellar: Conjugate convex functions in optimal control and the calculus of variations, J. Math. Anal. Appl. 32 (1970) 174-222.https://doi.org/10.1016/0022-247X(70)90324-0Google Scholar
  13. R. T. Rockafellar: Existence and duality theorems for convex Bolza problems, Trans. Amer. Math. Soc. 159 (1971) 1-40.https://doi.org/10.1090/S0002-9947-1971-0282283-0Google Scholar

Suggested citation

G. Crasta, A. Malusa. “Euler-Lagrange Inclusions and Existence of Minimizers for a Class of Non-Coercive Variational Problems.” Journal of Convex Analysis 7 (2000), No. 1, 167–181. https://doi.org/10.68381/jca07008

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2000. Rights now held by Banach Press LLC.