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Selected new aspects of the calculus of variations in the large

Author(s): Ivar Ekeland; Nassif Ghoussoub
Journal: Bull. Amer. Math. Soc. 39 (2002), 207-265.
MSC (2000): Primary 35J60, 47J30, 58E05; Secondary 57R17
Posted: January 4, 2002
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Abstract: We discuss some of the recent developments in variational methods while emphasizing new applications to nonlinear problems. We touch on several issues: (i) the formulation of variational set-ups which provide more information on the location of critical points and therefore on the qualitative properties of the solutions of corresponding Euler-Lagrange equations; (ii) the relationships between the energy of variationally generated solutions, their Morse indices, and the Hausdorff measure of their nodal sets; (iii) the gluing of several topological obstructions; (iv) the preservation of critical levels after deformation of functionals; (v) and the various ways to recover compactness in certain borderline variational problems.

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Additional Information:

Ivar Ekeland
Affiliation: CEREMADE, Université Paris-Dauphine, Paris, France
Email: Ivar.Ekeland@dauphine.fr

Nassif Ghoussoub
Affiliation: Pacific Institute for the Mathematical Sciences, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2
Email: nassif@math.ubc.ca

DOI: 10.1090/S0273-0979-02-00929-1
PII: S 0273-0979(02)00929-1
Received by editor(s): January 2001
Received by editor(s) in revised form: June 13, 2001
Posted: January 4, 2002
Additional Notes: The second author was partially supported by a grant from the Natural Science and Engineering Research Council of Canada (NSERC)
Copyright of article: Copyright 2002, American Mathematical Society

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