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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Selected new aspects of the calculus of variations in the large

Authors: Ivar Ekeland and Nassif Ghoussoub
Journal: Bull. Amer. Math. Soc. 39 (2002), 207-265
MSC (2000): Primary 35J60, 47J30, 58E05; Secondary 57R17
Published electronically: January 4, 2002
MathSciNet review: 1886088
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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

Nassif Ghoussoub
Affiliation: Pacific Institute for the Mathematical Sciences, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2

Received by editor(s): January 1, 2001
Received by editor(s) in revised form: June 13, 2001
Published electronically: 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)
Article copyright: © Copyright 2002 American Mathematical Society