Selected new aspects of the calculus of variations in the large

Ekeland, Ivar; Ghoussoub, Nassif

doi:10.1090/S0273-0979-02-00929-1

Selected new aspects of the calculus of variations in the large
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by Ivar Ekeland and Nassif Ghoussoub PDF

Bull. Amer. Math. Soc. 39 (2002), 207-265 Request permission

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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