Selected new aspects of the calculus of variations in the large

By Ivar Ekeland, Nassif Ghoussoub. 2002

161 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/de180149ca4f7a1e6ccc6d73a8dd9c4a0e13d3e0

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Existence, non-existence and regularity of radial ground states for p-Laplacian equations with singular weights (2008, background): “1) was studied in several papers for general p (see, for instance, [17,24]) and for p = 2 (see, for example, [10,27]).”
The Allen–Cahn equation on closed manifolds (2016, methodology): “on this Introduction. For some other semilinear elliptic equations results like these have been proven. This is a vast subject, we refer the reader to the surveys by Ambrosetti [3], Ekeland-Ghoussoub [20], Rabinowitz [53] and the references therein. However, we should not expect some of these method”
Existence and non-existence of solutions to elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities (2007, unknown): “More results can be found in [7, 15, 18, 19, 20, 22].”
A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms $ (2003, background): “O juj ; 8uAH 0 ðOÞ: ð2Þ Due to the invariance of H 0 -norm, L 2 -norm and R O mu jxj with respect to rescaling u/ur 1⁄4 r N 2 2 uðrð ÞÞ and the existence of nontrivial entire solution of the limiting problem (see [4,8,7,9]) Du 1⁄4 mu jxj þ juj 2 u; xAR ; u-0; jxj-N; 8< : ð3Þ”