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Multiple solution of H-systems and Rellich’s conjecture. (English) Zbl 0537.49022
Cet article contient les démonstrations des résultats annoncés par les auteurs [C. R. Acad. Sci., Sér. I 295, 615-618 (1982; Zbl 0505.49019)]. Pour surmonter la difficulté essentielle - défaut de campacité - les auteurs font appel à une méthode semistable à celle utilisèe par Th. Aubin pour le problème de Yamabe [J. Math. Pures Appl. IX. Sér. 55, 269-296 (1976; Zbl 0336.53033)] et reprise ensuite par H. Brezis et L. Nirenberg [Commun. Pure Appl. Math. 36, 437-478 (1983)]. La meilleure constante dans l’inégalité isopérimétrique joue un rôle important.

MSC:
49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
52A40 Inequalities and extremum problems involving convexity in convex geometry
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