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Mixed volumes and connected variational problems. (English) Zbl 0701.49044

Miniconference on operators in analysis, Proc. Miniconf., Sydney/Australia 1989, Proc. Cent. Math. Anal. Aust. Natl. Univ. 24, 109-114 (1990).
Summary: [For the entire collection see Zbl 0699.00025.]
The recent achievements concerning m-curvature equations [see, e.g., the author, Sov. Math. Dokl. 37, No.2, 322-325 (1988); translation from Dokl. Akad. Nauk SSSR 299, No.1, 35-38 (1988); Mat. Sb. 180, No.7, 867-887 (Russian) (1989; Zbl 0695.35074); L. Caffarelli, L. Nirenberg and J. Spruck, Commun. Pure Appl. Math. 41, No.1, 47-70 (1988; Zbl 0672.35028); N. S. Trudinger, “A priori bounds for solutions of prescribed curvature equations” (Preprint CMA 1989)] gave a new point of view to the geometrical theory of mixed volumes of convex bodies which was developed by A. D. Aleksandrov in 1938-1940. A principal goal of this paper is to pose corresponding variational problems correctly and to formulate sufficient conditions for the existence of minimizers.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49J40 Variational inequalities