ESTIMATES FOR SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS CONNECTED WITH PROBLEMS OF GEOMETRY "IN THE LARGE"

Abstract

The following questions are presented in this paper: A geometric method for obtaining two-sided estimates for general quasilinear elliptic equations and its applications to problems of the calculus of variations and the problem of recovering a hypersurface from its mean curvature in spaces of constant curvature. Estimates of the modulus of the gradient for a hypersurface with boundary in a Riemannian space by means of its mean curvature and the metric tensor of the space. Estimates of the modulus of the gradient of a hypersurface depending on the distance of a point from the boundary and its mean curvature in Euclidean space. Estimates of these three types are of independent interest and play a fundamental role in the proofs of existence theorems for a hypersurface with prescribed mean curvature in Riemannian spaces. Bibliography: 3 items.