Geometric problems in quasilinear elliptic equations

In their survey reports A. D. Aleksandrov and A. V. Pogorelov [1] and N. V. Efimov [2] give a detailed account of the deep relationships between the theory of surfaces and the theory of partial differential equations; they also highlight the main results and research problems on the boundary of geometry and analysis connected with Gaussian curvature of surfaces and its various generalizations. More recently problems on the boundary of geometry and the theory of quasilinear equations have been intensively investigated in various directions. In this article we shall be concerned with just two closely related problems from among the diverse questions that have been studied in this field: the geometric methods of estimating the solution to the Dirichlet problem for a quasilinear elliptic equation, and the construction of a non-parametric hypersurface with given mean curvature in Riemannian space. Even for the case of zero mean curvature (minimal surfaces) these questions present considerable difficulties. We shall not pay special attention to the minimal surface situation, but we draw attention to the very full presentation of the research in this field in the surveys by R. Osserman [3], [40] and J. C. C. Nitsche [4]. This article is a considerably enlarged account of the lectures given by the author in the Second and Third All-Union symposia on geometry in the large in 1967 and 1969`[15], [35].