Investigations in modem nonlinear analysis rely on ideas, methods and prob lems from various fields of mathematics, mechanics, physics and other applied sciences. In the second half of the twentieth century many prominent, ex emplary problems in nonlinear analysis were subject to intensive study and examination. The united ideas and methods of differential geometry, topology, differential equations and functional analysis as well as other areas of research in mathematics were successfully applied towards the complete solution of com plex problems in nonlinear analysis. It is not possible to encompass in the scope of one book all concepts, ideas, methods and results related to nonlinear analysis. Therefore, we shall restrict ourselves in this monograph to nonlinear elliptic boundary value problems as well as global geometric problems. In order that we may examine these prob lems, we are provided with a fundamental vehicle: The theory of convex bodies and hypersurfaces. In this book we systematically present a series of centrally significant results obtained in the second half of the twentieth century up to the present time. Particular attention is given to profound interconnections between various divisions in nonlinear analysis. The theory of convex functions and bodies plays a crucial role because the ellipticity of differential equations is closely connected with the local and global convexity properties of their solutions. Therefore it is necessary to have a sufficiently large amount of material devoted to the theory of convex bodies and functions and their connections with partial differential equations. Front Matter....Pages I-XXI Front Matter....Pages 1-1 Convex Bodies and Hypersurfaces....Pages 3-53 Mixed Volumes. Minkowski Problem. Selected Global Problems in Geometric Partial Differential Equations....Pages 54-107 Front Matter....Pages 109-111 Generalized Solutions of N -Dimensional Monge-Ampere Equations....Pages 113-181 Variational Problems and Generalized Elliptic Solutions of Monge-Ampere Equations....Pages 182-203 Non-Compact Problems for Elliptic Solutions of Monge-Ampere Equations....Pages 204-226 Smooth Elliptic Solutions of Monge-Ampere Equations....Pages 226-283 Front Matter....Pages 285-285 Geometric Concepts and Methods in Nonlinear Elliptic Euler-Lagrange Equations....Pages 287-339 The Geometric Maximum Principle for General Non-Divergent Quasilinear Elliptic Equations....Pages 339-495 Back Matter....Pages 497-510 This book is suitable as a graduate text and reference work in the areas of convex functions and bodies, global geometric problems, and nonlinear elliptic boundary value problems with special emphasis on Monge-Ampere equations. The theory of convex functions and bodies is presented first so that it can be used to study the other areas. In fact, the author makes a point of emphasizing the interrelationship of all the areas mentioned above. This enables the reader to obtain a working knowledge of the material. Specific topics of the book include the Minkowski problem, mixed volumes of convex bodies, the Brunn-Minkowski inequalities, geometric maximum principles, the normal mapping of convex hypersurfaces, the R-curvature of convex functions.
