Synopses & Reviews
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.
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.
I. Bakelman was an expert in the study of nonlinear elliptic partial differential equations by methods of differential and convex geometry. In Russia he is also recognized as a reformer of mathematical education at both school and university levels. This book represents much of Bakelman's work of the last ten years (until his death in 1992). Much of his work was devoted to boundary value problems for mean curvature and Monge-Ampere equations in more than two variables and their generalizations. The book is suitable as a text book and reference work for graduate students and scientists (mathematicians but also physicists) working in the areas of convex functions and bodies, global geometric problems and nonlinear elliptic boundary value problems.