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AuthorBakelman, Ilya J. author
TitleConvex Analysis and Nonlinear Geometric Elliptic Equations [electronic resource] / by Ilya J. Bakelman
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1994
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Descript XXI, 510 p. online resource


Investigations in modem nonlinear analysis rely on ideas, methods and probยญ lems from various fields of mathematics, mechanics, physics and other applied sciences. In the second half of the twentieth century many prominent, exยญ 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 comยญ 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 probยญ 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. Elements of Convex Analysis -- 1. Convex Bodies and Hypersurfaces -- 2. Mixed Volumes. Minkowski Problem. Selected Global Problems in Geometric Partial Differential Equations -- II. Geometric Theory of Elliptic Solutions of Monge-Ampere Equations -- 3. Generalized Solutions of N-Dimensional Monge-Ampere Equations -- 4. Variational Problems and Generalized Elliptic Solutions of Monge-Ampere Equations -- 5. Non-Compact Problems for Elliptic Solutions of Monge-Ampere Equations -- 6. Smooth Elliptic Solutions of Monge-Ampere Equations -- III. Geometric Methods in Elliptic Equations of Second Order. Applications to Calculus of Variations, Differential Geometry and Applied Mathematics. -- 7. Geometric Concepts and Methods in Nonlinear Elliptic Euler-Lagrange Equations -- 8. The Geometric Maximum Principle for General Non-Divergent Quasilinear Elliptic Equations

Mathematics Mathematical analysis Analysis (Mathematics) Differential geometry Physics Mathematics Analysis Differential Geometry Mathematical Methods in Physics Numerical and Computational Physics


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