Convex Analysis and Nonlinear Geometric Elliptic Equations
Convex Analysis and Nonlinear Geometric Elliptic Equations
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DOI：
10.1007/978-3-642-69881-1
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发表时间：
1994-12
期刊：
影响因子：
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通讯作者：
I. Bakelman
中科院分区：
文献类型：
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作者：
I. Bakelman
现代非线性分析的研究依赖于数学、力学、物理学和其他应用科学各个领域的思想、方法和问题。在世纪后半叶，非线性分析中的许多突出的、典型的问题受到了深入的研究和检验。微分几何、拓扑学、微分方程、泛函分析等数学研究领域的统一思想和方法被成功地应用于非线性分析中复杂问题的完整解决。在一本书中不可能包含所有与非线性分析有关的概念、思想、方法和结果。因此，我们将限制自己在本专着非线性椭圆边值问题以及全球的几何问题。为了研究这些问题，我们提供了一个基本工具：凸体和超曲面理论。在这本书中，我们系统地介绍了从世纪后半叶到现在所获得的一系列具有中心意义的结果。特别注意在非线性分析中的各个部门之间的深刻的相互联系。由于微分方程的椭圆性与其解的局部凸性和整体凸性密切相关，因此凸函数和凸体理论起着至关重要的作用。因此，有必要有足够大量的材料致力于理论的凸体和功能及其连接与偏微分方程。
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.