Convexity of domain functionals
Convexity of domain functionals
复制标题
域泛函的凸性
DOI：
10.1007/bf02825640
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发表时间：
1952
期刊：
影响因子：
--
通讯作者：
M. Schiffer
中科院分区：
文献类型：
--
作者：
P. Garabedian;M. Schiffer
翻译后摘要：一个严格的理论发展变化的域函数在3维空间以及在平面上。讨论了空间中经典的Hadamard变分公式，并将所谓的内变分方法推广到三维空间。利用依赖于小参数ε的微分映射定义了三维区域D的内部变分。由D的这种变化引起的绿色函数、Neumann函数和本征值的依Epsilon的一阶位移，通过无穷小映射将所有变化的量返回到原始D来严格计算。由于证明了变域函数可以展开为埃伯利的幂，因此采用摄动法计算这些函数的二次变分。第二变分表达式获得的能力，虚拟质量，和对应于各种特定的方式，其中D可以被转移的特征值。变分理论是专门的情况下，两个独立的变量，以显示存在的涡面在轴对称，无旋流动的不可压缩流体。在没有任何对称性的三维空间中，涡面的外部特征被概略地描绘出来。在振动膜的本征函数和本征值的研究中，第二变分被用来表明，在依赖于适当参数的区域的某些共形映射下，该区域的主频率的平方反比成为该参数的凸函数。
Abstract : A rigorous theory is developed for variation of domain functions in a space of 3 dimensions as well as in the plane. The classical Hadamard variational formulas in space are discussed, and the so-called interior variational method is generalized to 3 dimensions. Interior variations of a 3- dimensional domain D are defined by means of differential mappings of D which depend on a small parameter Epsilon. The first-order shifts in terms of Epsilon of the Green's function, Neumann's function, and eigen-values, which result from this variation of D, are calculated rigorously by referring all varied quantities back to the original D through the infinitesimal mappings. Since proof is possible that the varied domain functions can be expanded in powers of Epsilon, the perturbation method is employed to calculate the second variations of these functions. Second variation expressions are obtained for the capacity, virtual mass, and eigenvalues corresponding to various particular ways in which D can be shifted. The variational theory is specialized to the case of 2 independent variables to show the existence of vortex sheets in axially symmetric, irrotational flow of an incompressible fluid. An external characterization of vortex sheets in 3-dimensional space without symmetry of any kind is sketched heuristically. In a study of the eigen functions and eigenvalues of the vibrating membrane, the second variation is used to show that under certain conformal mappings of a domain, which depend on a suitable parameter, the inverse square of the principal frequency of the domain becomes a convex function of the parameter.