计算数学 1979, 1(3) 199-208 DOI:     ISSN: 0254-7791 CN: 11-2125/O1

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PubMed

组合流形上的椭圆方程与组合弹性结构

冯康,

中国科学院计算中心,

摘要

在偏微分方程的通常理论中,人们讨论空间 R~n 中的均匀维数的区域Ω,在其上规定了微分方程,在 n—1维的边界Ω上则规定边界条件,这种边界条件通常在性质上要比微分方程本身简单些.很自然地期望把这样的问题框架推广到不均匀维数的区域,它是由不同维数的片块适当地连结而成,在每一片块上规定了微分方程,它们是通过交接关系相互耦合着的,最终还可以在剩下的边界上规定边界条件.许多工程问题中的数学性状

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ELLIPTIC EQUATIONS ON COMPOSITE MANIFOLD AND COMPOSITE ELASTIC STRUCTURES

Feng Kang (Computing Center,Chinese Academy of Sciences)

Abstract:

In the usual theory of partial differential equations,one considers in R~n a domain of homogeneous dimension n,on which the differential equations are defined,and the boundary of dimension n-1,on which the boundary conditions are prescribed.It is natural and desirable to extend such a setting to the case where the domain is of heterogeneous dimensions,i.e.,it consists of a finite number of pieces of different dimensions,suitably connected together,with differential equations on each piece cou- pled through incidence relations and eventually supplemented by boundary conditions on the remaining boundaries.This is actually the mathematical situation in many engineering problems,and the great geometrical and analytical complexities herein encountered should be trickled in a proper mathematical way. In section 1 we define a composite manifold as a closed complex of cells of dif- ferent dimensions,each cell being a connected smooth orientable Riemannian mani- fold with piecewisc smooth boundary,i.e.,the boundary consists of a finite number of cells of 1 dimension lower.In a composite manifold,a subcomplex is defined as a composite structure Ω when its closure coincides with the underlying composite manifold and certain strong connectedness property is satisfied;and another subcom- plex is suitably defined as the boundary Ω of the composite structure Ω.The cou- pled differential equations will be defined on each cell of Ω and the boundary condi- tions will be prescribed on each cell of Ω. In section,2,a product of Sobolev spaces of order 1 corresponding to all the cells of a composite structure is introduced and a closed subspacc is specified by cer- tain link condition.For this subspace,some injection theorems in sense of Sobolev can be established and a standard elliptic variational problem is introduced and leads to a system of coupled Poissou equations on a conlpositc manifold in R~n which is a natural extension of the classical Poisson equation and is applicable to the heat transfer,diffusion on complex structures. In section 3,considerations analogous to those of section 2 lead to another product of Sobolev spaces for a composite structure in R~3 and the corresponding injection theorems.This gives a precise mathematical foundation for the composite elastic structures. Differential equations on composite manifolds seem to have wide applications. Some relevent theoretial problems worthy of further study are indicated.

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