engleski

On some projections of the homogenised coefficients in stationary diffusion equation

We consider the stationary diffusion equation $$ -{\rm div}(A\nabla u)=f\, , $$ where $A$ is conductivity matrix function corresponding to a mixture of two phases (possibly anisotropic). The mathematical theory of homogenisation introduces the notion of composite materials, as fine mixture limits of different phases. Given the local proportion $\theta$ of the first material, the set of all possible composite materials is denoted by ${\cal K}(\theta)$. For some applications in optimal shape design problems, optimisation over the set ${\cal K}(\theta)$ (which is in fact unknown in some situations), could be changed to optimisation over a simpler set, knowing the characterisation of the set ${\cal K}(\theta)e$ for any vector $e$. This set actually corresponds to the first column of effective conductivity matrix. We address the question of describing two columns of these matrices, or more precisely the set $\{(A e, A f):A \in{\cal K}(\theta)\}$, for two vectors $e$ and $f$. Thanks to geometric interpretation, it is possible to solve the problem, although the solution involves tedious computations. Some interesting properties of the set under consideration are proved.

stationary diffusion equation; homogenization

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