Download Amenability by Paterson A.L.T. PDF

By Paterson A.L.T.

ISBN-10: 0821809857

ISBN-13: 9780821809853

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Let us find the relation between the homogenized matrices AO and Co. Consider the Dirichlet problems h h \1ep. A"\1u" dx = \1ep. AO\1uo dx = h h f(x)ep(x) dx, Vep E HJ(Q); u" E HJ(Q); f(x)ep(x) dx, Vep E HJ(Q); UO E HJ(Q); where f E L 2 (Q), Q = {x E lRm, variables x = uy, we obtain h h Ixl :::; \1 yep. C"\1 yU" dy = I} is the unit ball. After changing the h f(uy)ep(y) dy , \1yep. uAou- 1\1yuOdy = hf(uy)ep(Y)dY , Vep(y) E HJ(Q) . 6. 61) where 0' is the rotation by 7r / 2 in the plane of the coordinates Xi, X j , i f:j, i, j = 1, ...

In analogy with the methods of nonlinear mechanics (as shown by Bachvalov [1], [2], SanchezPalencia [1], and many other authors concerned with applications), it is natural to seek the first approximation in the form where Y = c 1x, and the function Ul(X, y) is periodic with respect to y. In the sequel we shall have to calculate an explicit expression for A[UI = div (A(ClX)\7u~). As a preparation, it will be useful to find out how the operator A[ acts on composite functions of the form

Assume that Al is the conductivity tensor of an anisotropic monocrystal. Then the formula A2 = u- l Alu = (Al)-l det Al defines the conductivity tensor of the monocrystal rotated by the angle 7r /2. 59) is satisfied with k = det AI. This allows us to calculate the homogenized conductivity tensor for the simplest periodic polycrystal that is formed by the cells 01, 02, 03, 04, occupied by monocrystals with alternating orientation of the principal axes. Let AI, A2 denote the eigenvalues of the tensor AI; then the corresponding homogenized tensor has the form VAlA2 I.

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