A unified theory of elastic degradation and damage based on by Carol I.

By Carol I.

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2)+ (U x IRq). 39 is part of the results of [17]. It can be proved by first verifying the corresponding (y,7])-wise result on the half-axis, cf. 37, and finally applying an argument on representing symbols as asymptotic sums of symbols that are homogeneous for large IT, 7]1. 34. Concerning more details of the proof cf. 36]. 27) for arbitrary p(y, t, 7], T) E Sfr(U x 1R+ x IRq+1) which are independent of t for large t and g(y, 7]) E R'//(U x IRq). 41 (i) Show that for a symbolp(y,t,7),T) E S-OO(Ux1R+xIRq+1) which is independent oft for large t, we have op+(p)(y, 7)) E Rcoo,0(U x IRq).

L, d) - ~. L)+. Proof. Since the entries of A, except for the upper left corner, are of finite rank it suffices to consider the case j_ = j+ = O. ); then we have d = O. ). Then op+(a) op+(p) = 1 + G r for an element G r E rO(1R+), d. 19. -'(1R+). L, 0) - ~. At the same time we see that op+(p) is a parametrix. 30 concerns the upper left corner An of A, since the other entries are of finite rank. 32 If A is elliptic, kernel and cokernel are independent of s > max(/J, d) - ~. More precisely, we have ker A c S(IR+) EB Cj-, and there is a finite-dimensional subspace W c S(IR+) EB Cj+ such that imA n W = {O} and im A + W = HS-I-'(IR+) EB Cj+ for all s > max(/J, d) - ~.

The next point is to characterise the behaviour of operators op+(a) under compositions (say, as maps in S (1R+) ). 20) g(t, t')u(t')dt', o g(t, t') E S(1R+ x 1R+) (:= S(JR x JR) IlR+xi)' is called a Green's operator of type O. The space of all these operators is denoted by rO(1R+). Moreover, rd(1R+) for dEN is defined to be the set of all operators d dj G= LGjdtj j=O for arbitrary Gj E r O(1R+); these are called Green's operators of type d. Let L2(JR+ x JR+) = {uIIR+XIR+: u E L2(JR2)} be endowed with the standard scalar product.

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