By Makhmudov O. I.

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Extra resources for A Cauchy Problem for the System of Elasticity Equations

Example text

T,-0} = O. _j . YJ (~) = E ~ _ z3k ' k k aj G C 3 , Z3k 9 ~:)J( V ) . 113) 52 3. 9. 113) such that w1 D M . y~ - T M . 114) W2 - Z~ D M . y 2 - T M . yx = u2(~) ~ 7{ 3 . Proof. For 9 E {L,T,O} and z C ~2, \ {VL,VT,~'o, vs} such that T,(z) # L,,, we define the 3 • 3 matrices t,(z) by t, (z) = d m -1 (T, ( z ) ) t m , (z). 65)). 9 for 9 = 0, z = ~o cos:~, To(z) = ~o cosr sin . . 5. The Functional Equation for the Spectral Function 53 because the third column of the m a t r i c e s {L,T are zero.

3. Some Properties of the Operators D M , 47 TM F1 -vo ' -~ - VL-"~ . . 0 L . . . J . . . . . l . . . . . VT J___>. Vo F i g . 6. The contour F1 F0 t vL UT F i g . 7 Deformation of the contour Fo onto the contour 0~2+ 48 3. The Spectral Function 0 ~- v o cos -v 0 \ \ \ \ / 0 -- v 0 0 J --u0 Fig. 8. 3. 98) For H(~), the contour F0 id deformed onto the contour c9/2o (Fig. 7). The deformation of c o s ~ + sin~@(() is pictured on Fig. 7. If z E /2+, the residue is 1 Do(~, z)tmo(z) . v - ( _ To(z)tmo(z).

59) ~-~lyl if t~l = -0. In all cases, h~ ~ S(N-), hence no interface-wave equation does exist in this case. 60) or equivalently dm(ei~ ~ = 0. 3. 8), is an holomorphic function in the domain 38 3. 62) v = c \ {z c R, nzl ~ ~L = 1}. Its determinant is get(din(z)) = A(z) - Q ~L~T [1_4#z2+4#2z2Q+p~o] and the inverse matrix is din(z)-1 = A DB AAC] 1 D C 1 - AB . 64) The matrix din(z) is well defined for z such that 4. (z) ~ 0, * e {L, T, 0}, or equivalently for z ~ {+1 = +YL, • • The functions ~L,T,O(~) being even functions in ~ E R, A(~) is also a even function.

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A Cauchy Problem for the System of Elasticity Equations by Makhmudov O. I.


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