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Tham khảo tài liệu 'wave propagation 2011 part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | A Volume Integral Equation Method for the Direct Inverse Problem in Elastic Wave Scattering Phenomena 19 Note that Fr in Eq. 106 is the Rayleigh function given by FK er r 2e2- 2-4e27p where r ỵ ri2 T ực - cr cui2 Lemma 2 For fi e L2 R and n e C B Ui x3 gij x3 y3 r v fj y3 dy3 L2 R Jr Proof First fix i and j and define Vi X3 9ij x3 y3 r rf fj y3 dy3 7r Then the following is obtained by means of the Schwarz inequality v 3 gij 3 y3 r ĩ fj y3 2dy3 .JR_ _ . lỡp- 313 2 3 77 I j ỉ 3 2dỉ 3 _ R_ _ _ - 1 2 1 1 2 lổíj 3 ỉ 3 ír ĩ dỉ 3 _ R_ _ Ml 1 1 2 where Ml sup 3 eR_ _ y3 r Ti dy3 1 2 As a result the following is obtained I i ar3 2da 3 M2 I giAx3 y3 r ri fj .y3 2dy3 dx3 JR_ _ R_ _ JR_ _ M12M2 j 2 R where M2 sup gij X3 y3Ar V dX3 yaSR-l- JR_ _ Equation 113 concludes the proof. Theorem 1 The operator J J with the domain D - i is self-adjoint. Proof It is sufficient to prove that Vfi e T2 R there exist Ii . u e Df Ff satisfying 107 108 109 110 111 112 113 114 20 Wave Propagation in Materials for Modern Applications 3 fi x3 115 116 where p is a positive real number. This fact is based on the results of a previous study Theorem 3.1 Berthier 1982 . For the construction of i define 117 where n is chosen such that n2 ip. Note that n e C B. The following equation 118 yields Eq. 115 where rz R is the Schwartz space. During the derivation of Eq. 118 the following equation stfji ipy o5ij pi x3 gjk .x3 y3 r ri dx3 JR L J 119 is based on the following properties of gij x3 K3 r n at x3 y3 gik y3 e 2 3 r 7 l H9jk X3 y3 77 Ss y3-e gik .V3 - e y3 r Tf dik 120 In addition the following is obtained ijUj res 3 I g3k x3 y3 tr ri fk ys dy3 R_ _ ijgjk x3 y3 r n fk .y3 dy3 J R_ _ 121 The order of the integral and differential operators of the properties of function gij are changed such that V feCz a Pi 6 R A Volume Integral Equation Method for the Direct Inverse Problem in Elastic Wave Scattering Phenomena 21 lim K AaZaC -3. VcL-. Mte - 0 122 r 3- v for an arbitrary positive integer n. According to
