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# The limiting absorption principle for the elastic equations

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### The limiting absorption principle for the elastic equations

1. 1. The Limiting Absorption Principle for theElasticity Operator in HomogeneousAnisotropic Media Alejandro Domínguez-Torres1AbstractThe spectral theory is studied and the principle of limiting absorption is proved for the elasticity operatorderived from the wave equations in infinite elastic homogeneous anisotropic media.The Wave Equation for Elastic Homogeneous Anisotropic MediaFor a medium obeying the generalized Hooke´s Law, the linear stress-strain relations in rectangularCartesian coordinates xi ; i  1,2,3; are [Achenbach p. 52 and Fedorov pp. 8-9](1)  ij  Cijlm lm ; i , j , l , m  1,2,3.Here  ij is the stress tensor,  lm is the strain tensor, and C ijlm are the elastic coefficients satisfying thesymmetry conditions [Achenbach p. 52 and Fedorov pp. 12-15]:(2) Cijlm  C jilm  Cijml  Clmij .In this way, only 21 independent constants are involved. Moreover, the convention summation forrepeated suffixes is assumed.The strain tensor may be expressed in terms of the displacement vector ui ; i  1,2,3; by 1  u u (3)  ij   i  j  . 2  x j xi   In the absence of body forces and considering that the density of the medium is constant and equal toone, the equation of motions are [Fedorov pp. 85-86] 2ui 2um(4)  C ijlm . t 2 x j xlIt is also assumed that the constants C ijlm have numerical values such that the strain energy W ispositive definite for symmetric stress components  ij   ji , where:1 This research paper is derived from the thesis dissertation (Spectral Theory for Elasticity Equations) presented th(August 24 , 1989) by the author for obtaining the degree of Master of Science (Physics) at the School of Sciencesin the Universidad Nacional Autónoma de México. This paper never was published. Page 1 de 14
2. 2. 1(5) W  Cijlm ij  lm . 2Solutions of Eq. (4) are the monochromatic plane waves of the form(6) u  x , t   Aei  px t  ;where A   A1 , A2 , A3   is a constant vector, p   p1 , p2 , p3   0 , p  p1  p2  p3  1 , and 3 3 2 2 2 2p  x  p1 x1  p2 x2  p3 x3  .From Eqs. (4) and (6) if follows the relation(7) C ijlm pj pl  2 im  um  0 .Let   p  be a 3  3 matrix defined by(8)   p   C ijlm p j pl .Therefore, Eq. (7) may be expressed as(9)   p   I  u  0;    2 .  Eq. (9) is known in literature as “Christoffel Equation”. If u 3 0 , then Eq. (9) is equivalent to(10)    , p     p   I  0 .From the above assumption and derivations, the following Lemma is immediate.Lemma 1. Matrix   p  holds the following properties: i.   p  is symmetric for u 3 0 ; ii.   p  is positive definite; iii. All the roots of    , p   0 are positive.The Elasticity Operator and Its Spectral FamilyLet L2,3  3 , 3  be the linear space of all 3  1 column matrix defined on 3 and 3 -valued, such that ifu L2,3  3 , 3  then u is Lebesgue measurable and square integrable. For x 3 and u L2,3  3 , 3 ,let L2,3  3 , 3  be equipped with the norm 3 u  x  dx    u x 2  2 2(11) u  n dx . 3 3 n 1Since Eq. (11) satisfies the parallelogram law, then for f , g L2,3  3 , 3  the scalar product is defined by Page 2 de 14
3. 3. 3(12)  f , g    f  x   g  x  dx    fn  x gn  x  dx  3 3 n 1Here * denotes the complex conjugate operation. The linear space L2,3  3 , 3  equipped with scalarproduct (12) defines a Hilbert space which is again denoted as L2,3  3 , 3 .For u L2,3  3 , 3  , define the Fourier Transform operator as F : L2,3  3 , 3 L  2,3 3 , 3 ;(13) 1  Fu  p   u  p   ˆ s  lim  u  x e  ip x dx.  2  3 R  2 2 x R 2Here s  lim denotes the limit in the strong topology of L2,3  3 , 3  . In a similar way, the inverse FourierTransform is defined as F 1 : L2,3  3 , 3 L  2,3 3 , 3 ; F u   x  (14) 1  u  p e 1 ip x ˆ s  lim ˆ dp.  2  3 R  2 2 p R 2 u  2uLemma 2. If u, ,  L2,3  3 , 3  , where derivatives are taken in distribution sense, then the x j xi x jFourier Transform defined by Eq. (13) holds the following properties ux   ux dx   u  p  dp  u  p  2 2 2 2(15) ˆ (Parseval Identity); 3 3   u   u (16) F  x   p     x   ip j  Fu   ip ju; j  1,2,3 ;  ˆ  j   j     2u    2u (17) F  x x   p     x x    pi p j  Fu    pi p j u; i , j  1,2,3 .  ˆ  i j   i j On the other hand, let    be the operator defined in L2,3  3 , 3  by(18)  D       f  L2,3  3 , 3  /    f  L  2,3 3 , 3 ;     f   p     p  f  p  .Lemma 3. Operator    defined in Eq. (18) is a self-adjoint operator with respect to the (usual) scalarproduct of L2,3  3 , 3 . Page 3 de 14
4. 4. Proof. Since the set D3  3 , 3   C  ,  C  ,  C   0 3  0 3  0 3 ,   D   it follows that D    is a dense set in L2,3  3 , 3  . The proof will be completed if it is proved that        and       , from it follows that        .From Lemma 1 it follows that   p  is symmetric and positive definite, then for all u, v  D         u,v      p  u  p  v  p  dp   u  p    p  v  p  dp   u  p    p v  p  dp †  † †  †    3 3 3   u†  p    p  v  p   dp   u†  p    p  v  p   dp  u ,    v  .       3 3This result proves that        .For proving that        , let v D   ; i.e., v L2,3    3 , 3  and     u , v    u ,  , for some L2,3  3 , 3  and for all u  D     . Notice that vector  is equal to   p  v  p  by definition, then itfollows that    u,v    u  p    p  v  p     † dp   u†  p    p  dp  u ,  . 3 3Since D      is dense in L2,3  3 , 3  ;   v   ; i.e., v  D     . This proves that if v D   , then v  D      and    v       v ; this means that        . 2Eqs. (8), (18) and (13) to (17) allows to express the action of “elasticity operator” H  C ijlm as x j xl(19) Hu  F 1   Fu .Of course, the domain of elasticity operator is defined by(20) D  H   u  L2,3   3 , 3  /    u  L  ˆ 2,3 3 , 3  .Obviously D  H   H2  3 , 3  , the Sobolev space of all 3  1 column matrixes defined in L2,3  3 , 3 such that the first and second partial derivatives belong to L 2,3  3 , 3 .Theorem 4. The elasticity operator H defined in L2,3  3 , 3  with domain H  2 3 , 3  is a self-adjointoperator with respect to the usual scalar product in L2,3  3 , 3 .Proof. The proof follws immediately if it is noticed that the Fourier Transform F is a unitary mapping  from L2,3 3 , 3 to itself and that operator    is self-adjoint by Lemma 3.The plane wave solution (6) for Eqs. (4) generates the following problem for H Page 4 de 14
5. 5. (21) H   x     x  .Here       /   0 and for a given constant vector A 3 ,   x  is defined as   x   Ae ip x ; p  1 . 2(22)From Lemma 1, it may be considered three values of    such that(23) 1  2  3 .Moreover, it is to prove that for any   ,   p      p  .Let  n  x  be the function associated to n ; n  1,2,3 ; i.e.,(24)  n  x   An e ipx ; n  1,2,3 .Substituting Eq. (24) into Eq. (21), it is obtained(25)   p   nI  An  0 .  Thus An is the eigenvector corresponding to eigenvalue n . These eigenvectors may be takenorthonormal among them if relationship (23) holds., i.e., Ai  A j  0 unless i   j .Without loss of generality, solutions (24) may be written as 1(26) n  x , p  An eipx ; p  3 0 ; n  1,2,3 . 2  3 2Since n , n  1,2,3 , do not belong to L2,3  3 , 3  , they will be called the “improper eigenfunctions” ofoperator H . However, the spectral properties may be obtained by building, in formal sense, a set of integral transforms of functions f L2,3 3 , 3 with the improper eigenfunctions  f  p   f  x    n   x , p  dx; n  1,2,3 . n (27) 3Lemma 5. For every f L2,3  3 , 3  the following limits exist in the strong topology of L  2,3 3 , 3  f f  x     n   x , p  dx ; n  1,2,3 .   p   sM  n(28)  lim 2 x  M2 3Here x   xk . 2 2 k 1Proof. From relation (26) it follows that Page 5 de 14
6. 6. 3   f  x     n   x , p  dx   1 1(29)  2 f  x   Ane  ipx dx     f j  x  e  ipx dx  An . j 1   2  2  j 2  2 3 3   2 2 2 x M2 x M x M2Since A n are constants for j  1,2,3 ; and fj L2 j  , then from the Plancharel’s Theorem [Bochner  3 ,and Chandrasekaran pp.112-113] it follows that the integral (29) converges in the norm of L  ,  and 2 3the limit belongs to the same linear space.For f L2,3  3 , 3  , Lemma 5 associates to it a vector  f 1  , f 2 , f 3 , where f n L2  3 ,  , n  1,2,3 .Moreover, the following result holds.Lemma 6. For each f L2,3  3 , 3  , it follows the Parseval Identity 3  fn 2 2(30) f . n 1 L2  3 , Here f n , n  1,2,3 ; are defined by expression (28).Proof. From relations (26) and (27) it follows that(31) ˆ  ˆ ˆ ˆ f n  f  A1 , f  A2 , f  A3  fA ;  ˆwhere A is a 3  3 matrix whose columns are formed by An , n  1,2,3 ; and f  Ff . Moreover, A is anorthogonal matrix since its columns are orthonormal vector, thus A1  A† .On the other hand,   3 3 3 3   fn, fn    f n  p  dp    f  p  f  p  fˆ  p  A 2 2 2 2 2 fn n  dp  dp n 1 2 L  3 ,  n 1 2 L 3 ,  n 1 3 3 n 1 3 3  fˆ  p   f x 2 2 2  dp  dx  f . 3 3The previous to last equality follows from Parseval’s identity for Fourier Transforms [Bochner andChandrasekaran p. 113].Define the following linear operator  : L2,3   L  3 , 3 2,3 3 , 3 (32) f  f   f , f , f  . 1 2 3Therefore, from Lemma 6 the next identity holds(33) f  f , f L2,3  3 , 3 .This means that Page 6 de 14
7. 7. (34)   I .Therefore  is a partial isometry and   P is the orthogonal projection of  L2,3   3 , 3   , therange of  .(35)  f  f  A1 , f  A2 , f  A3 ; f  L2,3  ˆ ˆ ˆ  3 , 3 . ˆThis means that the components of f are the projections of f on each An ; n  1,2,3 . Therefore, thevector base formed by An ; n  1,2,3 ; vector f may be expressed as   f   f  An An ; f  L2,3   3(36) ˆ 3 , 3 n 1From this interpretation and from Plancherel´s Theorem [Bochner and Chandrasekaran pp.112-113] it isobtained the following result.Lemma 7. For each f L2,3  3 , 3  , the following limits exist on For each L  2,3 3 , 3 . 3  f  s  lim   f  p   x , p  dp . n(37) n M  p M2 n 1 2Here fn ; n  1,2,3 ; are the components of f .From Eq. (37), it may be seen that(38)  f  F 1 Af .Theorem 8. The operator  defined in (32) is a unitary linear operator; i.e.,(39)   I   .Proof. The first equality of (39) follows from (34). On the other hand, since A is orthonormal, then it isthe matrix of a bijective linear transformation. Thus, if f L2,3 3 , 3 then g  Af L2,3 3 , 3 .    Moreover, since F is a unitary linear transformation from L2,3  3 , 3  to itself, then F 1 g L2,3  3 , 3 .Therefore, if h   f ,(40)  f     f   h  h† A . ˆFrom Eq. (38) it follows that  h†   f    FF 1 Af    Af   fA† . † † †(41) ˆ  Combination of Eqs. (40) and (41) gives  f  fA† A  fI  f . Page 7 de 14
8. 8. Eq. (39) is the eigenfunction expansion in abstract form; i.e., for each, it follows the next representation(42) f   f .This eigenfunction expansion may be used to obtain a representation for the elasticity operator H .Theorem 9. The operator  whose action is given by Eqs. (32), (35), and (36) defines a spectralrepresentation for H in the sense(43)   Hf  1 f 1 , 2 f 2 , 3 f 3 ; f  D  H  .Proof. Since D3  3 , 3  C   0 3 ,  C   0 3 ,  C   0 3 ,  is a dense set on L2,3  3 , 3  , thus iff  D  H  and g  D3   , then 3  Hf , g   Hf ,  g   F   Ff ,F Ag      Ff ,FF Ag     Ff , Ag   1 1 1(44)   A   Ff , g    A   IFf , g    A   AA Ff , g    A   Af , g  . † † † † †Matrix A†    A is a diagonal matrix whose components are the eigenvalues of    :(45)   A†    A  i    ij     .Thus Eq. (44) becomes(46)  Hf , g    A   Af , g      f , g  . †Eq. (43) follows immediately from Eq (46) since D3  3 , 3  is a dense set on L  2,3 3 , 3  .Notice that Eq. (43) implies that(47) H   .This means that operator  diagonalizes operator H .Let Pn   be the orthogonal projection on the corresponding eigenspace of n   ; n  1,2,3 . Then Pn   isgiven by [Kato] 1 dz(48) Pn  p   p   p   z ; n  1,2,3 . 2 i Cn  Here C n  p  is a simple closed curve around n  p  ; n  1,2,3 . From Lemma 6, Lemma 7, Theorem 8, andTheorem 9, the following corollary is proved.Corollary 10. Operators    , H , and Pn   hold the following properties on L2,3  3 , 3  Page 8 de 14
9. 9. 3(49)  P  p   I; p  n 1 n 3 0 ; 3(50)   p   A n  p  Pn  p  A† ; p  3 0 ; n 1 3 3(51) H  F 1 A n  p  Pn  p  A†    n  p  Pn  p  ; p  3 0 ; n 1 n 1 3(52)  P  I; P n 1 n n  Pn   .In order to find out more properties of the spectrum of the elasticity operator, return to Christoffelequation (9) and its corresponding associated equation (10). For a fixed  , Eq. (10) defines a two-dimensional surface on the vector spaced defined by vector p . Since the second term in Eq. (10) is pproportional to    2 and p   p1 , p2 , p3   3 0 , with p  p1  p2  p3  1 ; let q  1 be the 2 2 2 2  2 2 1“slowness vector” [Achenbach p. 126], then it follows that q  . In this way and in virtue of Eq. (8), Eq. (9) becomes   p   I  u  C ijlm pl pm   im  u  C ijlm ql qm   im  u   C ijlmql qm   im  u  0 .        This last equation in turn implies that(53)  1, q     q   I  0 .Eq. (53) describes an inverse velocity two-dimensional surface known in literature as “slowness surface”. 1Notice that this surface is independent of    2 and only depends on the direction of propagationvector q .An alternative way of describing the slowness surface is by noticing that Eq. (53) is a polynomial of thirddegree that in turn may be factorized in a unique way as(54)  1, q   Q1 1, q  Q2 1, q  Q3 1, q   0 .The locus described by each Qn 1, q  ; n  1,2,3 ; is given by the following set(55) Sn    3 / n    1; n  1,2,3 .Therefore, the slowness surface is given by 3(56) S Sn . n 1This description for the slowness surface permits defining a system of generalized radial coordinates on 3 : Page 9 de 14
10. 10. : 3 0    , Sn  ;(57)  1      :  n 2   , 1  ; n  1,2,3.  n 2     Moreover(58)  1   ,n   n ; n  1,2,3 . 3Let dS be the two-dimensional measurable infinitesimal surface on the unitary sphere in , then(59) d   2d dn .Here   dn  n dS  n 3(60)  .   n  2Obviously Eq. (60) defines a finite measure on . Let L2,3 Sn , 3  be the Hilbert space of all 3 -valued measurable functions taking values on Sn and aresquare integrable with respect to the measure dn . Define the following Hilbert spaces(61)  Hn    L2,3  Sn , 3  / P         ; n  1,2,3 ; n n n n(62) H  H1  H2  H3 ;(63)   L2   , d  , H  .Define the following linear operator U : L2,3  3 , 3   ;(64) 3 U   ,     Pn n   n  ;  1 ,2 ,3  . n 1From Theorem 9 and Corollary 10, it follows the next corollary.Collorary 11. i. Operator U is a unitary operator. ii. Operator UHU 1 :    is the multiplication operator by  :(65) UHU 1  I .For each Borel set   , let     be the characteristic function of for each   , and let E bethe spectra family of operator H , then Corollary 11 implies Page 10 de 14
11. 11. (66) E   f      U 1     Uf    ;(67) UHE   f         Uf .Theorem 12. i. Operator H is absolutely continuous; ii. The spectrum of operator H ,   H  , is   0 . Proof. Let  be a Borel set. From Eq. (66) it follows  E   f , f   U 1  Uf , f     Uf ,Uf    Uf ,Uf H d  . If  0 (the Lebesgue measure of ), then  E   f , f   0 . This proves (i).Since operator H is absolutely continuous, then the singular spectrum of H is equal to zero. Therefore,(68)  H     0 .In this way, the interval   ,0  belongs to the resolvent of operator H .The Limiting Absorption Principle for the Elasticity OperatorFor z    H  , let R  z    H  z  be the resolvent for the elasticity operator H . Let investigate 1when R  z  takes, in some sense, limit values on the positive real axis  when these values areobtained as limits on R  z  as z     for z    z  /  Im  z   0 .Since from Theorem 12     H  , it follows that those limits do not exist in the uniform topology ofthe all bounded operators from L2,3  3 , 3  to itself. However, as it will be seen later, those limits existif R  z  is considered as a function taking values on an optimum topology of linear bounded operators.This result is known in literature as “the limiting absorption principle”.For   , define the following Hilbert space  / 1  x  (69) L2   3 ,    f L   2 3 , 2 2 f  L2  3 ,  .   For f , g L2   3 ,  , define its scalar product as  1  x  2 (70)  f , g L  2  3 ,  f  x  g  x  dx . 3For   , let H  3 ,  be the Hilbert space given by the closure of C   0 3 ,  in the norm Page 11 de 14
12. 12.    F 1  p  Ff ; f C 0  . 1 2 2  3(71) f H  3 , , L2  3 , Here F and F 1 denote the forward Fourier Transform and inverse Fourier Transform operators defined  on L2 3 , , respectively.Theorem 13. For   1 2 ,     0 , and Sn defined by Eq. (55), there is a trace bounded operatorTn    ; n  1,2,3 ; from space H  3 ,  to space L  S ,  such that if  2 n n is given by Eq. (58), then(72) T            ; C  n n n  0 3 ,  H   3 , Moreover, Tn   is a Hölder continuous mapping from   0 to the space of bounded linear operatorsfrom H  3 ,  to L  S ,  with exponent 2 n  1 3    2 , if   2 ;   3(73)   1   , if   , with   0 arbitrary small;  2  3  1, if   2 . Proof. This theorem is a particular case of the Trace Theorem proved by Weder [Weder].Define the following linear operator for n  1,2,3 ; Bn ,    : L2,3  3 H ; , 3 n(74) Bn ,    f  P   T      f    . n n n  n    2  2Here     0 ,   1 2 , and   1  x . Thus, from Theorem 13 is Hölder continuous withexponent  given by (73).Now define the linear operator(75) B    f  B1,    f  B2,    f  B3,    f .It follows that this last operator is also Hölder continuous with exponent  given by (73).In this way, from Eq. (65) the next results are immediate for f  D  H  ,(76) U f   ,    B    f    ;(77) UH f   ,     B    f    . Page 12 de 14
13. 13. Moreover, for z   and for each compact interval I    0 , such that  belongs to I , it followsthat if R  z  is the resolvent operator of H and I denotes the relative complement of I with respect to C   0 , then B    B    (78)  R  z     d    R  z  P  I   . C I  zLet L2,3    L   . If H2   3 3 3 2 3 3  ,  , , denotes the Sobolev space of all functions belonging to 1L2  3 ,  such that its first and second generalized derivatives belong to also to L  2 3 ,  , let the space  / 1  x  (79) H  2 3    f H   2 3 , 2 2 f H2  3 ,  .   Define the norm of this space as    2 2(80) f H 2  3 ,   1 x f . H2  3 , Finally, define H    H  . 3 2,3 3 3 2 3(81) ,  , 1Theorem 14 (The Limiting Absorption Principle for H ). For each     0 , the limits(82) R     i 0   limR    i  ;  0exist in the topology of the space of bounded linear operators from L2,3   3 , 3  2,3 to H  3 , 3  for  1 2 . Moreover, the functions R  z  , if z     ; (83) R  z     R  z  i 0  , if z  ;  are locally Hölder continuous on the space of bounded linear operators from L2,3   3 , 3  toH  2,3 3 , 3  with exponent  if z   , and analytic if 1  2  3 and if Im  z   0 .Proof. The existence of limits (82) in the topology of the space of bounded linear operators from   2,3   L2,3 3 , 3 to H 3 , 3 follows from the fact that B   is locally Hölder continuous. Moreover, B    B      R     i 0    p.v. d   i B    B      R    P  I  .  C I   Page 13 de 14
14. 14. The Hölder continuity of (83) follows from Privalov-Plemelj’ Theorem [Weder]. The analyticity followsfrom the analyticity of the slowness surface [Weder].ReferencesAchenbach, J. D. (1975). Wave propagation in elastic solids. North Holland Publishing Company.Bochner, S. and Chandrasekaran, K. (1949). Fourier Transforms. Princeton University Press.Fedorov, F. I. (1968). Theory of elastic waves in crystals. Plenum Press.Kato, T. (1976). Perturbation theory for linear operators. Springer Verlag.Weder, R. (1985). Analyticity of the scattering matrix for elastic waves in crystals. J. Math. Pures et Appl. 64; pp. 121-148. Page 14 de 14