This article was downloaded by: [University of Guelph] On: 13 June 2012, At: 05:20 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Inverse Problems in Science and Engineering Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gipe20 The body force in a three-dimensional Lamé system: identification and regularization a b Dang Duc Trong , Phan Thanh Nam & Phung Trong Thuc a a Department of Mathematics, Vietnam National University, Ho Chi Minh City, Vietnam b Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark Available online: 14 Dec 2011 To cite this article: Dang Duc Trong, Phan Thanh Nam & Phung Trong Thuc (2012): The body force in a three-dimensional Lamé system: identification and regularization, Inverse Problems in Science and Engineering, 20:4, 517-532 To link to this article: http://dx.doi.org/10.1080/17415977.2011.638068 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-andconditions This article may be used for research, teaching, and private study purposes Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material Inverse Problems in Science and Engineering Vol 20, No 4, June 2012, 517–532 The body force in a three-dimensional Lame´ system: identification and regularization Dang Duc Tronga, Phan Thanh Namb and Phung Trong Thuca* a Department of Mathematics, Vietnam National University, Ho Chi Minh City, Vietnam; Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark b Downloaded by [University of Guelph] at 05:20 13 June 2012 (Received 14 June 2011; final version received 23 September 2011) Let a three-dimensional isotropic elastic body be described by the Lame´ system with the body force of the form F(x, t) ¼ ’(t)f (x), where ’ is known We consider the problem of determining the unknown spatial term f (x) of the body force when the surface stress history is given as the overdetermination This inverse problem is ill-posed Using the interpolation method and truncated Fourier series, we construct a regularized solution from approximate data and provide explicit error estimates Keywords: body force; elastic; ill-posed problem; interpolation; Fourier series AMS Subject Classifications: 35L20; 35R30 Introduction Let  ¼ (0, 1)  (0, 1)  (0, 1) represent a three-dimensional isotropic elastic body and let T > be the length of the observation time For each x :¼ (x1, x2, x3) , we denote by u(x, t) ¼ (u1(x, t), u2(x, t), u3(x, t)) the displacement, where uj is the displacement in the xj-direction As known, u satisfies the Lame´ system (see, e.g [13]) @2 u Du  ỵ ịrdivuịị ẳ F, x , t 0, T Þ, ð1Þ @t2 where F(x, t) ¼ (F1(x, t), F2(x, t), F3(x, t)) is the body force and div(u) ¼ r u ẳ@u1/@x1 ỵ @u2/ @x2 ỵ @u3/@x3 The Lame´ constants  and  satisfy  > and  ỵ 2 > The system (1) is associated with the initial condition < ðu ðxÞ, g2 ðxÞ, g3 ðxÞÞ, x ,  ðx, 0Þ, u2 x, 0ị, u3 x, 0ịị ẳ g @u1 @u2 @u3 2ị x, 0ị, x, 0ị, x, 0ị ẳ ðh1 ðxÞ, h2 ðxÞ, h3 ðxÞÞ, x , : @t @t @t and the Dirichlet boundary condition ðu1 ðx, tị, u2 x, tị, u3 x, tịị ẳ 0, 0, 0Þ, x @, t ð0, T Þ, namely the boundary of the elastic body is clamped *Corresponding author Email: phungtrongthuc@vanlanguni.edu.vn ISSN 1741–5977 print/ISSN 1741–5985 online ß 2012 Taylor & Francis http://dx.doi.org/10.1080/17415977.2011.638068 http://www.tandfonline.com ð3Þ 518 D.D Trong et al The direct problem is to determine u from u(0, x), ut(0, x) and F We are, however, interested in the inverse problem of determining both (u, F ) Of course, to ensure the uniqueness of the solution we shall require some additional information (the overdetermination) Similar to [4,5], we shall assume that the surface stress is given on the boundary of the body, i.e 10 1 1 12 13 n1 X1 @ 21 2 23 A@ n2 A ¼ @ X2 A, x @, t ð0, T Þ, ð4Þ 31 32 3 n3 X3 Downloaded by [University of Guelph] at 05:20 13 June 2012 where n ¼ (n1, n2, n3) is the outward unit normal vector of @ and the stresses  and  are defined by   @uj @uk @uj ỵ , j, k f1, 2, 3g: , j ẳ  divuị þ 2 jk ¼  @xk @xj @xj Grasselli et al [4] showed that the body force of the form F(x, t) ¼ ’(t)f (x) is uniquely determined from (1)–(4) provided that ’ C1([0, T ]) is given such that ’(0) 6¼ and the time of observation T > is large enough In spite of the uniqueness, the problem of determining the spatial term f is still ill-posed, i.e a small error of data may cause a large error of solutions Therefore, it is important in practice to find a regularization process, namely to construct an approximate solution using approximate data Recently, the regularization problem was solved partially in [5], where a regularized solution for the time-independent term f is produced using further information on the final condition u(x, T ) The final condition plays an essential role in [5] since it enables the authors to find an explicit formula for the Fourier transform of f, and then use this information to recover f It was left as an open problem in [5] (see their Conclusion) to find a regularization process without using this technical condition The aim of this article is to solve this problem completely, i.e to find a regularization process of f using only the data in (1)–(4) We follow the interpolation method introduced in [6] where the authors constructed a regularized solution for the heat source of a heat equation More precisely, lacking the final condition, we are only able to find an approximation for the Fourier transform b f ðÞ with jj large The idea is that because b f ðÞ is an analytic function (since f has compact support), we can use some interpolation process to recover b f ðÞ with jj small This article is organized as follows In Section 2, we shall set some notations and state our main results Then we prove the uniqueness in Section and the regularization in Section Finally, in Section we test our regularization process on an explicit numerical example Main results Recall that our aim is to recover the spatial term f xị ẳ f1 xị, f2 xị, f3 ðxÞÞ, x , of the body force F(x, t) ¼ ’(t)f (x) from the system (1)–(4) The Lame´ constants always satisfy  > and  ỵ 2 > 0, and the data I ¼ (’, X, g, h) is allowed to be non-smooth, À Á I L1 ð0, T Þ, ðL1 ð0, T, L1 ð@ÞÞÞ3 , ðL2 ðÞÞ3 , ðL2 ðÞÞ3 : 519 Inverse Problems in Science and Engineering Forpffiffiffiffiffiffiffiffiffiffi ¼ (ffi 1, 2, 3),  ¼ (1, 2, 3) C3, we set  Á  ẳ 11 ỵ 22 ỵ 33, jj ẳ jj0 ¼ j Á j For  C3 and x R3, denote pffiffiffiffiffiffiffiffi  Á  and Gð, xÞ ¼ G11 ð, xÞ ¼ G22 ð, xÞ ¼ G33 , xị ẳ cos1 x1 ị cos2 x2 ị cos3 x3 Þ, G12 ð, xÞ G13 ð, xÞ G23 ð, xị ẳ G21 , xị ẳ sin1 x1 ị sin2 x2 ị cos3 x3 ị, ẳ G31 , xị ẳ sin1 x1 ị cos2 x2 ị sin3 x3 ị, ẳ G32 , xị ẳ cos1 x1 ị sin2 x2 ị sinð3 x3 Þ: Downloaded by [University of Guelph] at 05:20 13 June 2012 Sometimes we shall write Gjk instead of Gjk ð, xÞ if there is no confusion We start with the following lemma LEMMA If u (C2([0, T ]; L2()) \ L2(0, T; H2()))3 and f (L2())3 satisfy the system (1)–(4) with data I ¼ (’, X, g, h), then Z E1j Iịị ỵ E1j ị E2j Iịị ỵ E2j ị ỵ , j f1, 2, 3g fj G dx ẳ D1 Iịị D2 Iịị  for all  ¼ (1, 2, 3) C3 such that jj20 ẳ 21 ỵ 22 ỵ 23 ị ! and D1 Iịị 6ẳ 0, D2 Iịị 6ẳ 0, where p   ỵ 2ị jj0 T tị p  dt, D1 I ịị ẳ jj20 tị cosh  ỵ 2ị jj0 T p  ZT sinh  jj0 ðT À tÞ pffiffiffiffi  dt, D2 I ịị ẳ jj20 tị cosh  jj0 T Z p E1j Iịị ẳ  ỵ 2 jj0 j 1 g1 Gj1 ỵ 2 g2 Gj2 þ 3 g3 Gj3 dx  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Z À 1 h1 Gj1 ỵ 2 h2 Gj2 ỵ 3 h3 Gj3 dx  ỵ 2 jj0 T j    ffi Z T Z sinh pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ỵ 2 jj0 T tị p  j 1 X1 Gj1 ỵ 2 X2 Gj2 ỵ 3 X3 Gj3 d! dt, À @ cosh  þ 2 jj0 T pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  þ 2 jj Àpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j E1j ị ẳ cosh  ỵ 2 jj0 T Z  1 u1 x, TịGj1 ỵ 2 u2 x, TịGj2 ỵ 3 u3 x, TịGj3 dx, Z sinh T  p E2j Iịị ẳ  jj0 Z Á Àjj20 gj Gjj À j ð1 g1 Gj1 ỵ 2 g2 Gj2 ỵ 3 g3 Gj3 dx  Àpffiffiffiffi Á À  jj0 T Z Á jj20 hj Gjj j 1 h1 Gj1 ỵ 2 h2 Gj2 ỵ 3 h3 Gj3 dx  520 D.D Trong et al ZT Z ỵ @ p sinh ð  jj0 ðT À tÞ Á pffiffiffiffi cosh ð  jj0 T  Á  jj20 Xj Gjj þ j ð1 X1 Gj1 þ 2 X2 Gj2 þ 3 X3 Gj3 d! dt, Z pffiffiffiffi  jj Àpffiffiffiffi jj20 uj x, TịGjj E2j ị ẳ cosh  jj0 T  Downloaded by [University of Guelph] at 05:20 13 June 2012   À j 1 u1 x, TịGj1 ỵ 2 u2 x, TịGj2 ỵ 3 u3 ðx, TÞGj3 Þdx: Note that Ễ1j and Ễ2j in Lemma depend on u(x, T ) instead of the data I ¼ (’, X, g, h) Therefore, in general these terms are unknown However, our observation is that with jj large, EÃ1j and EÃ2j are relativelyRsmall in comparison with E1j and E2j, and can be relaxed when computing the integrals  fj G dx which are the Fourier coefficients of fj’s So we introduce some convenient notations Definition (Information from data) For I ¼ (’, X, g, h) and  C3 such that  Á  < 0, denote (using notations of Lemma 1) < E1j ðIÞ E2j Iị ỵ , if D1 Iịị D2 Iịị 6ẳ 0, Hj Iịị ẳ D1 Iị D2 Iị : 0, if D1 Iịị D2 Iịị ẳ 0: Definition (Fourier co-efficients) For  ¼ (1, 2, 3) C3, w L2(), denote Z Z F wịị ẳ wðxÞ cosð1 x1 Þ cosð2 x2 Þ cosð3 x3 Þdx: wxịG, xịdx ẳ   Note that any function w L2() admits the representation wxị ẳ X m, n, pÞF ðwÞðm, n, pÞ cosðmx1 Þ cosðnx2 Þ cosð px3 Þ, ð5Þ m,n, p!0 where (m, n, p) :¼ (1 þ 1{m6¼0})(1 þ 1{n6¼0})(1 þ 1{p6¼0}) As we have explained above, we hope to approximate F ( fj)() by Hj(I )() with jj large To this, we need some lower bounds on jD1(I )()j and jD2(I )()j We shall require the following assumptions on ’ and T (W1) There exist Ã(’) (0, T ) and C(’) > such that either ’(t) ! C(’) for a.e t (0, Ã(’)) or ’(t) ÀC(’) for a.e t (0, Ã(’)) pffiffiffi ffi ffi g, or (W20 ) T 12  maxfp1ffiffiffi , pffiffiffiffiffiffiffiffi g (W2) T maxfp1 , p ỵ2 ỵ2 Remark If ’ is continuous at t ¼ then the condition (W1) is equivalent to ’(0) 6¼ The conditions (W2) and (W20 ) mean that the observation time must be long enough The condition (W2) is enough for the uniqueness, while the stronger condition (W20 ) is required in our regularization These conditions should be compared to similar conditions (2.7) and (2.8) in [4] THEOREM (Uniqueness) Assume that (W1) and (W2) hold true Then the system (1)–(4) has at most one solution À Á ðu, f Þ C2 ẵ0, T ; L2 ịị \ L2 0, T; H2 ðÞÞÞ3 , ðL2 ðÞÞ3 : 521 Inverse Problems in Science and Engineering The main point in our regularization is to recover F ( f )() with jj small from approximate values of F ( f )() with jj large As in [6] we shall use the Lagrange interpolation polynomial Definition (Lagrange interpolation polynomial) Let A ¼ {x1, x2, , xn} be a set of n mutually distinct complex numbers and let w be a complex function Then the Lagrange interpolation polynomial L[A; w] is ! n Y z À xk X À Á w xj : LẵA; wzị ẳ x xk jẳ1 k6ẳj j Downloaded by [University of Guelph] at 05:20 13 June 2012 THEOREM (Regularization) Assume that (u0, f 0) is the exact solution to the system (1)–(4) with the exact data I0 ¼ (’0, X0, g0, h0), where the conditions (W1) and (W20 ) are satisfied Let " > 0, consider the inexact data I" ¼ (’", X", g", h") such that, for all j {1, 2, 3},   ’ À ’"  L ",     " Xj À X0j  L1 ",     " gj À g0j  L2 ",     " hj À h0j  L2 ": Construct the regularized solution f"j from I" ¼ (’", X", g", h") by X ðm, n, pÞF " ðm, n, pịGm, n, pị, f"j ẳ m, n, p r" where  ! ln"1 ị ln"1 ị r" ẳ Z \ , ỵ1 , 60 60 ẩ ẫ Br" ẳ ặ5r" ỵ jị : j ẳ 1, 2, , 24r" ,  à F " ðm, n, pị ẳ L Br" , Hj I" ịi:, n, pÞ ðimÞ: Then we have, for all j {1, 2, 3}, (i) (Convergence) f"j C1 ðR3 Þ and f"j ! fj0 in L2() as " ! (ii) (L2-estimate) If fj0 H1 ðÞ then f"j ! fj0 in H1() and there exist constants "0 > and C0 > depending only on the exact data such that for all " (0, "0),    À ÁÀ12   " C0 ln "À1 : fj À fj0  L ð Þ (iii) (H -estimate) If fj0 H2 ðÞ then for all " (0, "0),    À ÁÀ14   " C0 ln "À1 : fj À fj0  H ðÞ Remark In our construction, the convergence in H2() is not expected even if fj0 C1 ðÞ since @f"j =@n ¼ on @ Uniqueness In this section we shall prove Theorem We start with the proof of Lemma by using the argument in [5] 522 D.D Trong et al Proof of Lemma Fix  ¼ (1, 2, 3) C3 with jj20 ẳ 21 ỵ 22 ỵ 23 ị ! For any j {1, 2, 3}, get the inner product (in L2()) of the k-th equation of the system (1) with Gjk (k ¼ 1, 2, 3), then using the integral by part and the boundary conditions (3) and (4), we have Z Z À Á d2 j 2 u G dx ỵ   ỵ  ỵ  uk Gjk dx k k dt2   Z 1 u1 Gj1 ỵ 2 u2 Gj2 ỵ 3 u3 Gj3 dx ỵ  ỵ Þk  Z Z j Xk Gk d! À ’ðtÞ fk Gjk dx: Ak ị ẳ  Downloaded by [University of Guelph] at 05:20 13 June 2012 @ Multiplying (Ak) with jk, and then getting the sum of k ¼ 1, 2, 3, we have Z À Á d2 j 1 u1 Gj1 ỵ j 2 u2 Gj2 ỵ j 3 u3 Gj3 dx dt  Z À Á j 1 u1 Gj1 ỵ j 2 u2 Gj2 þ j 3 u3 Gj3 dx À ð þ 2Þjj0  Z j j 1 X1 G1 ỵ j 2 X2 Gj2 ỵ j 3 X3 Gj3 d! ẳ @ Z j 1 f1 Gj1 ỵ j 2 f2 Gj2 ỵ j 3 f3 Gj3 dx: ỵ tị 6ị  Choosing k ẳ j in (Ak), then multiplying the result by jj20 , and adding to (6), we obtain Z   d2 j j jÁ u G ỵ   u G ỵ   u G ỵ   u G dx jj j j 1 j 2 j 3 dt2  Z   ỵ jj20 jj20 uj Gjj j 1 u1 Gj1 ỵ j 2 u2 Gj2 ỵ j 3 u3 Gj3 dx  Z   jj20 Xj Gjj ỵ j 1 X1 Gj1 ỵ j 2 X2 Gj2 ỵ j 3 X3 Gj3 d! ¼ @ Z  Á jj20 fj Gjj ỵ j 1 f1 Gj1 ỵ j 2 f2 Gj2 ỵ j 3 f3 Gj3 dx: 7ị ỵ tị  We can consider (6) and (7) as the differential equations of the form y00 tị 2 ytị ẳ hðtÞ, ð8Þ where  > is independent of t Getting the inner product (in L2(0, T )) of (8) with sinhððTÀtÞÞ coshðTÞ , we have ZT  sinhððT À tịị yTị ẳ dt: 9ị htị coshTị coshTị p pffiffiffiffi Applying (9) to (6) and (7) with  ¼  ỵ 2 jj0 and  ẳ  jj0 , respectively, we get Z À Á à E1j ðIÞðÞ ỵ E1j ị ẳ D I ịị j 1 f1 Gj1 ỵ 2 f2 Gj2 ỵ 3 f3 Gj3 dx, Àjj0  Z   j j j jÁ j j D ð I Þ ð  ị f G   f G ỵ  f G ỵ  f G E2j I ịị ỵ E2j ị ẳ  j j 1 2 3 j dx: jj20  y0 0ị tanhT ị y0ị ỵ Inverse Problems in Science and Engineering 523 It follows from the latter equations that Z fj G dx ¼  E1j Iịị ỵ E1j ị E2j Iịị ỵ E2j ị þ : D1 ðIÞðÞ D2 ðIÞðÞ g This is the desired result Downloaded by [University of Guelph] at 05:20 13 June 2012 The following lemma gives a lower bound for jDj(I )()j (defined in Lemma 1) when ’ satisfies the condition (W1) LEMMA Let ’ L1(0, T ) satisfy the condition (W1), then there exists a constant R(’) > such that for all  ¼ (1, 2, 3) C3,  Á  < and jj0 ! R(’), & '   Dj ðI ÞðÞ ! jj0 Cð’Þ p1ffiffiffiffi , pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi for j ẳ 1, 2:   ỵ 2 p p Proof Denote k1 ẳ  ỵ 2 and k2 ¼  By the triangle inequality, one finds that     D ðI ÞðÞ  Z T sinhðkj jj0 ðT À tÞÞ     j À dt tị  ẳ     jj20   cosh kj jj0 T  Z  Ãð’Þ sinhðkj jj0 ðT À tÞÞ   À Á dt ! ’ðtÞ   cosh kj jj0 T  Z  T sinhðkj jj0 ðT À tÞÞ   À Á dt: À ’ðtÞ   Ãð’Þ cosh kj jj0 T We have, with jj0 large,  Z  T sinhðkj jj0 ðT À tÞÞ   dt ’ðtÞ    Ãð’Þ coshðkj jj0 TÞ   sinhðkj jj0 ðT À Ãð’ÞÞÞ ’ L coshðkj jj0 TÞ Cð’Þ : maxfk1 , k2 gjj0 On the other hand, the condition (W1) implies that  Z Z à ð’ Þ  Ãð’Þ sinhðkj jj0 ðT À tÞÞ  sinhðkj jj0 ðT À tÞÞ  À Á dt ! Cð’Þ À Á dt ’ðtÞ    cosh kj jj0 T cosh kj jj0 T ! Cị cosh kj jj0 T ịịị Cị ẳ ! kj jj0 coshðkj jj0 TÞ 2kj jj0 with jj0 large The desired result follows immediately from the above inequalities g The proof of the uniqueness below follows the argument in [6] We shall need a useful result of entire functions (see, e.g [7, Section 6.1]) 524 D.D Trong et al LEMMA (Beurling) Let  be a non-constant entire function satisfying the condition: there exists a constant k > such that M(r) ker, for all r > 0, where M (r) ¼ j(z)j: jzj ¼ r Then   lnðrÞ ! À1: lim sup r r!1 Proof of Theorem Suppose that (u1, f1) and (u2, f2) are two solutions to the system (1)–(4) with the same the data I ¼ (’, X, g, h) Then (u, f ) :¼ (u1 À u2, f1 À f 2) is a solution to (1)–(4) with data (’, 0, 0, 0) We shall show that (u, f) ¼ Assume that f 6¼ 0, namely fj 6¼ for some j {1, 2, 3} For any n, p N [ {0}, let us consider the entire function Z z n,p zị ẳ fj xị cosizx1 ị cosnx2 ị cosð px3 Þdx: Downloaded by [University of Guelph] at 05:20 13 June 2012  Because d n,p dz Z ix1 fj xị sinmx1 ị cosnx2 ị cospx3 ịdx imị ẳ  and {sin(mx1)cos(nx2)cos( px3)}m2N,n,p2N[{0} is an orthogonal basis on L2(), there exist some (n0, p0) such that n0 , p0 is non-constant On the other hand, recall from Lemma that Z E1j r ị E2j r ị ỵ , 10ị rị ẳ f xịG , xịdx ẳ n0 ,p0 j r D1 ðI Þðr Þ D2 ðI Þðr Þ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi where r ¼ (ir, n0, p0) Fix " > such that minfT  ỵ 2, T g ỵ " Then it is straightforward to see, from the explicit formulas in Lemma and the lower bound in Lemma 2, that jỄej ðr Þj C1 e1ỵ"ịr , jDe Iịr ịj ! C2 r, e ẳ 1, 2, with r > large, where C1 > and C2 > are independent of r Therefore, it follows from (10) that j n0 ,p0 rịj 1ỵ"ịr À1 2C1 CÀ1 r e for r > large This yields lim sup r!1  ln   n , p rị r ỵ "Þ, which is a contradiction to Lemma Thus f  Now following the proof of Lemma up to Equations (6) and (7) (2 :ẳ 21 ỵ 32 þ 33 need not be negative at that time) we obtain Z d2 j 1 u1 Gj1 ỵ j 2 u2 Gj2 ỵ j 3 u3 Gj3 dx dt  Z À Á j 1 u1 Gj1 þ j 2 u2 Gj2 þ j 3 u3 Gj3 dx ẳ 0, 11ị ỵ  ỵ 2ị2  Inverse Problems in Science and Engineering d2 dt2 525 Z   2 uj G j 1 u1 Gj1 ỵ j 2 u2 Gj2 ỵ j 3 u3 Gj3 ị dx  Z   ỵ 2 2 uj Gjj j 1 u1 Gj1 ỵ j 2 u2 Gj2 ỵ j 3 u3 Gj3 ị dx ẳ 12ị  for all  C3 Note that y  is the unique solution to the differential equation 00 > < y tị ỵ  ytị ẳ 0, y0ị ẳ 0, > : y 0ị ẳ 0, where  ! is independent of t Applying this to (11) and (12) with  R3, we get Z j 1 u1 Gj1 ỵ j 2 u2 Gj2 ỵ j 3 u3 Gj3 dx ẳ Downloaded by [University of Guelph] at 05:20 13 June 2012  and Z   2 uj G À j 1 u1 Gj1 ỵ j 2 u2 Gj2 ỵ j 3 u3 Gj3 ị dx ẳ 0:  Adding the latter equations, we obtain Z uj ðx, :ÞGð, xÞdx ¼ for all  R3 ,  6¼ 0: ð13Þ  To get the same equation to (13) with  ¼ 0, we can simply use identity (Ak) with  ¼ to get Z d2 uj ðx, tịdx ẳ 0: dt2  Since Z ! d uj x, tịdx ẳ0ẳ dt  tẳ0 ! Z , uj x, tịdx  tẳ0 we have Z uj x, ịdx ¼ 0: ð14Þ  Putting (13) and (14) together, we arrive at Z Z uj G ẳ uj x, :ị cos1 x1 ị cos2 x2 ị cos3 x3 ịdx ẳ for all  R3 :   This implies that u  because {cos(mx1)cos(mx2)cos(nx3)}m,n,p!0 forms an orthogg onal basis of L2() Regularization Lemmas and allow us to compute F ( f )() approximately with jj large To recover F ( f)() with jj small, we shall need the following interpolation inequality This is an 526 D.D Trong et al adaption of Lemma in [6] The only difference is that we are working on a 3-dim problem, and hence we require more interpolation points LEMMA (Interpolation inequality) Let Br ẳ {ặzjjzj ẳ 5r þ j, j ¼ 1, 2, , 24r} for some integer r ! 50 Let !: C ! C be an entire function such that j!(z) j Aejzj} for some constant A Then for any function g: Br ! C, one has     sup !zị LẵBr , gzị Aer ỵ 48re30r sup!zị gðzÞ: jzj r z2Br Proof Fix z C with jzj {z} [ Br, r We have the following residue formula at 49 simple poles Downloaded by [University of Guelph] at 05:20 13 June 2012 Z 24r z2 À z2   !xị Y j dx ẳ 2i ! z ị L ẵ B , ! z ị : r 2 ẳfx2C:jxjẳ50rg x z j¼1 x À zj This gives the simple bound   !zị LẵBr , !zị ( ) 24r jz2 À z2 j j!ðxÞj Y j 50r sup : 2 x2 jx À zj j¼1 jx À zj j Now we bound the right-hand side of (15) For x C, jxj ¼ 50r, we have j!(x) j jx À zj ! (50 À )r and 24r jz2 À z2 j Y j j¼1 jx2 À z2j j 24r jzj2 ỵ z2 Y j jẳ1 jxj2 z2j 24r rị2 ỵ z2 Y j jẳ1 50rị2 z2j : 15ị Ae50r, 16ị We shall show that 24r rị2 ỵ z2 Y j 50 À  À51r e , 50 2 jẳ1 50rị zj 8r ! 50: 17ị rị ỵx Note that the function x vxị ¼ lnðð50rÞ Þ v(x) is increasing and concave in [0, (29r)2] Àx Using Jensen’s inequality, we get ! ! 24r   4hr 4hr   X X X X X 2 v zj ẳ v zj v z 4r 4r jẳ4h1ịrỵ1 j jẳ1 hẳ1 jẳ4h1ịrỵ1 hẳ1 ẳ 4r    X 31 v 24h ỵ 16h2 ỵ r ỵ ỵ 4hịr ỵ hẳ1    X 31 r2 r2 4r v 24h þ 16h þ r þ ð3 þ 4hÞ þ 50 502 hẳ1 ! 3ỵ4h X 2 ỵ 24h ỵ 16h2 ỵ 31 ỵ 50 þ 6Â502 ¼ 4r ln 3þ4h 502 À ð24h þ 16h2 þ 31 h¼1 þ 50 þ 6Â502 ị   50  51r ỵ ln : 50 Inverse Problems in Science and Engineering Thus (17) holds Replacing (16) and (17) into (15), we obtain   !zị LẵBr , !zị Aer , 8z C, jzj r: 527 ð18Þ Next, we observe that ! 24r Y z2 z2 z ỵ zj  X  k LẵBr , !zị LẵBr , gzị ẳ g zj ! z j z2 À z2k 2zj j¼1 k6¼j j ! 24r Y z2 À z2 z À zj  À Á X À  k ỵ ! zj g zj : 2 z À zk À2zj j¼1 k6¼j j Downloaded by [University of Guelph] at 05:20 13 June 2012 Therefore   jLẵBr , !zị LẵBr , gzị    24r Y 2 X z À z k  2  z2 À z2  , k j¼1  k6ẳj j 19ị where  ẳ supz2Br j!zị gðzÞj Now we bound the right-hand side of (19) We have    Y  z À z2k  Y jzj2 ỵ z2k Y jzj2 ỵ z2k z   À Á Á k  ¼  z2 À z2  À z2 j  jz z zk  z ỵ z z j j k k  k6¼j j k k k6¼j j k6¼j Y k6¼j z  k   zj À zk  5r ỵ 2ị5r ỵ 3ị 29rị j 1ị!24r jị! 5r ỵ 2ị5r ỵ 3ị 29rị ẳ: Jrị: 12r 1ị!12rị! Since J(1) e30 and, by direct expansion, Jr ỵ 1ị 29r ỵ 1ị29r ỵ 2ị 29r ỵ 29ị ẳ Jrị 5r ỵ 2ị 5r ỵ 6ị ẵ12r ỵ 1ị 12r ỵ 11ị2 12rị 12r ỵ 12ị 2929 e30 , 55 Á 1224 we conclude that J(r) e30r for all r ! Thus (19) reduces to   jL½Br , !zị LẵBr , gzị 48re30r : The desired result follows from (18), (20) and the triangle inequality ð20Þ g The last of our preparation to prove the regularization result is the following useful lemma on the truncated Fourier series The proof is elementary and we refer to Lemma in [7] for details (Trong et al [8] the authors dealt with the two-dimensional case, but the proof for three-dimensional case is essentially the same) For each w L2() and r > 0, we define X ðm, n, pÞF ðwÞðm, n, pÞ cosðmx1 Þ cosðnx2 Þ cosð px3 Þ: Àr ðwÞðxÞ ¼ LEMMA m,n,p r 528 D.D Trong et al Then (i) Àr(w) ! w in L2() as r ! (ii) If w H1() then Àr(w) ! w in H1() and   Àr ðwÞ À w L ðÞ pffiffi kwkH1 ðÞ :  r   Àr ðwÞ À w H ị p kwkH2 ị : ẵ4r (iii) If w H2() then Downloaded by [University of Guelph] at 05:20 13 June 2012 We are now ready to prove the main result Proof of Theorem We shall first estimate the error fj" À Àr" fj0 , where Àr is defined in Lemma Then we compare Àr" fj0 and f0j by employing Lemma The conclusion follows from the triangle inequality In the following, C, C1, C2 always stand for constants depending only on the exact data, and saying that " > small enough means that " (0, "0) for some constant "0 > depending only on the exact data Step Bound on jF"j ðm, n, pÞ À F ð fj0 Þðm, n, pÞj for m, n, p [{0, r"] Applying Lemma to gzị ẳ H"j z, n, pị and Z !zị ¼  fj0 ðx1 , x2 , x3 Þ cosðÀizx1 Þ cosðnx2 Þ cosðpx3 Þdx1 dx2 dx3 , we find that for m, n, p [0, r"] and " > small enough       à    " Fj ðm, n, pÞ À F fj0 m, n, pị ẳ jL Br" , g imị !imị   kfj0 kL1 ị er" ỵ 48 r" e30r" sup !ðzÞ À gðzÞ: ð21Þ z2Br" The error j!(z) À g(z)j can be bound by direct computation Indeed, for z Br" , we have       !zị gzị ẳ F f ðÀiz, n, pÞ À H" ðz, n, pÞ j j       F fj0 ðÀiz, n, pÞ À H0j ðz, n, pÞ     ỵ H0j z, n, pị H"j z, n, pÞ   à  E1j ðÀiz, n, pÞ Ễ2j ðÀiz, n, pÞ    D ðI ÞðÀiz, n, pị ỵ D I ịiz, n, pị   E1j ðI0 ÞðÀiz, n, pÞ E1j ðI" ịiz, n, pị  ỵ  D1 I0 ịiz, n, pÞ D1 ðI" ÞðÀiz, n, pÞ    E2j ðI0 ÞðÀiz, n, pÞ E2j ðI" ÞðÀiz, n, pÞ : ỵ  D2 I0 ịiz, n, pị D2 ðI" ÞðÀiz, n, pÞ  ð22Þ Inverse Problems in Science and Engineering 529 Downloaded by [University of Guelph] at 05:20 13 June 2012 It is straightforward to see from the explicit formulas of E1j, E2j, EÃ1j , EÃ2j that   E1j ðI" ÞðÀiz, n, pÞ À E1j ðI0 ÞðÀiz, n, pÞ C1 r3 e29r" ", "   E2j ðI" ÞðÀiz, n, pÞ À E2j ðI0 ÞðÀiz, n, pÞ C1 r3 e29r" ", "   E1j ðI0 ÞðÀiz, n, pÞ C1 r3 e29r" , "   E2j ðI0 ÞðÀiz, n, pÞ C1 r3 e29r" , "   C1 r3" e29r"   à , E1j ðÀiz, n, pị pp e ỵ2 T r"  C r3 e29r"    à " pffiffipffiffiffi : E2j ðÀiz, n, pÞ e  T r" On the other hand, Lemma ensures that C2 r" jDj ðI0 ÞðÀiz, n, pÞj C1 r2" and     Dj ðI" ÞðÀiz, n, pÞ ! Dj ðI0 ÞðÀiz, n, pÞ   À Dj ðI" ÞðÀiz, n, pÞ À Dj ðI0 ÞðÀiz, n, pÞ ! C2 r" À C1 r2" ", for all z Br" and " > small enough Therefore, from (21) and (22) we get       " Fj ðm,n,pÞ À F fj0 ðm, n,pÞ " # 3   1 C1 ð30 lnð"À1 ÞÞ C1 ð30 lnð"À1 ÞÞ 4C21 ð30 lnð"À1 ÞÞ e59 "60  0 pp ỵ : ỵ fj  "60 ỵ 48 pp 1 1 L  ị C "60 ỵ2 T59ị C2 "60  TÀ59Þ C2 ðC2 À 30 C1 lnð" Þ"Þ Moreover, the condition (W20 ) gives  1 pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5  ỵ 2T 59 , 60 60  1 pffiffiffipffiffiffiffi 5 T À 59 : 60 60 Thus we obtain       " Fj ðm, n, pÞ À F fj0 ðm, n, pÞ À Á Cðln "À1 Þ4 "60 ð23Þ for all " > small enough Step Conclusion (i) Using the Parseval equality and the estimate (23), one gets 2    " fj À Àr" fj0  L  ị ẳ X m, n, p r" 2      ðm, n, pÞF"j ðm, n, pÞ À F fj0 ðm, n, pÞ : Cðlnð"À1 ÞÞ11 "30 : ð24Þ Since kfj" À Àr" fj0 kL2 ðÞ ! by (24) and kÀr" fj0 À fj0 kL2 ðÞ ! by Lemma 5, we obtain k fj" À fj0 kL2 ðÞ ! as " ! by the triangle inequality (ii) Assume that fj0 H1 ðÞ Then by Lemma one has kÀr" fj0 À fj0 kH1 ðÞ ! and kÀr" fj0 À fj0 kL2 ðÞ Cðlnð"À1 ÞÞÀ2 : Combining the latter estimate with (24), we conclude 530 D.D Trong et al using the triangle inequality that     "  fj À fj0  L ð Þ À Á Cðln "À1 ÞÀ2 for " > small enough Moreover, using (23) and the Parseval equality for H1, we have 2 2     X    "  m,n,pị1 ỵ 2 m2 þ n2 þ p2 ÞÞF"j ðm,n,pÞ À F fj0 ðm,n,pÞ : fj À Àr" fj0  ¼ H ð Þ m, n, p r" Downloaded by [University of Guelph] at 05:20 13 June 2012 Cðlnð"À1 ÞÞ13 "30 ! 0: ð25Þ Thus k fj" À fj0 kH1 ðÞ ! by the triangle inequality (in H1-norm) (iii) Assume that fj0 H2 ðÞ Then by Lemma one has     Cðlnð"À1 ÞÞÀ1=4 : Àr" fj0 À fj0  H ðÞ This estimate together with (25) and the triangle inequality yield     " Cðlnð"À1 ÞÞÀ1=4 fj À fj0  H ð Þ for " > small enough g Numerical example In this section, we test our regularization process in an explicit example Choose  ¼ 1,  ¼ À1, T ¼ 30 and consider the system (1)–(4) with the exact data I0 ¼ (’0, X0, g0, h0) given by tị ẳ232 costị, h01 ẳ h02 ẳ h03 ẳ 0, X01 x1 , x2 , x3 , tị ẳ cosðtÞðÀ4 sinð2x2 Þ sinð2x3 Þn1 À 2 sinð4x1 Þ sinð2x3 Þn2 À 2 sinð4x1 Þ sinð2x2 Þn3 Þ, X02 ðx1 , x2 , x3 , tị ẳ costị2 sin4x2 ị sinð2x3 Þn1 À 4 sinð2x1 Þ sinð2x3 Þn2 À 2 sinð2x1 Þ sinð4x2 Þn3 Þ, X03 ðx1 , x2 , x3 , tị ẳ costị2 sin2x2 ị sin4x3 ịn1 2 sinð2x1 Þ sinð4x3 Þn2 À 4 sinð2x1 Þ sinð2x2 Þn3 Þ, g01 ðx1 , x2 , x3 Þ ¼ sinð4x1 Þ sinð2x2 Þ sinð2x3 Þ, g02 ðx1 , x2 , x3 ị ẳ sin2x1 ị sin4x2 ị sin2x3 ị, g03 x1 , x2 , x3 ị ẳ sin2x1 ị sinð2x2 Þ sinð4x3 Þ: 531 Inverse Problems in Science and Engineering Note that in this example the conditions (W1) and (W20 ) are satisfied The exact solution of the system (1)–(4) can be computed explicitly u01 ðx1 , x2 , x3 , tị ẳ costị sin4x1 ị sin2x2 ị sin2x3 Þ, u02 ðx1 , x2 , x3 , tÞ ¼ cosðtÞ sinð2x1 Þ sinð4x2 Þ sinð2x3 Þ, u03 ðx1 , x2 , x3 , tị ẳ costị sin2x1 ị sin2x2 Þ sinð4x3 Þ, f10 ðx1 , x2 , x3 Þ ¼ sinð4x1 Þ sinð2x2 Þ sinð2x3 Þ, f20 ðx1 , x2 , x3 ị ẳ sin2x1 ị sin4x2 ị sin2x3 Þ, f30 ðx1 , x2 , x3 Þ ¼ sinð2x1 Þ sinð2x2 Þ sinð4x3 Þ: Downloaded by [University of Guelph] at 05:20 13 June 2012 Now, we consider the disturbed data, for n N, j {1, 2, 3}, n ẳ , hnj ẳ 0, 2 costị p sin2nx2 ị sin2nx3 ịn1 n ỵ sin2nx1 ị sin2nx3 ịn2 ỵ sinð2nx1 Þ sinð2nx2 Þn3 Þ, sinð2nx1 Þ sinð2nx2 Þ sinð2nx3 ị gnj ẳ g0j ỵ : n2 Xnj ẳ X0j ỵ The disturbed solution of the system (1)(4) with the disturbed data is cosðtÞ sinð2nx1 Þ sinð2nx2 Þ sinð2nx3 ị p , n3 ỵ 12n2 ị sin2nx1 ị sin2nx2 ị sin2nx3 ị p : fnj ẳ fj0 ỵ 23 n3 unj ẳ u0j ỵ We can see that the error of the data is small pffiffiffi ¼ pffiffiffiffiffi , n3     n gj À g0j  L ð Þ     n Xj À X0j  L1 ð0, T, L1 ð@ÞÞ 360 pffiffiffi n (in fact, they even converge in the uniform norm) However, the error of the solution is large since    n fj À fj0  L ð Þ pffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 144n4 À 24n2 þ !1 ¼ 529n3 as n ! Thus the problem is ill-posed and a regularization is necessary Now, we apply our regularization procedure for " ¼ 0.01 and the disturbed data with n ¼ 10 The resulting regularized solution is f"1 ðx1 , x2 , x3 Þ % 0:035 À 0:063 cosðx1 Þ À 0:156 cosðx2 Þ 0:455 cosx3 ị ỵ 0:033 cosx1 ị cosx2 ị þ 0:027 cosðx1 Þ cosðx3 Þ þ 0:15 cosðx2 Þ cosðx3 Þ À 0:005 cosðx1 Þ cosðx2 Þ cosðx3 Þ: 532 D.D Trong et al The error between the regularized solution and the exact solution is "  f À f 2 % 0:273: 1 L ð Þ To see the effect of our regularization, note that the corresponding disturbed solution (with data error " ¼ 0.01) causes an extremely large error kf10 j À fj kL2 ðÞ % 3:4  10 Acknowledgements The work was done when P.T Thuc was a Master student in Ho Chi Minh City University of Education Downloaded by [University of Guelph] at 05:20 13 June 2012 References [1] M.H Sadd, Elasticity Theory, Applications, and Numerics, Elsevier, Amsterdam, 2005 [2] S Timoshenko and J.N Goodier, Theory of Elasticity, McGraw-Hill, New York, 1970 [3] K Aki and P.G Richards, Quantitative Seismology Theory and Methods, Vol I, Freeman, New York, 1980 [4] M Grasselli, M Ikehata, and M Yamamoto, An inverse source problem for the Lame´ system with variable co-efficients, Appl Anal 84(4) (2005), pp 357–375 [5] D.D Trong, A.P.N Dinh, P.T Nam, and T.T Tuyen, Determination of the body force of a two-dimensional isotropic elastic body, J Comput Appl Math 229(1) (2009), pp 192–207 [6] D.D Trong, A.P.N Dinh, and P.T Nam, Determine the spacial term of a two-dimensional heat source, Appl Anal 88(3) (2009), pp 457–474 [7] B.Y Levin, Lectures on Entire Functions, Transactions of the Mathematical Monographs, Vol 150, AMS, Providence, RI, 1996 [8] D.D Trong, M.N Minh, A.P.N Dinh, and P.T Nam, Hoălder-Type approximation for the spatial source term of a backward heat equation, Numer Funct Anal Optim 31(12) (2010), pp 1386–1405  ... important in practice to find a regularization process, namely to construct an approximate solution using approximate data Recently, the regularization problem was solved partially in [5], where a. ..Inverse Problems in Science and Engineering Vol 20, No 4, June 2012, 517–532 The body force in a three-dimensional Lame system: identification and regularization Dang Duc Tronga, Phan Thanh... small This article is organized as follows In Section 2, we shall set some notations and state our main results Then we prove the uniqueness in Section and the regularization in Section Finally,