Wavelets and regularization of the Cauchy problem for the Laplace equation

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1 J. Math. Anal. Appl Wavelets and regularization of the Cauchy problem for the Laplace equation Chun-Yu Qiu, Chu-Li Fu School of Mathematics and Statistics, Lanzhou University, Lanzhou , PR China Received 6 January 007 Available online 8 June 007 Submitted by R.H. Torres Abstract In this paper, a Cauchy problem for two-dimensional Laplace equation in the strip 0 <x 1 is considered again. This is a classical severely ill-posed problem, i.e., the solution if it exists) does not depend continuously on the data, a small perturbation in the data can cause a dramatically large error in the solution for 0 <x 1. The stability of the solution is restored by using a wavelet regularization method. Moreover, some sharp stable estimates between the exact solution and its approximation in H r R)- norm is also provided. 007 Elsevier Inc. All rights reserved. Keywords: Cauchy problem; Laplace equation; Meyer wavelet; Regularization 1. Introduction Many physical and engineering problems like geophysics and seismology require the solution of a Cauchy problem for the Laplace equation [1]. For example certain problems related to the search for mineral resources, which involve interpretation of the earth s gravitational and magnetic fields; the continuation of the gravitational potential observed on the surface of the earth in a direction away from the sources of the field, are equivalent to the following Cauchy problem for the Laplace equation in a strip: { uxx + u yy 0, 0 <x 1, <y<, u0,y) gy), <y<, 1.1) u x 0,y) 0, <y<. Problem 1.1) is a classical ill-posed problem: the solution if it exists) does not depend continuously on the initial data. C. Vani and A. Avudainayagam have investigated this problem in [1] and proposed a wavelet regularization method similar to the procedure given by Regińska [] to obtain a regularized solution for the problem 1.1). But The project is supported by the NNSF of China No ), the NSF of Gansu Province of China No. 3ZS051-A5-015) and the Fundamental Research Fund for Physics and Mathematics of Lanzhou University No. Lzu05005). * Corresponding author. address: fuchuli@lzu.edu.cn C.-L. Fu) X/$ see front matter 007 Elsevier Inc. All rights reserved. doi: /j.jmaa

2 C.-Y. Qiu, C.-L. Fu / J. Math. Anal. Appl there exists a insurmountable defect in the proof of the main conclusion of [], i.e., the convergence of the regularized approximation in the interval [0,e ] cannot be obtained, where e is a small constant see [3]). In addition, in [1], there is not explicit estimate of the convergence rate for the regularized approximate solution. In this paper, by using Meyer wavelet, some sharp stable estimates in H r R) and L R) are obtained. The rate of convergence of the regularized solution is faster and the convergence of the regularization solution at x 1 is obtained. Let SR) be the Schwartz space, and S R) be its dual. For a function ϕ SR), its Fourier transform ˆϕ is defined by ˆϕξ) 1 π ϕt)e itξ dt, 1.) while the Fourier transform of a tempered distribution f S R) is defined by f,ϕ) ˆ f, ˆϕ), ϕ SR). For s R, the Sobolev space H s R) consists of all tempered distributions f S R) for which fξ)1 ˆ + ξ ) s a function in L R). The norm on this space is given by is f H s : fξ) ˆ ) s dξ. 1.3) It is easy to see that H 0 R) L R), and L R) H s R) for s 0. As a solution of problem 1.1) we understand a function ux,y) satisfying 1.1) in the classical sense, and for every fixed x [0, 1], the function ux, ) belongs to L R). Now we are ready to examine 1.1) in frequency space. Let ux,y) be the solution of problem 1.1), the Fourier transform ûx, ξ) of ux, y) about variable y satisfies û xx x, ξ) ξ ûx, ξ), ξ R, û0,ξ)ĝξ), 1.4) û x 0,ξ) 0. The solution of problem 1.4) is ûx, ξ) ĝξ)coshxξ). 1.5) Denote f ) : u1, ), then ˆ fξ)ĝξ)coshξ), 1.6) and the solution ux,y) of problem 1.1) can be expressed by ux, t) 1 π e iξt coshxξ)ĝξ)dξ. 1.7) Because ûx, ) L R) for x [0, 1], so from 1.5) we know that ĝξ), which is the Fourier transform of exact data function gt), must decay rapidly as ξ. Small errors in high frequency components can blow up and completely destroy the solution for 0 <x 1. Such a decay is not likely to occur in the Fourier transform of the measured data g m y) at x 0, its Fourier transform ĝ m ξ) is merely in L R). The Meyer wavelet has a very good local property in frequency domain, i.e., for fixed index J, the Fourier transform of the scaling functions in V J and the wavelet functions in W J have common compact support, respectively. Problem 1.1) will become well-posed in the scale space V J. So Meyer wavelet will be applied to formulate a regularized solution of problem 1.1) in Section, by appropriate choice of J, which converges to the exact one when data error tends to zero.

3 144 C.-Y. Qiu, C.-L. Fu / J. Math. Anal. Appl Meyer wavelet and auxiliary results Let ϕ be the Meyer scaling function defined by its Fourier transform [4] π) 1, ξ 3 π, ˆϕξ) π) 1 cos[ π ν 4π 3 ξ 1)], 3 π ξ 4 3 π, 0, otherwise, where ν is a C k function 0 k ) with { 0, x 0, νx) 1, x 1, and νx) + ν1 x) 1. Then ˆϕ is a C k function and the corresponding wavelet function ψ is given by π) 1 e iξ/ sin[ π ˆψξ) ν 3 π ξ 1)], 3 π ξ 4 3 π, π) 1 e iξ/ cos[ π ν 4π 3 ξ 1)], 4 3 π ξ 8 3 π, 0, otherwise. The supports of ˆϕ and ˆψ are Supp ˆϕ [ 43 π, 43 ] π, Supp ˆψ [ 83 ] [ π, 3 π 3 π, 8 ] 3 π. From [4], we see that the functions.1).) ψ jk t) : j ψ j t k ), j,k Z, constitute an orthonormal basis of the L R). It is easy to see that ˆψ jk ξ) j e ik j ξ ˆψ j ξ ), and Supp ˆψ jk [ 83 πj, 3 ] [ πj 3 πj, 8 ] 3 πj, k Z..3) The multiresolution analysis MRA) {V j } j Z is generated by and V j {ϕ jk : k Z}, ϕ jk t) j ϕ j t k ),j,k Z, Supp ˆϕ jk [ 43 πj, 43 ] πj, k Z..4) The orthogonal projection of a function f on the space V J is given by P J f : k Zf, ϕ Jk )ϕ Jk, f L R), where, ) denotes L -inner product, while Q J f : k Zf, ψ Jk )ψ Jk, f L R), denotes the orthogonal projection on the wavelet space W J with V J +1 V J W J. It is easy to see by.4) that P J fξ) 0 for ξ 4 3 πj.5)

4 C.-Y. Qiu, C.-L. Fu / J. Math. Anal. Appl and for j>jit follows from.3) that Q j fξ) 0 for ξ < 4 3 πj..6) Since I P J )f j J Q j f, and from.6) we have I PJ )f )ˆ ξ) Q J fξ) for ξ < 4 3 πj..7) The following inequality for differential operators D k, k N is known [5]: Lemma.1. Let {V j } j Z be Meyer s MRA and D k dk be differential operator. Suppose J N, r R, then for all dt h V k J we have D k h H r C J 1)k h H r, k N,.8) where C is a positive constant. Define an operator T x : gy) ux,y) by 1.5): T x g ux,y), 0 <x 1, or equivalently, T x gξ) ĝξ)coshxξ), 0 <x 1..9) Then we have Lemma.. Let {V j } j Z be Meyer s MRA and suppose J N, r R, 0 x 1. Then for all h V J we have T x h H r C exp J 1) x ) h H r, where C is the same as in Lemma.1..10) Proof. For h V J, by definition 1.3), Parseval equality, formula.9) and Lemma.1, we have T x h H r T x hξ) ) r dξ ĥξ) coshxξ) ) r dξ k0 k0 x k k)! ξ k 1 ĥξ) + ξ ) r dξ k0 x k k)! x k D k h k)! H r iξ) k ĥ ) r dξ

5 1444 C.-Y. Qiu, C.-L. Fu / J. Math. Anal. Appl C k0 x k k)! J 1)k h H r C cosh J 1 x ) h H r C exp { J 1 x } h H r. 3. Regularization and error estimates coshxξ) e x ξ ) In this section we suppose that the function g ) L R) is exact data and g m is measured one with g g m H r for some r 0. Since g m belongs, in general, to L R) H r R) for r 0, so r should not be positive. We will give an approximation of exact solution ux, ) for 0 <x 1. For that we need an additional condition, i.e. fy): u1,y)belongs to H s R) for some s r, and f H s E. Let T x,j : T x P J, we can show that it approximates to T x in a stable way for an appropriate choice of J N. In fact, we have Theorem 3.1. For s r, suppose condition 3.1) holds and a priori bound 3.) also holds for r min{0,s}, ifwe take E J : [log ln ln E ) s r) )))], 3.3) where [a] with square bracket denotes the largest integer less than or equal to a R, then there holds the following stability estimate { T x g T x,j g m H r C + C + 1) ln E ) ln E + lnln E ) s r) where constant C is the same constant appearing in Lemmas.1 and.. Proof. 3.1) 3.) ) s r } ln E ) s r)x E x 1 x, 3.4) T x g T x,j g m H r T x g T x,j g H r + T x,j g T x,j g m H r. 3.5) By Lemma. and condition 3.1) we can see that the second term of the right-hand side of 3.5) satisfies T x,j g T x,j g m H r T x P J g g m ) H r C exp { J 1 x } PJ g g m ) H r C exp { J 1 x }. 3.6) For the first term of the right-hand side of 3.5), from.5) we have T x g T x,j g H r Tx I P J )g H r coshxξ) I PJ )g )ˆ ξ) ) r dξ coshxξ)ĝξ) ) r dξ ξ 4 3 πj

6 C.-Y. Qiu, C.-L. Fu / J. Math. Anal. Appl coshxξ) I PJ )g )ˆ ξ) ) r dξ ξ < 4 3 πj : I 1 + I. Note that 1.6) and cosh xξ e x ξ, cosh ξ 1 e ξ we have I 1 ξ 4 3 πj coshxξ) ˆ fξ) coshξ) ) r dξ e 1 x) ξ ˆ fξ) ) r dξ 3.7) ξ 4 3 πj sup e 1 x) ξ 1 ξ 4 ) s r)/ 3 πj sup e 1 x) 4 3 π J 1 ξ πj 3 πj ) f s r H s ξ 4 3 πj ˆ fξ) ) s dξ Js r) exp { 1 x) J } E. 3.8) From.7), Lemma., and noting that Q J g W J V J +1, it is easy to see that the integral I in 3.7) satisfies I coshxξ) Q J gξ) ) r dξ Tx Q J g H r C exp { J x } Q J g H r. ξ < 4 3 πj Let χ J be the characteristic function of interval [ 3 πj, 3 πj ], we introduce the operator M J defined by the following equation M J g 1 χ J )ĝ. Noting that g L R), the Parseval formula, and.3) we know Q J g g, ψ JK )ψ JK ĝ, ˆψ JK )ψ JK ) 1 χj )ĝ, ˆψ JK ψjk M J g,ψ JK )ψ JK k Z k Z k Z k Z Q J M J g. So, by 1.6) there holds Q J g H r Q J M J g H r M J g H r ξ > 3 πj ξ > 3 πj ξ > 3 πj 1 fξ) coshξ) ˆ ) r dξ ĝξ) ) r dξ 1 e ξ ˆ ) s r)/ fξ) ) s dξ sup e ξ 1 ξ > ξ s r f H s 3 πj Js r) exp { J } E.

7 1446 C.-Y. Qiu, C.-L. Fu / J. Math. Anal. Appl Therefore I C Js r) exp { 1 x) J } E. Together with 3.8) we get T x g T x,j g H r C + 1) Js r) exp { 1 x) J } E. 3.10) Combining 3.10) with 3.6) we obtain T x g T x,j g m H r C exp { J 1 x } + C + 1) Js r) exp { 1 x) J } E, 3.11) where the constant C is the same constant appearing in.10) of Lemma.. We can obtain a stable estimate of Hölder-type by using 3.11). For that we choose J as in 3.3), and note E log ln ln E ) s r) ))) E 1 + log ln ln E ) s r) )) we get exp { J 1 x } { E exp x ln exp { 1 x) J } exp { 1 x) ln ln E ) s r) )} E 1 x ln E ) s r)1 x) E 1 x), E ln E ) s r) ) x 1 x ln E ) s r)x E x, ln E ) s r) ))} and J E s r) ln ln E ) s r) )) s r) ) 1 s r) ln E + lnln E. ) s r) Summarizing the above estimates we get the final estimate T x g T x,j g m H r C 1 x ln E ) s r)x E x + C + 1) { C + C + 1) The proof is completed. ln E ) ln E + lnln E ) s r) Remark. When s r 0 in 3.4), we can obtain a L -estimate 1 ln E + lnln E ) s r) ) s r } ln E ) s r)x 1 x E x. E ln E ) s r) ) 1 x) ) s r 1 x ln E ) s r)1 x) E x T x g T x,j g m L 3C + )E x 1 x : C 1 E x 1 x. 3.1) This result is order optimal [6]. Therefore we cannot expect to find a numerical method for approximating solution of 1.1) that satisfies a better estimate in L -sense, except for more delicate choice of coefficient C 1.This suggests that wavelets must be useful for solving the considered ill-posed problem. Acknowledgment 3.9) The authors thank the referees for their careful reading and valuable comments and suggestions on the manuscript which led to the improvement version.

8 C.-Y. Qiu, C.-L. Fu / J. Math. Anal. Appl References [1] C. Vani, A. Avudainayagam, Regularized solution of the Cauchy problem for the Laplace equation using Meyer wavelets, Math. Comput. Modelling [] T. Regińska, Sideways heat equation and wavelets, J. Comput. Appl. Math ) [3] C.L. Fu, C.Y. Qiu, Y.B. Zhu, A note on Sideways heat equation and wavelets and constant e, Comput. Math. Appl [4] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 199. [5] Dinh Nho Hào, A. Schneider, H.-J. Reinhardt, Regularization of a non-characteristic Cauchy problem for a parabolic equation, Inverse Problems [6] U. Tautenhahn, Optimal stable solution of Cauchy problem for elliptic equation, J. Anal. Appl. 15 4) 1996)