HP estimation for the Cauchy problem for nonlinear elliptic equation

oblem for a ND nonlinear elliptic equation in a bounded domain. As we know, the problem is severely ill-posed. We apply the Fourier truncation method to regularize the problem. Error estimates between the regularized solution and the exact solution are established in p H space under some priori assumptions on the exact solution.

TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ T5- 2017 HP estimation for the Cauchy problem for nonlinear elliptic equation  Le Duc Thang University of Science, VNU- HCM Ho Chi Minh City Industry and Trade College (Received on 5th December 2016, accepted on 28 th November 2017) ABSTRACT In this paper, we investigate the Cauchy Error estimates between the regularized solution p problem for a ND nonlinear elliptic equation in a and the exact solution are established in H space bounded domain As we know, the problem is under some priori assumptions on the exact severely ill-posed We apply the Fourier solution truncation method to regularize the problem Key words: nonlinear elliptic equation, ill-posed problem, regularization, truncation method INTRODUCTION In this paper, we consider the Cauchy problem for a nonlinear elliptic equation in a bounded domain The problem has the form u F (x u (x , x N ) u (x ,T ) ux (x ,T ) , x N , u ( x , x N )), 0, (x ), 0, x N ( x , xN ) ( x , xN ) x , is a positive constant, , N is a natural number and N , the function L ( ) is known and F is called the source function It is well-known the above problems is severely ill-posed in the sense of Hadamard In fact, for a given final data, we are not sure that a solution of the problem exists In the case a solution exists, it may not depend continuously on the final data The problem has many various applications, for example in electrocardiography [7], astrophysics [6] and plasma physics [15, 16] (0, N ) (1) T Where ( 0,T ), ( 0,T ), In the past, there have been many studies on the Cauchy problem for linear homogeneous elliptic equations, [1, 5, 9, 10, 12] However, the literature on the nonlinear elliptic equation is quite scarce We mention here a nonlinear elliptic problem of [13] with globally Lipschitz source terms, where authors approximated the problem by a truncation method Using the method in [13,14], we study the Cauchy problem for nonlinear elliptic in multidimensional domain The paper is organized as follows In Section 2, we present the solution of equation (1) In Section 3, we present the main results on regularization theory for local Lipschitz source function We finish the paper with a remark SOLUTION OF THE PROBLEM Assume that problem (1) has a unique solution u(x , x N ) By using the method of separation of variables, we can show that solution of the problem has the form Trang 193 Science & Technology Development, Vol 5, No.T20- 2017 u(x , x N ) n1 n2 nN T x N ) n12 n12 xN Indeed, let u(x , x N ) n1n2 nN nN2 ) nN2 un n n nN 1 2 (x ) n22 n22 n1 n2 orthonormal basis n22 nN2 ) n1n2 nN 1 sinh(( + x N ) n12 cosh((T N Fn n n N 1 (u )( )d n1n2 nN (x ) (2) (x N ) n1n2 nN (x ) be the Fourier series in L2 ( ) with N sin(n1x )sin(n2x ) sin(nN 1x N ) From (1), we can obtain the following ordinary differential equation d2 N dx un n un n nN nN T T where Fn n n N 1 n12 n1n2 nN d u dx N n1n2 nN un n n n22 nN2 un n n N xN Fn n nN u xN , 0,T , xN , (3) 0, (u)(x N ) u(x , x N ) N 1 xN F (x , x N , u(x , x N ) n1n2 nN n1n2 nN dx , n1n2 nN (x ) n1n2 nN (x )dx and (x )dx The equation (3) is ordinary differential equations It is easy to see that its solution is given by un n n N (x N ) cosh((T T x N ) n12 n22 x N ) n12 sinh(( n12 xN n22 nN2 n22 nN2 n1n2 nN nN2 1 Fn n n N (u )( )d (4) REGULARIZATION AND ERROR ESTIMATE FOR NONLINEAR PROBLEM WITH LOCALLY LIPSCHITZ SOURCE We know from (4) that, when n1, n2 , , nN cosh((T x N ) n12 n22 nN2 ) and become large , the terms sinh(( x N ) n12 n12 n22 n22 nN2 nN2 ) increase rather quickly Thus, these terms are the cause for instability In this paper, we use the Fourier truncated method The essence of the method is to eliminate all high frequencies from the solution, and consider the problem only for n1, n2 nN satisfying n12 n22 nN2 C Here C is a constant which will be selected appropriately as a regularization parameter which satisfies limC Trang 194 TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ T5- 2017 Let the function F : u, v [0,T ] and for any u, v satisfying such that: for each M M , there holds F (x , x N , u) F (x , x N , v) where (x , x N ) KF (M ) u v, (5) [0,T ] and F (x , x N , u) KF (M ) : sup F (x , x N , v) u v : u, v M, u We note that KF (M ) is increasing and lim KF (M ) v, (x ,x N ) , we approximate F by For all M M [0,T ] FM defined by F (x , x N , M ), F (x , x N , u(x , x N )), F (x , x N , M ), FM (x , x N , u(x , x N )) , we consider a parameter M For each u(x ,x N ) M , -M u(x ,x N ) M , u(x ,x N ) M We shall use the following well- as posed problem v PC FM x , x N , v x , x N , v x , xN x , xN x , xN 0, v x ,T 0,T , PC x , vx N x ,T 0,T , x 0, (6) where PC w w, n1 ,n2 ,nN 1 n12 n22 nN2 We show that the solution u , u , (x , x N ) 1 + L2 ( ) for all w of problem (6) satisfies the following integral equation x N ) n12 n22 n22 nN2 ) nN2 ) n1n2 nN C sinh(( x N ) n12 n12 xN 1 n12 n22 nN2 T n1n2 nN C cosh((T n1 ,n2 , ,nN n1n2 nN n22 nN2 (7) FM n1n2 nN (u )( )d n1n2 nN (x ), Lemma For u1 (x , x N ), u2 (x , x N ) , we have FM (x , x N , u2 (x , x N ) Proof If u1 (x , x N ) FM (x , x N , u1(x , x N ) M and u2 (x , x N ) M then FM (x , x N , u2 (x , x N ) If u1(x , x N ) M u2 (x , x N ) KF (M ) u2 (x , x N ) u1(x , x N ) FM (x , x N , u1(x , x N ) M then FM (x , x N , u2 (x , x N ) FM (x , x N , u1(x , x N )) FM (x , x N , u2 (x , x N ) FM (x , x N , M ) KF (M ) u2 (x , x N ) u1(x , x N ) If u1(x ', x N ) M M u2 (x ', x N ) then Trang 195 Science & Technology Development, Vol 5, No.T20- 2017 FM (x , x N , u2 (x , x N ) FM (x , x N , u1(x , x N )) FM (x , x N , M ) FM (x , x N , M ) KF (M ) u2 (x , x N ) u1(x , x N ) If M u1(x , x N ), u2 (x , x N ) M then FM (x , x N , u2 (x , x N ) FM (x , x N , u1(x , x N )) F (x , x N , u2 (x , x N ) F (x , x N , u1(x , x N )) KF (M ) u2 (x , x N ) u1(x , x N ) This completes the proof Lemma Let u be the exact solution to problem (1) Then we have the following estimate u , (x N ) PC u(x N ) exp(2(T L( ) x N )C ) L2 ( ) T +2K F2 (M )(T x N ) exp(2( x N )C ) u , ( ) u( ) L2 ( ) xN and u , we have Proof From the definition of u , u , (x N ) PC u(x N ) 2 L( ) +2 n1 ,n2 , ,nN 1 n12 n22 nN2 n1 ,n2 , ,nN n12 n22 n22 nN2 ( x N ) n12 sinh(( xN x N ) n12 cosh((T 2 n12 n22 nN T d n1n2 nN n1n2 nN 1 C n22 nN2 nN2 ) ((FM )n n n N (u , )( ) Fn n n N (u )( ))d C T 2exp(2(T -x N )C ) L( ) +2(T -x N ) exp(2( x N )C FM ( , u , ( )) F ( , u( )) xN d L2 ( ) (8) Since lim M , there exists M such that M , for a sufficiently small For M we have FM (x , x N , u(x , x N )) u L ([0,T ];L2 ( )) F (x , x N , u(x , x N )) Using the Lipschitz property of FM as in Lemma 1, we get FM ( , u , ( )) F ( , u( )) K F2 (M ) u , ( ) L2 ( ) u( ) L2 ( ) (9) Combining (8) and (9), we complete the proof of Lemma Theorem Let and let F be the function defined in (5) Then the problem (6) has a unique solution u , C ([0,T ]; L2 ( )) C ([0,T ]; L2 ( )) Put Proof We prove the equation (7) has a unique solution u , (u , )(x , x N ) (x , x N ) G(x , x ) where (x , x N ) cosh((T n1 ,n2 , ,nN 1 n12 n22 nN2 Trang 196 C ) x N ) n12 n22 nN2 ) n1n2 nN n1n2 nN (x ) TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ T5- 2017 and T n1 ,n2 , ,nN 1 n12 xN n12 n22 nN2 x N ) n12 sinh(( G(x , x N ) n22 n22 nN2 ) nN2 FM n1n2 nN (u , )( )d n1n2 nN (x ) C We claim that p p (v , )(x N ) (w )(x N ) , K F2 (M )T exp(2TC ) p v p! L2 ( ) w , (10) , for p , where is the sup norm in C ([0,T ]; L2 ( )) We shall prove the above inequality by induction For p , using the inequality T x N ) n12 sinh(( n xN n 2 n22 n nN2 ) d N exp(2 n12 n22 nN2 1T )T and using Lemma 1, we have (v , )(x N ) (w T n1 ,n2 , ,nN 1 n12 n22 nN2 , )(x N ) L2 ( ) x N ) n12 sinh(( n12 xN n22 n22 nN2 nN2 ) FM n1n2 nN 1 (v , )( ) FM T n1 n2 nN 1 n1n2 nN (v , )( ) T exp(2TC )T FM ( , v , ( )) FM ( , w , ( )) xN Thus (10) holds for p k We have , )(x N ) k p (w FM d )(x N ) , )( ) d K F2 (M )exp(2TC )T v L( ) p Suppose that (10) holds for , (w n1n2 nN w , k We prove that (10) holds for L ( ) exp(2TC )T FM ( , xN k (v ( ))) , k FM ( , (w T k (v , )( ) k (w , (v , )(x N ) (w , )(x N ) 2 L( ) F K (M )K 2k F exp(2TC )T exp(2TC k ) T )( ) xN k ( ))) , xN k , T K F2 (M ) exp(2TC )T k , FM xN (v )( ) d C exp(2TC )T k n1n2 nN (w L2 ( ) d L2 ( ) d k 1! v, w , Therefore, we get Trang 197 Science & Technology Development, Vol 5, No.T20- 2017 p p (v , )(x N ) (w )(x N ) , K F2 (M )T exp(2TC ) C ([0,T ]; L2 ( )) Let us consider : C ([0,T ]; L2 ( )) lim w , p0 (u) In fact, one has (u , ) u has a unique solution u , p0 ( (u (u )) , (11) C ([0,T ]; L2 ( )) It is easy to see that p0 As a consequence, there exists a positive integer number p0 such that the equation , , p p! p v p! L2 ( ) for all v , , w , K F2 (M )T exp(2TC ) p , C ([0,T ]; L2 ( )) We claim that p0 ) By the uniqueness of the fixed point of (u) u , , i.e., the equation is a contraction It follows that (u , ) u, , one has C ([0,T ]; L2 ( )) u has a unique solution u , To show error estimates between the exact solution and the regularized solution, we need the exact solution belonging to the Gevrey space Definition (Gevrey-type space) (see [2, 3]) The Gevrey class of functions of order s and index is denoted by G s /2 and is defined as G s /2 L2 ( ) : f (n12 n1 n2 nN n22 nN2 )s /2 exp(2 n12 n22 nN2 ) | f , n1n2 nN It is a Hilbert space with the following norm f (n12 G s /2 n1 n2 nN n22 nN2 )s /2 exp(2 n12 n22 nN2 ) | f , For a Hilbert space H , we denote L (0,T ; H ) f : [0,T ] n1n2 nN |2 H | ess sup|f(t)|H t T and f ess sup f (t ) L (0,T ;H ) H t T We consider some assumptions on the exact solution as the following: ess sup xN T (n12 n1 n2 nN ess sup xN (13) for all x N exp(2(x N nN n22 nN2 ) n12 n22 .nN2 )un2 n n N , I 1, I , are positive constants k G , we have the following inequality w Trang 198 nN2 ) exp 2x N n12 1 [0,T ] , where , Lemma For any w un2 n n n1 n2 n22 PC w L2 ( ) C ke C w Gk (x N ) I2 , N (x N ) I , (12) |2 TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ T5- 2017 G k , we get Proof For w w PC w 2 L( ) w, = n1 ,n2 , ,nN n12 C 2k e n22 e nN 1 C w Gk C n12 n22 nN2 k n12 exp n22 nN2 w, n12 n22 nN2 2k C n1 ,n2 , ,nN C n1n2 nN n1n2 nN C This completes the proof The following theorem provides some error estimates in the L2 norm when the exact solution belongs to the Gevrey space Theorem Assume that the problem (1) has a unique solution u which satisfies (12) If C and M are chosen such that lim eTC TC and lim exp(2KF2 (M )T )C lim exp(2KF2 (M )T ) e 0 , then we have u , (x N ) u(x N ) Proof Since u 2C L2 ( ) I 22 2e 2TC exp(2K F2 (M )T )e xNC (14) Gx then using Lemma 3, we get N u(x N ) PC u(x N ) C L ( ) 2x N C e w Gx N Lemma and the triangle inequality lead to u , (x N ) 2C e u(x N ) 2x N C 2 u , (x N ) L( ) u(x N ) Gx exp(2(T PC u(x N ) 2 u(x N ) L( ) PC u(x N ) L2 ( ) x N )C L2 ( ) N T +4K F2 (M )(T xN ) x N )C ) u exp(2( , ( ) u( ) xN Multiplying (15) by e e 2x N C 2x N C L2 ( ) d (15) and applying Gronwall’s inequality, we get u , (x N ) u(x N ) 2 L( ) 2C sup u(x N ) xN T Gx 4e 2TC exp 4K F2 (M )T , N which leads to the desired result u , (x N ) u(x N ) L2 ( ) 2C I 22 4e 2TC exp 2K F2 (M )T e xNC This completes the proof Trang 199 Science & Technology Development, Vol 5, No.T20- 2017 The next theorem provides an error estimate in the Hilbert scales {H p ( )}p which is equipped with a norm defined by f (n12 p H ( ) n1 n2 nN n22 nN2 )p f , n1n2 nN 1 Theorem Assume that the problem (1) has a unique u which satisfies (13) Let us choose C  and M such that lim C peTC )T and lim eKF (M 0 u , (x N ) u(x N ) 1)e ( p H ( ) e C KF2 (M )T KF2 (M )T Cp C e TC C pe lim e I3 2e nN2 KF2 (M )T TC , then we have xN C C pe e xN , [0,T ] Proof First, we have u , (x N ) PC u(x N ) n12 p H ( ) n1 ,n2 , ,nN n12 u , (x N ) PC u(x N ) n22 n22 1 nN2 u , (x , x N ) u(x , x N ), n1n2 nN (x ) C C 2p Hp( ) p u n1 ,n2 , ,nN , 1 n12 n22 nN (x , x N ) u(x , x N ), n1n2 nN (x ) C 2p u , (x N ) u(x N ) C It follows from theorem that u , (x N ) PC u(x N ) Hp( ) On the other hand, we consider the function G( ) p Since G ( ) D e i.e., 2(x N n12 n22 p e 2e C sup u(x N ) xN T D Gx0 nN2 p n22 exp nN2 2(x N nN2 2e 2TC (16) N D>0 , D ) , it follows that G is decreasing when D (p p , then for n12 )C xN C exp(2K F2 (M )T )C pe p Thus if e p (T ) C , we get ) n12 n22 nN2 C pe 2(x N )C , and u(x N ) PC u(x N ) Hp( ) n12 n1 ,n2 , ,nN n12 n22 C p exp n22 p nN2 C e (x ) C 2(x N )C )C ) n12 exp 2(x N 1 n12 n22 nN2 (x N n1n2 nN n1 ,n2 , ,nN 2p u(x , x N ), sup u(x N ) xN T Gx0 n22 nN2 u(x , x N ), n1n2 nN (x ) C N Therefore u(x N ) PC u(x N ) Hp( ) C pe (x N Combining (16) and (17), we get Trang 200 )C sup u(x N ) xN T Gx0 N (17) 2 L2 ( ) TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SOÁ T5- 2017 u , (x N ) u(x N ) u , (x N ) Hp( ) exp 2K F2 (M )T a2 The inequality b2 2e C PC u(x N ) sup u(x N ) xN T a u , (x N ) u(x N ) b for a, b p H ( ) 0, 2e 2TC sup u(x N ) xN T KF2 (M )T 1)e ln C e r ln 2T 2e T CONCLUSION C xNC C pe 2I 2 In this paper, we investigate the Cauchy problem for a ND nonlinear elliptic equation in a bounded domain We apply the Fourier truncation method for regularizing the problem Error estimates between the regularized solution and exact solution are established in HP space under some priori assumptions on the exact solution In future, we will TC e C pe 2 T xNC , TC lim exp(2KF2 (M )T ) e Then (14) becomes L2 ( ) e N KF2 (M )T ln u(x N ) Gx0 (0,1) and M such that , for I3 It is easy to check that lim exp(2KF2 (M )T )C u , (x N ) Hp( ) leads to T KF M for r Gx0 PC u(x N ) N ( Remark In theorem 2, let us choose C u(x N ) Hp( ) ln T ln r xN T consider the Cauchy problem for a coupled system for nonlinear elliptic equations in three dimensions Acknowledgment: The author thanks the anonymous referees for their valuable suggestions and comments leading to the improvement of the paper Đánh giá HP cho toán Cauchy cho phương trình elliptic phi tuyến  Lê Đức Thắng Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM Trường Cao Đẳng Cơng Thương TPHCM TĨM TẮT Trong báo này, chúng tơi nghiên cứu tốn Cauchy cho phương trình elliptic phi tuyến miền bị chặn không gian nhiều chiều Như biết, tốn khơng chỉnh Chúng sử dụng phương pháp chặt cụt Fourier để chỉnh hóa nghiệm tốn Đánh giá sai số nghiệm chỉnh hóa nghiệm xác thiết lập không gian HP với giả thiết cho trước tính trơn nghiệm xác Trang 201 Science & Technology Development, Vol 5, No.T20- 2017 Từ khóa: phương trình elliptic phi tuyến, tốn khơng chỉnh, chỉnh hóa, phương pháp chặt cụt TÀI LIỆU THAM KHẢO [1] L Bourgeois, J Dard, About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains, Appl Anal., 89, 1745–1768 (2010) [2] L Elden and F Berntsson, A stability estimate for a Cauchy problem for an elliptic partial differential equation, Inverse Problems, 21, 1643–1653 (2005) [3] X.L Feng, L Elden, C.L Fu, A quasi-boundaryvalue method for the Cauchy problem for elliptic equations with nonhomogeneous Neumann data, J Inverse Ill-Posed Probl., 18, 617–645 (2010) [4] J Hadamard, Lectures on the Cauchy Problem in Linear Differential Equations, Yale University Press, New Haven, CT (1923) [5] D.N Hao, N.V Duc, D Sahli, A non-local boundary value problem method for the Cauchy problem for elliptic equations, Inverse Problems, 25:055002 (2009) [6] Dan Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin Heildellberg, Berlin (1981) [7] V Isakov, Inverse Problems for Partial Differential Equations, volume 127 of Applied Mathematical Sciences Springer, New York, second edition, (2006) [8] Z Qian, C.L Fu, Z.P Li, Two regularization methods for a Cauchy problem for the Laplace equation, J Math Anal Appl., 338, 479–489 (2008) [9] T Reginska, R Kazimierz, Approximate solution of a Cauchy problem for the Helmholtz equation, Inverse Problems, 22, 975–989 (2006) Trang 202 [10] D.D Trong, N.H Tuan, Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation, Nonlinear Anal., 71, 4167–4176 (2009) [11] N.H Tuan, T.Q Viet, N.V Thinh, Some remarks on a modified Helmholtz equation with inhomogeneous source Appl Math Model, 37, 793–814 (2013) [12] N.H Tuan, D D Trong and P H Quan, A note on a Cauchy problem for the Laplace equation: regularization and error estimates, Appl Math Comput , 217, 2913–2922 (2010) [13] H Zhang, T Wei, A Fourier truncated regularization method for a Cauchy problem of a semi-linear elliptic equation, J Inverse Ill-Posed Probl., 22, 143–168 (2014) [14] N.H Tuan, L.D Thang, V.A Khoa, T Tran, On an inverse boundary value problem of a nonlinear elliptic equation in three dimensions J Math Anal Appl 426 , 1232–1261 (2015) [15] V.B Glasko, E A Mudretsova, V N Strakhov, Inverse problems in the gravimetry and magnetometry, Ill-Posed Prob-lems in the Natural Science ed A N Tikhonov and A V Goncharskii (Moscow: Moscow State University Press), C 89– 102 (1987) [16] L Bourgeois, A stability estimate for ill-posed elliptic Cauchy problems in a domain with corners, C R Math Acad Sci Paris, 345, 385–390 (2007)  ... investigate the Cauchy problem for a ND nonlinear elliptic equation in a bounded domain We apply the Fourier truncation method for regularizing the problem Error estimates between the regularized... r xN T consider the Cauchy problem for a coupled system for nonlinear elliptic equations in three dimensions Acknowledgment: The author thanks the anonymous referees for their valuable suggestions... a Cauchy problem for the Laplace equation, J Math Anal Appl., 338, 479–489 (2008) [9] T Reginska, R Kazimierz, Approximate solution of a Cauchy problem for the Helmholtz equation, Inverse Problems,