Research Article | Open Access
On an Inverse Boundary Value Problem for a Fourth Order Elliptic Equation with Integral Condition
An inverse boundary value problem for a fourth order elliptic equation is investigated. At first the initial problem is reduced to the equivalent problem for which the existence and uniqueness theorem of the solution is proved. Further, using these facts, the existence and uniqueness of the classic solution of the initial problem are proved.
The inverse problems are favorably developing section of up-to-date mathematics. Recently, the inverse problems are widely applied in various fields of science.
Different inverse problems for various types of partial differential equations have been studied in many papers. First of all we note the papers of Tikhonov , Lavrent’ev [2, 3], Denisov , Ivanchov , and their followers.
The goal of our paper is to prove the uniqueness and existence of the solution of a boundary value problem for a fourth order elliptic equation with integral condition.
2. Problem Statement and Its Reduction to Equivalent Problem
Consider the following equation: in the domain an inverse boundary problem with the boundary conditions the integral condition and with the additional condition where are the given functions and are the required functions.
Definition 1. The classic solution of problems (1)–(5) is the pair of the functions and possessing the following properties:(1)the function is continuous in together with all its derivatives contained in (1);(2)the function is continuous on ;(3)all the conditions of (1)–(5) are satisfied in the ordinary sense.
For investigating problems (1)–(5), at first consider the following problem: where is a given function, is a desired function, and under the solution of problems (6) and (7) we will understand a function satisfied in equation (6) and conditions (7).
The following lemma is valid.
Alongside with inverse boundary value problem, consider the following auxiliary inverse boundary value problem. It is required to determine the pair of the functions and possessing the properties and of definition of the classic solution of problems (1)–(5) from relations (1)–(3) and
The following lemma is valid.
Lemma 3. Let , let , and let the following consistency conditions be fulfilled: Then the following statements are true.(1)Each classic solution of problems (1)–(5) is the solution of problems (1)–(3), (10), and (11) as well.(2)Each solution of problems (1)–(3), (10), and (11) such that is the classic solution of problems (1)–(5).
Proof. Let be a solution of problems (1)–(5). Integrating equation (1) with respect to from 0 to 1, we have
Hence, by means of and (3) we obtain (10).
Substituting in (1), we find
Further assuming and differentiating (5) four times, we have
Taking into account last relation and condition (5) in (16) we obtain (11).
Now suppose that is a solution of problems (1)–(3), (10), and (11); moreover, (14) is fulfilled. Then, taking into account (3) and (10) in (15) we find
By (2) and (12), it is obvious that
Since by Lemma 2, problems (18), (19) have only a trivial solution, then ; that is, condition (4) is fulfilled.
Further, from (11) and (16) we get
By (2) and consistency conditions (13), we have
From (20) and (21), by Lemma 3 we conclude that condition (5) is fulfilled. The lemma is proved.
3. Investigation of the Existence and Uniqueness of the Classic Solution of the Inverse Boundary Value Problem
Proof. Let be any solution of problems (1)–(3), (10), and (11). Then, having multiplied the both sides of (1) by the function , integrating the obtained equality with respect to from 0 to 1, and using the relations
we get that (25) is satisfied.
Similarly, from (2) we get that condition (26) is fulfilled.
Thus, is a solution of problems (25), (26). Hence, it directly follows that the function satisfies in systems (28), (29). The lemma is proved.
Denote the set of all the functions of the form considered in , where each of the functions is continuous on and
In this set, we determine the operation of addition and multiplication by the number (real) in the usual way: under the zero element of this set we will understand the function on , and determine the norm in this set by the following formula:
Prove that all these spaces are Banach spaces. Indeed, the validity of the first two axioms of the norms is obvious, and validity of the third axiom of the norm is easily established by means of the summation inequality of Minkovsky; consequently, is a linear normalized space. Now prove its completeness. Let be any sequence fundamental in . Then for any there exists a number such that
Consequently, for any fixed ,
This means that for the sequence and for any fixed , the sequences are fundamental in and consequently by the completeness of they converge in the space :
Further, by (43), for any fixed number ,
Hence, by arbitrariness of (or the same, passing to limit as ), we get
Accept the denotation
Since and by (48) and also , we get that
Then, by (48) for any there exists a number such that
And this means that the sequence converges in to the element . This proves the completeness and consequently the Banach property of the space .
Denote by the space of the vector functions with the norm
It is known that and are Banach spaces.
It is easy to see that
Then from (56), we get where
From inequalities (57) we get where
So we can prove the following theorem.
Theorem 6. Let conditions (1)–(3) be fulfilled and
Proof. In the space consider
where the components of the operator are defined from the right sides of (32) and (35).
Consider the operator in the ball from . Similarly to (59), we get that for any the following estimates are valid:
Then taking into account (61) in (63) it follows that the operator acts in the ball and is contractive. Therefore, in the ball the operator has a unique fixed point that is a unique solution of (62) in the ball ; that is, it is a unique solution of systems (32), (35) in the ball .
The function as an element of the space is continuous and has continuous derivatives in .
From (25) it in easy to see that Hence it follows that is continuous in .
It is easy to verify that (1) and conditions (2), (3), (10), and (11) are satisfied in the ordinary sense.
Consequently, is a solution of problems (1)–(3), (10), and (11), and by Lemma 4 it is unique in the ball . The theorem is proved.
Theorem 7. Let all the conditions of Theorem 6 be fulfilled:
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
- A. I. Tikhonov, “On the stability of inverse problems,” Doklady Akademii Nauk SSSR, vol. 39, no. 5, pp. 195–198, 1943.
- M. M. Lavrent'ev, “On an inverse problem for the wave equation,” Doklady Akademii Nauk SSSR, vol. 157, no. 3, pp. 520–521, 1964.
- M. M. Lavrent'ev, V. G. Romanov, and S. P. Shishatsky, Ill-posed Problems of Mathematical Physics and Analysis, Nauka, Moscow, Russia, 1980.
- A. M. Denisov, Introduction to Theory of Inverse Problems, MSU, East Lansing, Mich, USA, 1994.
- M. I. Ivanchov, Inverse Problems for Equations of Parabolic Type, VNTL Publishers, Lviv, Ukraine, 2003.
- M. I. Īvanchov and N. V. Pabirivs'ka, “Simultaneous determination of two coefficients in a parabolic equation in the case of nonlocal and integral conditions,” Ukrainian Mathematical Journal, vol. 53, no. 5, pp. 589–596, 2001.
- M. I. Īvanchov, “An inverse problem with a free boundary for the heat equation,” Ukrainian Mathematical Journal, vol. 55, no. 7, pp. 901–910, 2003.
- A. I. Prilepko and A. B. Kostin, “Some inverse problems for parabolic equations with final and integral observation,” Matematichesky Sbornik, vol. 183, no. 4, pp. 49–68, 1992.
- A. I. Prilepko and D. S. Tkachenko, “Properties of solutions of a parabolic equation and the uniqueness of the solution of the inverse source problem with integral overdetermination,” Zhurnal Vychislitelnoi Matematiki and Matematicheskoy Fiziki, vol. 43, no. 4, pp. 562–570, 2003 (Russian).
- V. L. Kamynin, “On an inverse problem of determining the right-hand side of a parabolic equation with the integral overdetermination condition,” Matematicheskie Zametki, vol. 77, no. 4, pp. 522–534, 2005 (Russian).
- V. V. Solov'ev, “Inverse problems of source determination for the Poisson equation on the plane,” Zhurnal Vychislitelnoi Matematiki and Matematicheskoy fiziki, vol. 44, no. 5, pp. 862–871, 2004 (Russian).
- V. V. Solov'ev, “Inverse problems for elliptic equations on the plane. I,” Differentsial'nye Uravneniya, vol. 42, no. 8, pp. 1106–1114, 2006 (Russian).
- M. M. Hajiyev, “An inverse problem for a degenerate elliptic equation,” in Application of Functional Analysis Methods in Mathematical Physics Equations, pp. 66–71, Nauk SSSR, Novosibirsk, Russia, 1987.
- Y. T. Mehraliyev, “Inverse boundary value problem for a second order elliptic equation with additional integral condition,” Vestnik Udmurtskogo Universiteta Matematika, no. 1, pp. 32–40, 2012 (Russian).
- Y. T. Mehraliyev, “On solvability of an inverse boundary value problem for a second order elliptic equation,” Vestnic Tverskogo Gosudarstvennogo, no. 23, pp. 25–38, 2011 (Russian).
- Y. T. Mehraliyev, “An inverse boundary value problem for a fourth order elliptic equation,” Proceedings of Institute of Mathematics and Mechanics. National Academy of Sciences of Azerbaijan, vol. 36, pp. 75–90, 2012.
- K. I. Khudaverdiyev and A. A. Veliyev, Investigation of One-Dimensional Mixed Problem for a Class of Pseudohyperbolic equations of Third Order with Nonlinear Operator Right Side, Chashyoglu, Baku, Azerbaijan, 2010.
Copyright © 2014 Yashar T. Mehraliyev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.