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Well-posedness of determining the source term of an elliptic equation

Published online by Cambridge University Press:  17 April 2009

Wenhuan Yu
Wenhuan Yu
Affiliation:
Department of MathematicsTianjin University Tianjin300072 People'sRepublic of China
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Abstract

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In this paper the inverse problem for determining the source term of a linear, uniformly elliptic equation is investigated. The uniqueness of the inverse problem is proved under mild assumptions by use of the orthogonality method and an elimination method. The existence of the inverse problem is proved by means of the theory of solvable operators between Banach spaces, moreover, the continuous dependence of the solution to the inverse problem on measurement is also obtained.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 50 , Issue 3 , December 1994 , pp. 383 - 398
Copyright
Copyright © Australian Mathematical Society 1994

References

[l] Aminov, A. Kh., ‘Solvability of inverse problems’, (in Russian), Siberian Math. J. 28 (1988), 865–872.Google Scholar
[2]Cannon, J.R. and Rundell, W., ‘An inverse problem for an elliptic partial differential equation’, J. Math. Anal. Appl. 126 (1987), 329–340.CrossRefGoogle Scholar
[3]Denisov, A.M. and Solov'eva, S.I., ‘A problem for determining a coefficient of a nonlinear stationary heat equations’, (in Russian), J. Comput. Math. Math. Phys. 33 (1993), 1294–1304.Google Scholar
[4]Friedman, A. and Gustafsson, B., ‘Identification of the conductivity coefficient in an elliptic equation’, J. Math. Anal. Appl. 126 (1987), 777–787.Google Scholar
[5]Gilberg, D. and Trudinger, N.S., Elliptic partial differential equations of second order (Springer-Verlag, Berlin, Heidelberg, New York, 1977).CrossRefGoogle Scholar
[6]Hu, S.J. and Yu, W.H., ‘Identification of floated surface temperature in floated gyroscope’, in Proc. 3rd IFAC Symp. Control of Distributed Parameter Systems, Toulouce, France, (1982).Google Scholar
[7]Khaidarov, A., ‘A class of inverse problems for elliptic equations’, Soviet Math. Dokl. 30 (1984), 294–297.Google Scholar
[8]Khaidarov, A., ‘On estimates and the existence of solutions on a class of inverse problems for elliptic equations’, Soviet Math. Dokl. 35 (1987), 505–507.Google Scholar
[9]Korn, R.V. and Vogelius, M., ‘Determining conductivity by boundary measurements’, Comm. Pure Appl. Math. 37 (1984), 289–298.Google Scholar
[10]Lions, J.L. and Magenes, E., Non-homogeneous boundary value problems and applications, I and II (Springer-Verlag, Berlin, Heidelberg, New York, 1972).Google Scholar
[ll]Miranda, C., Partial differential equations of elliptic type (Springer-Verlag, Berlin, Heidelberg, New York, 1972).Google Scholar
[12]Orlovskii, D.G., ‘An inverse problem for a second-order differential equation in Banach spaces’, Differential Equations 25 (1989), 730–738.Google Scholar
[13]Prilepko, A.I., ‘On inverse problems in potential theory,’, Differential Equations 3 (1967), 14–20.Google Scholar
[14]Prilepko, A.I., ‘Inverse problems of potential theory’, Math. Notes of Academy of Sciences of USSR, Mat. Zameski 14 (1975), 990–996.Google Scholar
[15]Sun, Z., ‘On an inverse boundary value problem in two dimensions’, Comm. Partial Differential Equations 14 (1989), 1101–1113.CrossRefGoogle Scholar
[16]Sylvester, J. and Uhlmann, G., ‘Inverse boundary value problems at the boundary continuous dependence’, Comm. Pure Appl. Math. 41 (1988), 197–219.CrossRefGoogle Scholar
[17]Taylor, A.E. and Lay, D.C., Introduction to functional analysis (John Wiley and Sons, New York, 1980).Google Scholar
[18]Vabishchevich, P.N., ‘Inverse problems of finding the second member of an elliptic equation and its numerical solution’, Differential Equations 21 (1985), 201–207.Google Scholar
[19]Yu, W.H., ‘On well-posedness of an inverse problem for an elliptic system’, (in Chinese), J. Math. 7 (1987).Google Scholar
[20]Yu, W.H., ‘On determination of source terms in the 2nd order linear partial differential equations’, Acta Math. Sci. 13 (1993), 23–32.CrossRefGoogle Scholar