Property C for ODE and applications to an inverse problem for a heat equation

• Autorzy: Ramm A. G.
• Języki publikacji: EN

Abstrakty

EN

Let lj := -d2 /dx2 + k2qj(x), k = const > 0, j = 1,2, 0 < ess inf qj(x) ≤ ess sup qj(x) < ∞. Suppose that (∗) [integral of, between limits 1 and 0] p(x)u1(x, k)u2(x, k) dx = 0 for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0, 1], which changes sign finitely many times, and uj solves the problem ljuj = 0, 0 ≤ x ≤ 1, u'j(0, k) = 0, uj(0, k) = 1. It is proved that (∗) implies p = 0. This result is applied to an inverse problem for a heat equation.

Słowa kluczowe

EN

property C; inverse problems; heat equation; discontinuous coefficients

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2009
• Tom: Vol. 57, no 3-4
• Strony: 243--249
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Ramm A. G.

• Department of Mathematics, Kansas State University, Manhattan, KS 66506-2602, U.S.A.,

Bibliografia

• [1] S. Gutman and J. Ha, Identifiability of piecewise constant conductivity in a heat conduction process, SIAM J. Control Optim. 46 (2007), 694-713.
• [2] N. S. Hoang and A. G. Ramm, An inverse problem for a heat equation with piecewise-constant thermal conductivity, J. Math. Phys. 50 (2009), 063512.
• [3] A. G. Ramm, On completeness of the products of harmonic functions, Proc. Amer. Math. Soc. 99 (1986), 253-256.
• [4] A. G. Ramm, Completeness of the products of solutions to PDE and uniqueness theorems in inverse scattering, Inverse Problems 3 (1987), L77-L82.
• [5] A. G. Ramm, Multidimensional inverse problems and completeness of the products of solutions to PDE, J. Math. Anal. Appl. 134 (1988), 211-253; Addenda, J. Math. Anal. Appl. 139 (1989), 302.
• [6] A. G. Ramm, Completeness of the products of solutions of PDE and inverse problems, Inverse Problems 6 (1990), 643-664.
• [7] A. G. Ramm, Necessary and sufficient condition for a PDE to have property C, J. Math. Anal. Appl. 156 (1991), 505-509.
• [8] A. G. Ramm, Property C for ODE and applications to inverse problems, in: Operator Theory and Its Applications, Amer. Math. Soc., Fields Inst. Comm. 25, Providence, RI, Amer. Math. Soc., 2000, 15-75.
• [9] A. G. Ramm, An inverse problem for the heat equation, J. Math. Anal. Appl. 264 (2001), 691-697.
• [10] A. G. Ramm, An inverse problem for the heat equation II, Appl. Anal. 81 (2002), 929-937.
• [11] A. G. Ramm, One-dimensional inverse scattering and spectral problems, Cubo Math. J. 6 (2004), 313-426.
• [12] A. G. Ramm, Inverse Problems, Springer, New York, 2005.
• [13] A. G. Ramm, Scattering by Obstacles, Reidel, Dordrecht, 1986.