An inverse problem for a heat equation with piecewise- constant thermal conductivity

The governing equation is ut= a x ux x, 0 x 1, t 0, u x ,0 =0, u 0, t =0, a 1 u 1, t = f t . The extra data are u 1, t =g t . It is assumed that a x is a piecewise-constant function and f 0. It is proved that the function a x is uniquely defined by the above data. No restrictions on the number of discontinuity points of a x and on their locations are made. The number of discontinuity points is finite, but this number can be arbitrarily large. If a x C2 0,1 , then a uniqueness theorem has been established earlier for multidimensional problem, x Rn ,n 1 see A. G. Ramm, Multidimensional inverse problems and completeness of the products of solutions to PDE, J. Math. Anal. Appl., 134, 211 2005 for the stationary problem with infinitely many boundary data. The novel point in this work is the treatment of the discontinuous piecewise-constant function a x and the proof of Property C for a pair of the operators 1 , 2 , where ja− d2 /dx2 +kqj 2 x , j =1,2, and qj 2 x 0 are piecewise-constant functions, and for the pair L1 ,L2 , where Ljua− aj x u x + u, j=1,2, and aj x 0 are piecewise-constant functions. Property C stands for completeness of the set of products of solutions of homogeneous differential equations see A. G. Ramm, Inverse Problems Springer, New York, 2005 . © 2009 American Institute of Physics. DOI: 10.1063/1.3155788