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

The governing equation is $u_t = (a(x)u_x)_x$, $0\le x\le 1$, $t>0$, $u(x,0)=0$, $u(0,t)=0$, $a(1)u'(1,t)=f(t)$. The extra data are $u(1,t)=g(t)$. It is assumed that $a(x)$ is a piecewise-constant function, and $f\not\equiv 0$. 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 $a(x)\in C^2[0,1]$, then a uniqueness theorem has been established earlier for multidimensional problem, $x\in \mathbb{R}^n, n>1$ (see MR1211417 (94e:35004)) 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 $\{\ell_1, \ell_2 \}$, where $\ell_j:= -\frac{d^2}{dx^2} + k^2 q_j^2(x)$, $j=1,2$, and $q_j^2(x)>0$ are piecewise-constant functions, and for the pair $\{L_1, L_2 \}$, where $L_ju:=-[a_j(x)u'(x)]'+λu$, $j=1,2$, and $a_j(x)>0$ are piecewise-constant functions. Property C stands for completeness of the set of products of solutions of homogeneous differential equations (see MR1759536 (2001f:34048))