An inverse heat equation in two space dimensions.
The author generalizes a result of [\textit{A. G. Ramm}, Proc. R. Soc. Edinb., Sect. A 123, No. 6, 973--976 (1993; Zbl 0804.35153)], to the two-dimensional case. The argument is applicable to the multidimensional case as well. The following inverse problem is studied: Consider a bounded domain \(D\) with a sufficiently smooth boundary \(S\), \(D=D_1\cup D_2\), and denote by \(\Gamma\) the sufficiently smooth surface, which is the common part of the boundary of \(D_1\) and \(D_2\). Let \(u_t=\nabla (a(x)\nabla u)\) in \(D\), \(u(x,0)=f(x)\), \(u=0\) on \(\partial D_1\setminus \Gamma\), \(u=g\) on \(\Gamma\). Denote \(\psi:=u\) on \(\partial D_2\setminus \Gamma\). Assume that \(a\) is strictly positive and sufficiently smooth, say \(C^1\). The inverse problem consists of finding \(\psi\) given \(a,f,\) and \(g\). The author proves that this problem has at most one solution.
scientific article; zbMATH DE number 2140798