Stability result for the inverse transmissivity problem (Q1206825)

The author considers the problem of identification of a smooth domain \(D\) in \(\mathbb{R}^ n\) entering the parabolic boundary value problem \[ u_ t=\text{div} ((a^ 0+a^ \# \chi (D)) \nabla u) \text{ in } \Omega \times(0,T) \] \[ u=g \text{ on } \partial \Omega \times(0,T),\;u=u_ 0 \text{ on } \Omega \times\{0\} \] where \(a^ 0\), \(a^ \#\) are diagonal constant matrices. To determine a smooth \(D\) one prescribes the additional Neumann data \[ a^ 0\nabla u \cdot \nu=h \text{ on } \partial \Omega \times(0,T). \] This inverse problem is fundamental in hydrology, but it has not been treated mathematically yet. Under some restrictive conditions (like \(\text{div}((a^ 0+a^ \# \chi(\Omega)) \nabla u^ 0) \in C^ \alpha)\) and \(a^ 0 \nabla u^ 0 \cdot \nu \neq 0\) on \(\partial D)\) the author obtains a (weak) stability for the inverse problem similar to those ones for elliptic equations established in the paper of \textit{H. Bellout}, \textit{A. Friedman} and \textit{V. Isakov} [Trans. Am. Math. Soc. 332, 271-296 (1992)].