A Note on the Identifiability of Distributed Parameters in Elliptic Equations
Carmen C. Chicone, Jürgen Gerlach · SIAM Journal on Mathematical Analysis · 1987
For u and f given smooth functions on a bounded domain $\Omega $ we consider solutions of the PDE–$ - {\operatorname{div}}(a abla u) = f$ for the parameter a. This problem arises in the identification of the flow of groundwater. We say a is identifiable if, for given u and f, a is unique. Our main result shows that a is identifiable on the points in $\Omega $ which are the closure of the interior of the set of points which stay in $\Omega $ for all positive time (or negative time) under the flow of the gradient field $ abla u$. We also show a is identifiable on $\Omega $ if the set of critical points of u has nonempty interior and the co-normal derivative of u is specified on $\partial \Omega $.