Identifiability of the Transmissivity Coefficient in an Elliptic Boundary Value Problem

Abstract

We deal with a coefficient inverse problem describing the filtration of ground water in a region $\Omega \subset R^n,n\ge 2$. Introducing a weak formulation of the problem, discretization and regularization methods can be constructed in a natural way. These methods converge to the normal solution of the problem, i.e. to a transmissivity coefficient of a minimal $L^2(\Omega )$-norm. Thus a question about $L^2$-identifiability (identifiability among functions of the class $L^2(\Omega )$ of the transmissivity coefficient arises. Our purpose is to describe subregions of $\Omega$ where the transmissivity coefficient is really $L^2$-identifiable or even $L^1$-identifiable. Thereby we succeed introducing physically realistic conditions on the data of the problem, e.g. piecewise smooth surfaces in $\Omega$ are allowed where the data of the inverse problem may have discontinuities. With some natural changes, our results about the $L^1$-identifiability extend known results about the identifiability among more smooth functions given by G. R. Richter [4], C. Chicone and J. Gerlach [1], and K. Kunisch [3].