Inherent identifiability of parameters in elliptic differential equations

Abstract

together with appropriate boundary conditions for u on the boundary r of Q. For given a and f this is a well-known equation in the unknown U. We shall consider the inverse problem which, givenf, consists in determining a from U. More precisely we study the question of uniqueness and continuous dependence of a in terms of U. The uniqueness problem is generally referred to as the identifiability problem. Well-known contributions are the Borg and the Gelfand Levitan theories, which require knowledge of spectral data of (1.1) (see [5, 61). For recent work an identifiability of a from Neumann boundary data for all g in the elliptic equation -V(aVu) =O, ~1 f = g, we refer to [S]. The problem of continuous dependence of a with respect to u is of equal importance as identifiability but has received much less attention so far. In [ 121 the study of the inverse problem associated with (1.1) is based on the observation that (1.1) is a hyperbolic equation in a. Employing the method of characteristics a continuous dependence analysis is given under combinations of the assumptions \Vul > 0 and ldul > 0. In this paper we concentrate on uniqueness and continuous dependence results without a priori assumptions on the sign of )Vul. The basic and very simple idea for which we give an analytical justification is that a in (1.1)