ON THE CONDITIONS FOR UNIQUENESS AND EXISTENCE OF THE SOLUTION TO AN ACOUSTIC INVERSE PROBLEM I: THEORY

This paper which is Part I of a sequence deals with the problem of determining a radially dependent coefficient n (r) in the equation ∆ u − n² (r) u = 0, in the unit disk Ω from the Dirichlet–Neumann data pair . We prove that the sufficiency condition for uniqueness established in Ref. 2 is, in some instances, also a necessity for uniqueness. We also discuss the solvability of this inverse problem. In Part II numerical experiments will be presented which illustrate the theory developed here.