Property C and applications to inverse problems

Abstract

Let j:=-d2dx2+k2qj(x), k=const>0, j=1,2, 0<c0≤ qj(x)≤ c1, %q∈ BV([0,1]), q has finitely many discontinuity points xm∈ [0,1], and is real-analytic on the intervals [xm,xm+1] between these points. The set of such functions q is denoted by M. Only the following property of M is used: if qj∈ M, j=1,2, then the function p(x):=q2(x)-q1(x) changes sign on the interval [0, 1] at most finitely many times. Suppose that (*) ∫01p(x)u1(x,k)u2(x,k)dx=0, ∀ k>0, where p∈ M is an arbitrary fixed function, and uj solves the problem juj=0, 0≤ x≤ 1, u'j(0,k)=0, uj(0,k)=1. If (*) implies h=0, then the pair \1,2\ is said to have property C on the set M. This property is proved for the pair \1,2\. Applications to some inverse problems for a heat equation are given. the set M. This property is proved for the pair