Property C and applications to inverse problems

2009/09/02 by A. G. Ramm, А. Г. Рамм, Ramm, A. G.
Computer Science · Mathematics · Physics and Astronomy · #34E05 #35R30 #74J25 #Advanced Mathematical Modeling in Engineering #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Numerical methods in inverse problems #math-ph #math.MP #msc:34E05 #msc:35R30 #msc:74J25

paper · pdf · doi:10.48550/arxiv.0909.0523

arxiv created 2009/09/02 · openalex publication_date 2009/09/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ℓ_j:=-(d²)/(dx²)+k²q_j(x), k=const>0, j=1,2, 0<c₀≤ q_j(x)≤ c₁, %q∈ BV([0,1]), q has finitely many discontinuity points x_m∈ [0,1], and is real-analytic on the intervals [x_m,x_m+1] between these points. The set of such functions q is denoted by M. Only the following property of M is used: if q_j∈ 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 u_j solves the problem ℓ_ju_j=0, 0≤ x≤ 1, u'_j(0,k)=0, u_j(0,k)=1. If (*) implies h=0, then the pair \ℓ₁,ℓ₂\ is said to have property C on the set M. This property is proved for the pair \ℓ₁,ℓ₂\. Applications to some inverse problems for a heat equation are given. the set M. This property is proved for the pair

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