Estimation of coefficients in a hyperbolic equation with impulsive inputs

DOI number:10.1515/156939406779768283

Journal:Journal of Inverse and Ill-Posed Problems

Abstract:For the solution to ∂^2_{t}u (x, t) − ∆u (x, t) + q(x)u (x, t) = δ(x_1)δ'(t) and u|_{t<0} = 0, we consider an inverse problem of determining q(x), x ∈ Ω from data f = u|_{S_T} and g = (∂u/∂ν)|_{S_T}. Here Ω ⊂ {(x_1, . . ., x_n) ∈ R^n|x_1 > 0}, n ≥ 2, is a bounded domain, S_T = {(x, t) | x ∈ ∂Ω, x_1 < t < T + x_1} and T > 0. For suitable T > 0, we prove an L^2(Ω)-size estimation of q: ||q||_{L^2(Ω)} ≤ C{||f||_{H^1(S_T)} + ||g||_{L^2(S_T)}}, provided that q satisfies a priori uniform boundedness conditions. We use an inequality of Carleman type in our proof.

First Author:S. Li

Indexed by:Journal paper

Discipline:Natural Science

Document Type:J

Volume:14

Issue:9

Page Number:891-904

Translation or Not:no

Date of Publication:2006-09-01

Included Journals:EI

Links to published journals:https://www.degruyter.com/document/doi/10.1515/156939406779768283/html