A Stability Estimate for an Inverse Problem of Determining a Coefficient in a Hyperbolic Equation with a Point Source

DOI number:10.1007/s40304-016-0091-4

Journal:Communications in Mathematics and Statistics

Key Words:Inverse problem; Stability; Carleman estimate; Hyperbolic equation

Abstract:For the solution to ∂^2_t u(x, t)−Δu(x, t)+q(x)u(x, t) = δ(x, t) and u |_{t<0}=0, consider an inverse problem of determining q(x), x ∈Ω from data f = u |_{S_T} and g = (∂u/∂n) |_{S_T} . Here Ω⊂{(x_1, x_2, x_3) ∈ R^3 | x_1 > 0} is a bounded domain, S_T = {(x, t) | x ∈ ∂Ω, |x| < t < T + |x|}, n = n(x) is the outward unit normal n to ∂Ω, and T > 0. For suitable T > 0, prove a Lipschitz stability estimation: ||q_1 − q_2||_{L^2(Ω)} ≤ C{||f_1 − f_2||_{H^1(S_T) }+ ||g_1 − g_2||_{L^2(S_T)}}, provided that q1 satisfies a priori uniform boundedness conditions and q2 satisfies a priori uniform smallness conditions, where uk is the solution to problem (1.1) with q = q_k , k = 1, 2.

Co-author:Shumin Li

First Author:Xue Qin

Indexed by:Journal paper

Discipline:Natural Science

Document Type:J

Volume:4

Issue:3

Page Number:403-421

ISSN No.:2194-6701

Translation or Not:no

Date of Publication:2016-09-12

Links to published journals:https://link.springer.com/article/10.1007/s40304-016-0091-4