Conditional stability for a single interior measurement

• Published in: Inverse problems Vol. 30; no. 5; pp. 55001 - 19
• Main Authors: Honda, Naofumi, McLaughlin, Joyce, Nakamura, Gen
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.05.2014
• Subjects: Boundaries; critical set; Estimates; hydrology; Inverse problems; Mathematical analysis; Mathematical models; MRE; Names; Scalars; Stability; stratification; Uniqueness
• ISSN: 0266-5611; 1361-6420
• DOI: 10.1088/0266-5611/30/5/055001

An inverse problem to identify unknown coefficients of a partial differential equation by a single interior measurement is considered. The equation considered in this paper is a strongly elliptic second order scalar equation which can have complex coefficients in a bounded domain with C2 boundary. We are given a single interior measurement. This means that we know a given solution of the forward equation in this domain. The equation includes some model equations arising from acoustics, viscoelasticity and hydrology. We assume that the coefficients are piecewise analytic. Our major result is the local Hölder stability estimate for identifying the unknown coefficients. If the unknown coefficient is a complex coefficient in the principal part of the equation, we assumed a condition which we name admissibility assumption for the real part and imaginary part of the difference of two complex coefficients. This admissibility assumption is automatically satisfied if the complex coefficients are real valued. For identifying either the real coefficient in the principal part or the coefficient of the 0th order of the equation, the major result implies global uniqueness for the identification.