Conditional Stability for Single Interior Measurement

TL;DR

This paper establishes local stability estimates for identifying unknown coefficients in a strongly elliptic PDE using a single interior measurement, with results applicable to complex coefficients and implications for uniqueness.

Contribution

It provides the first local Hölder stability estimate for this inverse problem with a single interior measurement, including complex coefficients and a new admissibility condition.

Findings

Proves local Hölder stability for coefficient identification.

Establishes global uniqueness for certain coefficient reconstructions.

Introduces an admissibility condition for complex coefficients.

Abstract

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 $C^2$ boundary and single interior measurement means that we know a given solution of the 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 H\"older stability estimate for identifying the unknown coefficients. If the unknown coefficients is a complex coefficient in the principal part of the equation, we assumed a condition which we named admissibility assumption for the real part and imaginary part of the difference of the 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 0-th order of the equation, the major result implies the global uniqueness for the identification.