Conditional Stability for Single Interior Measurement

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.