How does the optimal constant of the Carleman estimate depend on the size of the observation boundary?

Abstract

Abstract In this paper, we would like to elucidate the question of the Carleman estimate constants dependency on the size of the observation boundary. This allows to quantify the effect of measurements uncertainty with respect to the size of the observation boundary in many inverse and control problems. We take an example of an inverse source problem for a parabolic equation and explicitly calculate Lipschitz stability constants that appears in the L 2 -estimate of the source function. We deliberately construct a class of space dependent weight functions that depend on the measurement boundary size. Then we identify the optimal weight function that allows to minimize the stability constant. Using our approach, we found that when the measurement boundary covers 80% or more of the domain boundary, we can explicitly provide the formula of the optimal constant. When the measurement boundary is less than 80%, we are not able to find the explicit formula of the optimal expression, but we are able to numerically approximate the optimal constant. This paper presents meticulous calculations that require a rigorous and careful reading to fully comprehend the technical intricacies of the analysis.