Propagation of smallness and size estimate in the second order elliptic equation with discontinuous complex Lipschitz conductivity

In this paper, we would like to derive three-ball inequalities and propagation of smallness for the complex second order elliptic equation with discontinuous Lipschitz coefficients. As an application of such estimates, we study the size estimate problem by one pair of Cauchy data on the boundary, that is, a pair of the Neumann and Dirichlet data of the solution on the boundary. The main ingredient in the derivation of three-ball inequalities and propagation of smallness is a local Carleman proved in our recent paper [E. Francini, S. Vessella, J.-N. Wang, Carleman estimate for complex second order elliptic operators with discontinuous Lipschitz coefficients, J. Spectr. Theory 12 (2) (2022) 535–57].