Propagation of smallness for an elliptic PDE with piecewise Lipschitz coefficients

In this paper we derive a propagation of smallness result for a scalar second elliptic equation in divergence form whose leading order coefficients are Lipschitz continuous on two sides of a C² hypersurface that crosses the domain, but may have jumps across this hypersurface. Our propagation of smallness result is in the most general form regarding the locations of domains, which may intersect the interface of discontinuity. At the end, we also list some consequences of the propagation of smallness result, including stability results for the associated Cauchy problem, a propagation of smallness result from sets of positive measure, and a quantitative Runge approximation property.