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In this paper we derive a propagation of smallness result for a scalar secondelliptic equation in divergence form whose leading order coefficients areLipschitz continuous on two sides of a $C^2$ hypersurface that crosses thedomain, but may have jumps across this hypersurface. Our propagation ofsmallness result is in the most general form regarding the locations ofdomains, which may intersect the interface of discontinuity. At the end, wealso list some consequences of the propagation of smallness result, includingstability results for the associated Cauchy problem, a propagation of smallnessresult from sets of positive measure, and a quantitative Runge approximationproperty.

Propagation of smallness for an elliptic PDE with piecewise Lipschitz coefficients