Initial Data Identification for Conservation Laws with Spatially Discontinuous Flux

We consider a scalar conservation law with a spatially discontinuous flux at a single point $x=0$, and we study the initial data identification problem for $AB$-entropy solutions associated to an interface connection $(A,B)$. This problem consists in identifying the set of initial data driven by the corresponding $AB$-entropy solution to a given target profile~$\omega^T$, at a time horizon $T>0$. We provide a full characterization of such a set in terms of suitable integral inequalities, and we establish structural and geometrical properties of this set. A distinctive feature of the initial set is that it is in general not convex, differently from the case of conservation laws with convex flux independent on the space variable. The results rely on the properties of the $AB$-backward-forward evolution operator introduced in~\cite{talamini_ancona_attset}, and on a proper concept of $AB$-genuine/interface characteristic for $AB$-entropy solutions provided in this paper.