  In our previous work, we have established the existence of transonic
characteristic discontinuities separating supersonic flows from a static gas in
two-dimensional steady compressible Euler flows under a perturbation with small
total variation of the incoming supersonic flow over a solid right-wedge. It is
a free boundary problem in Eulerian coordinates and, across the free boundary
(characteristic discontinuity), the Euler equations are of elliptic-hyperbolic
composite-mixed type. In this paper, we further prove that such a transonic
characteristic discontinuity solution is unique and $L^1$--stable with respect
to the small perturbation of the incoming supersonic flow in Lagrangian
coordinates.
