  We consider the stability of transonic contact discontinuity for the
two-dimensional steady compressible Euler flows in a finitely long nozzle. This
is the first work on the mixed-type problem of transonic flows across a contact
discontinuity as a free boundary in nozzles. We start with the Euler-Lagrangian
transformation to straighten the contact discontinuity in the new coordinates.
However, the upper nozzle wall in the subsonic region depending on the mass
flux becomes a free boundary after the transformation. Then we develop new
ideas and techniques to solve the free-boundary problem in three steps: (1) we
fix the free boundary and generate a new iteration scheme to solve the
corresponding fixed boundary value problem of the hyperbolic-elliptic mixed
type by building some powerful estimates for both the first-order hyperbolic
equation and a second-order nonlinear elliptic equation in a Lipschitz domain;
(2) we update the new free boundary by constructing a mapping that has a fixed
point; (3) we establish via the inverse Lagrangian coordinates transformation
that the original free interface problem admits a unique piecewise smooth
transonic solution near the background state, which consists of a smooth
subsonic flow and a smooth supersonic flow with a contact discontinuity.
