  The linear stability with variable coefficients of the vortex sheets for the
two-dimensional compressible elastic flows is studied. As in our earlier work
on the linear stability with constant coefficients, the problem has a free
boundary which is characteristic, and also the Kreiss-Lopatinskii condition is
not uniformly satisfied. In addition, the roots of the Lopatinskii determinant
of the para-linearized system may coincide with the poles of the system. Such a
new collapsing phenomenon causes serious difficulties when applying the
bicharacteristic extension method. Motivated by our method introduced in the
constant-coefficient case, we perform an upper triangularization to the
para-linearized system to separate the outgoing mode into a closed form where
the outgoing mode only appears at the leading order. This procedure results in
a gain of regularity for the outgoing mode which allows us to overcome the loss
of regularity of the characteristic components at the poles, and hence to close
all the energy estimates. We find that, analogous to the constant coefficient
case, elasticity generates notable stabilization effects, and there are
additional stable subsonic regions compared with the isentropic Euler flows.
Moreover, since our method does not rely on the construction of the
bicharacterisic curves, it can also be applied to other fluid models such as
the non-isentropic Euler equations and the MHD equations.
