  We consider the free boundary problem for the incompressible elastodynamics
equations. At the free boundary moving with the velocity of the fluid particles
the columns of the deformation gradient are tangent to the boundary and the
pressure vanishes outside the flow domain. We prove the local existence of a
unique smooth solution of the free boundary problem, under the mixed type
stability condition that some regions of the initial free boundary satisfy the
Rayleigh-Taylor sign condition, while the remaining boundary satisfy the
non-collinearity condition. In particular, we solve an open problem proposed by
Y. Trakhinin in the paper \cite{Trak_18}.

Analysis of PDEs