Well-posedness of the free boundary problem in elastodynamics with mixed stability condition

In this paper, we prove the local well-posedness of the free boundary problem in incompressible elastodynamics under a mixed type stability condition, i.e., for each point of the free boundary, at least one of the Taylor sign condition $-{\mathrm{\partial }}_{n}p>0$ and the non-collinearity condition holds. This gives an affirmative answer to a problem raised by Trakhinin in [30].