Local Well-posedness of the Free-boundary Problem in Incompressible Elastodynamics with Surface Tension

We prove the local well-posedness of the 3D free-boundary incompressible elastodynamics with surface tension describing the motion of an elastic medium in a periodic domain with a moving graphical surface. The deformation tensor is assumed to satisfy the neo-Hookean linear elasticity. We adapt the idea in arXiv:2312.11254 to generate an approximate problem with artificial viscosity indexed by $\kappa > 0$ to boost the boundary regularity, which recovers the original system as $\kappa\to 0$, and the energy estimates yield no regularity loss.