  We prove existence of a solution for a polymer crystal growth model
describing the movement of a front $(\Gamma(t))$ evolving with a nonlocal
velocity. In this model the nonlocal velocity is linked to the solution of a
heat equation with source $\delta_\Gamma$. The proof relies on new regularity
results for the eikonal equation, in which the velocity is positive but merely
measurable in time and with H\"{o}lder bounds in space. From this result, we
deduce \textit{a priori} regularity for the front. On the other hand, under
this regularity assumption, we prove bounds and regularity estimates for the
solution of the heat equation.
