Journal of the
Korean Mathematical Society JKMS

In this paper, we consider the regularity theory and the existence of smooth solution of a degenerate fully nonlinear equation describing the evolution of the rolling stones with nonconvex sides:
$$
\begin{cases} M(h)= h_t - F(t,z,z^\alpha h_{zz}) & \mbox{ in }\{ 0 < z \le 1 \} \times [0,T]\\
h_t(z,t)=H(h_z(z,t),h) & \mbox{ on } \{ z=0 \}.
\end{cases}
$$ We establish the Schauder theory for $C^{2,\alpha}$-regularity of $h$.

Keywords: rolling stone, degenerate fully nonlinear equations, free boundary problem

MSC numbers: 35K20