Regularity for a quasilinear continuous casting problem

In this paper study the regularity of continuous casting problem \begin{equation} \hbox{div}(|\nabla u|^{p-2}\nabla u-{\bf v} \beta(u))=0\tag{$\sharp$} \end{equation} for prescribed constant velocity $\bf v$ and enthalpy $\beta(u)$ with jump discontinuity at $u=0$. We establish the following estimates: local log-Lipschitz $p>2$ for $u$ (and BMO for $\nabla u$) for two phase, Lipschitz $p>1$ for one phase and linear growth up-to boundary near the contact points. We also prove that the free boundary is continuous curve in the direction of $\bf v$ in two spatial dimensions. The proof is based on a delicate argument exploiting Sard's theorem for $W^{2, 2+\eta}, \eta>0$ functions and circumventing the lack of comparison principle for the solutions of ($\sharp$).