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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 $\bfv$ and enthalpy $\beta(u)$ with jump discontinuity at $u=0$. We establish thefollowing 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 boundarynear the contact points. We also prove that the free boundary is continuouscurve in the direction of $\bf v$ in two spatial dimensions. The proof is basedon a delicate argument exploiting Sard's theorem for $W^{2, 2+\eta}, \eta>0$functions and circumventing the lack of comparison principle for the solutionsof ($\sharp$).

Regularity for a quasilinear continuous casting problem