Regularity for a quasilinear continuous casting problem

Abstract

In this paper study the regularity of continuous casting problem \begin{equation} \hbox{div}(|\nabla u|^{p-2}\nabla u-{\bf v} β(u))=0\tag{$♯$} \end{equation} for prescribed constant velocity $\mathbf{v}$ 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 $\mathbf{v}$ 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 ($♯$).