Regularity for a quasilinear continuous casting problem

In this paper study the regularity of continuous casting problem div(|∇u|p−2∇u−vβ(u))=0(♯) for prescribed constant velocity v and enthalpy β(u) with jump discontinuity at u=0. We establish the following estimates: local log-Lipschitz p>2 for u (and BMO for ∇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 v in two spatial dimensions. The proof is based on a delicate argument exploiting Sard's theorem for W2,2+η,η>0 functions and circumventing the lack of comparison principle for the solutions of (♯).