Lipschitz continuity of free boundary in the continuous casting problem with divergence form elliptic equation

In this paper we are concerned with the regularity of weak solutions u to the one phase continuous casting problem div(A(x)∇u(X))=div[β(u)v(X)], X∈CL in the cylindrical domain CL=Ω×(0,L) where X=(x,z),x∈Ω⊂RN−1,z∈(0,L),L>0 with given elliptic matrix A:Ω→RN2,Aij(x)∈C1,α0(Ω),α0>0, prescribed convection v, and the enthalpy function β(u). We first establish the optimal regularity of weak solutions u≥0 for one phase problem. Furthermore, we show that the free boundary ∂ {u > 0} is locally Lipschitz continuous graph provided that v=eN, the direction of xN coordinate axis and ∂zu≥0. The latter monotonicity assumption in z variable can be easily obtained for a suitable boundary condition.