LIPSCHITZ CONTINUITY OF FREE BOUNDARY IN THE CONTINUOUS CASTING PROBLEM WITH DIVERGENCE FORM ELLIPTIC EQUATION

作者：Karakhanyan Aram L^*

来源：Discrete and Continuous Dynamical Systems, 2016, 36(1): 261-277.

DOI：10.3934/dcds.2016.36.261

In this paper we are concerned with the regularity of weak solutions u to the one phase continuous casting problem
div (A (x) del u(X)) = div [beta(u)v(X)], X is an element of e(L)
in the cylindrical domain e(L) = Omega x (0, L) where X = (x, z), x is an element of Omega subset of RN-1, z is an element of (0, L), L > 0 with given elliptic matrix A : Omega R-N2, A(ij) (x) is an element of C-1 alpha 0 (Omega), alpha 0 > 0, prescribed convection v, and the enthalpy beta(u). We first establish the optimal regularity of weak solutions u >= 0 for one phase problem. Furthermore, we show that the free boundary partial derivative{u > 0} is locally Lipschitz continuous graph provided that v = eN, the direction of x(N) coordinate axis and partial derivative(z)u >= 0. The latter monotonicity assumption in z variable can be easily obtained for a suitable boundary condition.

• 出版日期2016-1