# American Institute of Mathematical Sciences

January  2016, 36(1): 261-277. doi: 10.3934/dcds.2016.36.261

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

 1 School of Mathematics, University of Edinburgh, King's Buildings, Mayfield Road, EH9 3JZ, Edinburgh, Scotland, United Kingdom

Received  January 2014 Revised  April 2015 Published  June 2015

In this paper we are concerned with the regularity of weak solutions $u$ to the one phase continuous casting problem $div (A(x) \nabla u(X)) = div [\beta (u) v(X)], X\in \mathcal{C}_L$ in the cylindrical domain $\mathcal{C}_L=\Omega\times (0,L)$ where $X=(x,z), x\in \Omega\subset \mathbb{R}^{N-1}, z\in(0,L), L>0$ with given elliptic matrix $A:\Omega \to \mathbb{R}^{N^2}, A_{ij}(x)\in C^{1,\alpha_0}(\Omega), \alpha_0 > 0$, prescribed convection $v$, and the enthalpy function $\beta(u)$. We first establish the optimal regularity of weak solutions $u\ge 0$ for one phase problem. Furthermore, we show that the free boundary $\partial$ {u > 0} is locally Lipschitz continuous graph provided that $v = e_N$, the direction of $x_N$ coordinate axis and $\partial_{z}u\geq 0$. The latter monotonicity assumption in $z$ variable can be easily obtained for a suitable boundary condition.
Citation: Aram L. Karakhanyan. Lipschitz continuity of free boundary in the continuous casting problem with divergence form elliptic equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 261-277. doi: 10.3934/dcds.2016.36.261
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