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

Aram L.  Karakhanyan

doi:10.3934/dcds.2016.36.261

Article Contents

2016, Volume 36, Issue 1: 261-277. Doi: 10.3934/dcds.2016.36.261

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

Aram L. Karakhanyan^1,

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

Received: December 31, 2013

Revised: March 31, 2015

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 35R35, 35B65, 80A22.

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