# 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 and Continuous Dynamical Systems, 2016, 36 (1) : 261-277. doi: 10.3934/dcds.2016.36.261
##### References:
 [1] J. Bear, Dynamics of fluids in porous media, Courier Dover Publications, 1988. [2] L. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problems, Graduate Studies in Mathematics, vol. 68 AMS, 2005. doi: 10.1090/gsm/068. [3] X. Chen and F. Yi, Regularity of the free boundary of a continuous casting problem, Nonlinear Anal., 21 (1993), 425-438. doi: 10.1016/0362-546X(93)90126-D. [4] E. DiBenedetto and M. O'Leary, Three-dimensional conduction-convection problems with change of phase, Arch. Rational Mech. Anal., 123 (1993), 99-116. doi: 10.1007/BF00695273. [5] A. Friedman, Variational Principles and Free Boundary Problems, John Wiley & Sons, 1982. [6] J. Frehse, Capacity methods in the theory of partial differential equations, Jahresbericht der Deutschen Math.-Ver., 84 (1982), 1-44. [7] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. [8] A. Karakhanyan, On the Lipschitz regularity of solutions of a minimum problem with free boundary, Interfaces Free Bound, 10 (2008), 79-86. doi: 10.4171/IFB/180. [9] A. Karakhanyan, Optimal regularity for phase transition problems with convection, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, in press, 2014. doi: 10.1016/j.anihpc.2014.03.003. [10] A. Karakhanyan and J.-F. Rodrigues, The Stefan problem with constant convection, preprint, available online at http://www.maths.ed.ac.uk/~aram/p13.pdf. [11] J.-F. Rodrigues, Variational methods in the Stefan problem, in Phase transitions and hysteresis (Montecatini Terme, 1993), Lecture Notes in Math., Springer, Berlin, 1584 (1994), 147-212, doi: 10.1007/BFb0073397. [12] J.-F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Mathematics Studies, 134. Notas de Matemática, 114. North-Holland Publishing Co., Amsterdam, 1987.

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##### References:
 [1] J. Bear, Dynamics of fluids in porous media, Courier Dover Publications, 1988. [2] L. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problems, Graduate Studies in Mathematics, vol. 68 AMS, 2005. doi: 10.1090/gsm/068. [3] X. Chen and F. Yi, Regularity of the free boundary of a continuous casting problem, Nonlinear Anal., 21 (1993), 425-438. doi: 10.1016/0362-546X(93)90126-D. [4] E. DiBenedetto and M. O'Leary, Three-dimensional conduction-convection problems with change of phase, Arch. Rational Mech. Anal., 123 (1993), 99-116. doi: 10.1007/BF00695273. [5] A. Friedman, Variational Principles and Free Boundary Problems, John Wiley & Sons, 1982. [6] J. Frehse, Capacity methods in the theory of partial differential equations, Jahresbericht der Deutschen Math.-Ver., 84 (1982), 1-44. [7] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. [8] A. Karakhanyan, On the Lipschitz regularity of solutions of a minimum problem with free boundary, Interfaces Free Bound, 10 (2008), 79-86. doi: 10.4171/IFB/180. [9] A. Karakhanyan, Optimal regularity for phase transition problems with convection, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, in press, 2014. doi: 10.1016/j.anihpc.2014.03.003. [10] A. Karakhanyan and J.-F. Rodrigues, The Stefan problem with constant convection, preprint, available online at http://www.maths.ed.ac.uk/~aram/p13.pdf. [11] J.-F. Rodrigues, Variational methods in the Stefan problem, in Phase transitions and hysteresis (Montecatini Terme, 1993), Lecture Notes in Math., Springer, Berlin, 1584 (1994), 147-212, doi: 10.1007/BFb0073397. [12] J.-F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Mathematics Studies, 134. Notas de Matemática, 114. North-Holland Publishing Co., Amsterdam, 1987.
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