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

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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

Aram L. Karakhanyan ^1,

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

Abstract
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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.

Keywords: free boundary, Stefan problem, convection., Continuous casting problem.

Mathematics Subject Classification: Primary: 35R35, 35B65, 80A2.

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.

show all references

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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