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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

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 & 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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Friedman, Variational Principles and Free Boundary Problems, John Wiley & Sons, 1982.  Google Scholar

[6]

J. Frehse, Capacity methods in the theory of partial differential equations, Jahresbericht der Deutschen Math.-Ver., 84 (1982), 1-44.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Karakhanyan and J.-F. Rodrigues, The Stefan problem with constant convection,, preprint, ().   Google Scholar

[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.  Google Scholar

[12]

J.-F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Mathematics Studies, 134. Notas de Matemática, 114. North-Holland Publishing Co., Amsterdam, 1987.  Google Scholar

show all references

References:
[1]

J. Bear, Dynamics of fluids in porous media, Courier Dover Publications, 1988. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Friedman, Variational Principles and Free Boundary Problems, John Wiley & Sons, 1982.  Google Scholar

[6]

J. Frehse, Capacity methods in the theory of partial differential equations, Jahresbericht der Deutschen Math.-Ver., 84 (1982), 1-44.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Karakhanyan and J.-F. Rodrigues, The Stefan problem with constant convection,, preprint, ().   Google Scholar

[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.  Google Scholar

[12]

J.-F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Mathematics Studies, 134. Notas de Matemática, 114. North-Holland Publishing Co., Amsterdam, 1987.  Google Scholar

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