American Institute of Mathematical Sciences

Advanced
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

[1]

Donatella Danielli, Marianne Korten. On the pointwise jump condition at the free boundary in the 1-phase Stefan problem. Communications on Pure & Applied Analysis, 2005, 4 (2) : 357-366. doi: 10.3934/cpaa.2005.4.357
[2]

Yang Zhang. A free boundary problem of the cancer invasion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021092
[3]

Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10
[4]

Michael L. Frankel, Victor Roytburd. Fractal dimension of attractors for a Stefan problem. Conference Publications, 2003, 2003 (Special) : 281-287. doi: 10.3934/proc.2003.2003.281
[5]

Lincoln Chayes, Inwon C. Kim. The supercooled Stefan problem in one dimension. Communications on Pure & Applied Analysis, 2012, 11 (2) : 845-859. doi: 10.3934/cpaa.2012.11.845
[6]

Piotr B. Mucha. Limit of kinetic term for a Stefan problem. Conference Publications, 2007, 2007 (Special) : 741-750. doi: 10.3934/proc.2007.2007.741
[7]

Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431
[8]

Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003
[9]

Naoki Sato, Toyohiko Aiki, Yusuke Murase, Ken Shirakawa. A one dimensional free boundary problem for adsorption phenomena. Networks & Heterogeneous Media, 2014, 9 (4) : 655-668. doi: 10.3934/nhm.2014.9.655
[10]

Yongzhi Xu. A free boundary problem model of ductal carcinoma in situ. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 337-348. doi: 10.3934/dcdsb.2004.4.337
[11]

Anna Lisa Amadori. Contour enhancement via a singular free boundary problem. Conference Publications, 2007, 2007 (Special) : 44-53. doi: 10.3934/proc.2007.2007.44
[12]

Shihe Xu. Analysis of a delayed free boundary problem for tumor growth. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 293-308. doi: 10.3934/dcdsb.2011.15.293
[13]

Chonghu Guan, Xun Li, Rui Zhou, Wenxin Zhou. Free boundary problem for an optimal investment problem with a borrowing constraint. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021049
[14]

Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087
[15]

Micah Webster, Patrick Guidotti. Boundary dynamics of a two-dimensional diffusive free boundary problem. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 713-736. doi: 10.3934/dcds.2010.26.713
[16]

Jan Prüss, Jürgen Saal, Gieri Simonett. Singular limits for the two-phase Stefan problem. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5379-5405. doi: 10.3934/dcds.2013.33.5379
[17]

Marianne Korten, Charles N. Moore. Regularity for solutions of the two-phase Stefan problem. Communications on Pure & Applied Analysis, 2008, 7 (3) : 591-600. doi: 10.3934/cpaa.2008.7.591
[18]

Karl P. Hadeler. Stefan problem, traveling fronts, and epidemic spread. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 417-436. doi: 10.3934/dcdsb.2016.21.417
[19]

Junde Wu. Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3399-3411. doi: 10.3934/dcds.2019140
[20]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences