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 & Continuous Dynamical Systems - A, 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 (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.  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.  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.   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, (1998).   Google Scholar

[8]

A. Karakhanyan, On the Lipschitz regularity of solutions of a minimum problem with free boundary,, Interfaces Free Bound, 10 (2008), 79.  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, (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, 1584 (1994), 147.  doi: 10.1007/BFb0073397.  Google Scholar

[12]

J.-F. Rodrigues, Obstacle Problems in Mathematical Physics,, North-Holland Mathematics Studies, 114 (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 (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.  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.  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.   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, (1998).   Google Scholar

[8]

A. Karakhanyan, On the Lipschitz regularity of solutions of a minimum problem with free boundary,, Interfaces Free Bound, 10 (2008), 79.  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, (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, 1584 (1994), 147.  doi: 10.1007/BFb0073397.  Google Scholar

[12]

J.-F. Rodrigues, Obstacle Problems in Mathematical Physics,, North-Holland Mathematics Studies, 114 (1987).   Google Scholar

[1]

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

[2]

Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021028

[3]

Huijuan Song, Bei Hu, Zejia Wang. Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 667-691. doi: 10.3934/dcdsb.2020084

[4]

Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124

[5]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

[6]

Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 401-414. doi: 10.3934/dcdsb.2020083

[7]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[8]

Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381

[9]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020398

[10]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, 2021, 14 (1) : 149-174. doi: 10.3934/krm.2020052

[11]

Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380

[12]

Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387

[13]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[14]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[15]

Giulio Ciraolo, Antonio Greco. An overdetermined problem associated to the Finsler Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021004

[16]

Lei Yang, Lianzhang Bao. Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1083-1109. doi: 10.3934/dcdsb.2020154

[17]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[18]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108

[19]

Kimie Nakashima. Indefinite nonlinear diffusion problem in population genetics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3837-3855. doi: 10.3934/dcds.2020169

[20]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]