# American Institute of Mathematical Sciences

January  2011, 10(1): 93-125. doi: 10.3934/cpaa.2011.10.93

## The heterogeneous dam problem with leaky boundary condition

 1 Fields Institute, 222 College Street, Toronto M5T 3J1

Received  January 2010 Revised  April 2010 Published  November 2010

We study the heterogeneous dam problem, assuming the ux at the bottoms of the reservoirs obeying to a nonlinear law called leaky boundary condition. The velocity and the pressure are related by a nonlinear Darcy's law. Under a general monotonicity hypothesis on the permeability matrix, we prove that the free boundary is represented locally by graphs of continuous functions. We also prove the uniqueness of minimal and maximal solutions. When the ow is given by a linear Darcy law and the permeability matrix is symmetric, we prove the uniqueness of the reservoirs-connected solution.
Citation: Samia Challal, Abdeslem Lyaghfouri. The heterogeneous dam problem with leaky boundary condition. Communications on Pure and Applied Analysis, 2011, 10 (1) : 93-125. doi: 10.3934/cpaa.2011.10.93
##### References:
 [1] J. Carrillo and M. Chipot, On the dam problem, J. Differential Equations, 45 (1982), 234-271. doi: doi:10.1016/0022-0396(82)90068-7. [2] J. Carrillo and M. Chipot, The dam problem with leaky boundary conditions, Applied Mathematics & Optimization, 28 (1993), 57-85. doi: doi:10.1007/BF01188758. [3] J. Carrillo and A. Lyaghfouri, The dam problem for nonlinear Darcy's laws and Dirichlet boundary conditions, Annali della Scuola Normale Superiore di Pisa Cl. Sci., 26 (1998), 453-505. [4] M. Chipot and A. Lyaghfouri, The dam problem for nonlinear Darcy's law and leaky boundary conditions, Mathematical Methods in the Applied Sciences, 20 (1997), 1045-1068. doi: doi:10.1002/(SICI)1099-1476(199708)20:12<1045::AID-MMA900>3.0.CO;2-3. [5] M. Chipot and A. Lyaghfouri, The dam problem with linear Darcy's law and leaky boundary conditions, Advances in Differential Equations, 3 (1998), 1-50. [6] S. Challal and A. Lyaghfouri, A Filtration problem through a heterogeneous porous medium, Interfaces and Free Boundaries, 6 (2004), 55-79. doi: doi:10.4171/IFB/91. [7] L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré, 15 (1998), 493-516. doi: doi:10.1016/S0294-1449(98)80032-2. [8] E. DiBenedetto, $C^1$ Local regularity of weak solutions of degenerate elliptic equations, Nonlinear Analysis, Theory, Methods & Applications, 7 (1983), 827-850. [9] A. Friedman and Shav-Yun Huang, The inhomogeneous dam problem with discountinuous permeability, Ann. Scu. Norm. Sup. Pisa, Ser. IV, 14 (1987), 49-77. [10] A. Lyaghfouri, The inhomogeneous dam problem with linear Darcy's law and Dirichlet boundary conditions, Mathematical Models and Methods in Applied Sciences, 6 (1996), 1051-1077. doi: doi:10.1142/S0218202596000432. [11] A. Lyaghfouri, On the uniqueness of the solution of a nonlinear filtration problem through a porous medium, Calculus of Variations and Partial Differential Equations, 6 (1998), 67-94. doi: doi:10.1007/s005260050082. [12] A. Lyaghfouri, A unified formulation for the dam problem, Rivista di Matematica della Universit\a di Parma, 1 (1998), 113-148. [13] A. Lyaghfouri, A free boundary problem for a fluid flow in a heterogeneous porous medium, Annali dell' Universita di Ferrara-Sez. VII-Sc. Mat., IL (2003), 209-262. [14] J. F. Rodrigues, On the dam problem with boundary leacky condition, Portugaliae Mathematica, 39 (1980), 399-411. [15] R. Stavre and B. Vernescu, Incompressible fluid flow through a nonhomogeneous and anisotropic dam, Nonlinear Analysis, 9 (1985), 799-810. doi: doi:10.1016/0362-546X(85)90019-7.

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##### References:
 [1] J. Carrillo and M. Chipot, On the dam problem, J. Differential Equations, 45 (1982), 234-271. doi: doi:10.1016/0022-0396(82)90068-7. [2] J. Carrillo and M. Chipot, The dam problem with leaky boundary conditions, Applied Mathematics & Optimization, 28 (1993), 57-85. doi: doi:10.1007/BF01188758. [3] J. Carrillo and A. Lyaghfouri, The dam problem for nonlinear Darcy's laws and Dirichlet boundary conditions, Annali della Scuola Normale Superiore di Pisa Cl. Sci., 26 (1998), 453-505. [4] M. Chipot and A. Lyaghfouri, The dam problem for nonlinear Darcy's law and leaky boundary conditions, Mathematical Methods in the Applied Sciences, 20 (1997), 1045-1068. doi: doi:10.1002/(SICI)1099-1476(199708)20:12<1045::AID-MMA900>3.0.CO;2-3. [5] M. Chipot and A. Lyaghfouri, The dam problem with linear Darcy's law and leaky boundary conditions, Advances in Differential Equations, 3 (1998), 1-50. [6] S. Challal and A. Lyaghfouri, A Filtration problem through a heterogeneous porous medium, Interfaces and Free Boundaries, 6 (2004), 55-79. doi: doi:10.4171/IFB/91. [7] L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré, 15 (1998), 493-516. doi: doi:10.1016/S0294-1449(98)80032-2. [8] E. DiBenedetto, $C^1$ Local regularity of weak solutions of degenerate elliptic equations, Nonlinear Analysis, Theory, Methods & Applications, 7 (1983), 827-850. [9] A. Friedman and Shav-Yun Huang, The inhomogeneous dam problem with discountinuous permeability, Ann. Scu. Norm. Sup. Pisa, Ser. IV, 14 (1987), 49-77. [10] A. Lyaghfouri, The inhomogeneous dam problem with linear Darcy's law and Dirichlet boundary conditions, Mathematical Models and Methods in Applied Sciences, 6 (1996), 1051-1077. doi: doi:10.1142/S0218202596000432. [11] A. Lyaghfouri, On the uniqueness of the solution of a nonlinear filtration problem through a porous medium, Calculus of Variations and Partial Differential Equations, 6 (1998), 67-94. doi: doi:10.1007/s005260050082. [12] A. Lyaghfouri, A unified formulation for the dam problem, Rivista di Matematica della Universit\a di Parma, 1 (1998), 113-148. [13] A. Lyaghfouri, A free boundary problem for a fluid flow in a heterogeneous porous medium, Annali dell' Universita di Ferrara-Sez. VII-Sc. Mat., IL (2003), 209-262. [14] J. F. Rodrigues, On the dam problem with boundary leacky condition, Portugaliae Mathematica, 39 (1980), 399-411. [15] R. Stavre and B. Vernescu, Incompressible fluid flow through a nonhomogeneous and anisotropic dam, Nonlinear Analysis, 9 (1985), 799-810. doi: doi:10.1016/0362-546X(85)90019-7.
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