Regularity results for a quasilinear free boundary problem
Abstract
In this paper we prove local interior and boundary Lipschitz continuity of the solutions of a quasilinear free boundary problem. We also show that the free boundary is the union of graphs of lower semi-continuous functions.
Keyword : A-Laplacian, free boundary, Lipschitz continuity
How to Cite
Challal, S., & Lyaghfouri, A. (2020). Regularity results for a quasilinear free boundary problem. Mathematical Modelling and Analysis, 25(3), 338-350. https://doi.org/10.3846/mma.2020.10659
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
S.J. Alvarez and J. Carrillo. A free boundary problem in theory of lubrication. Communications in Partial Differential Equations, 19(11-12):1743–1761, 1994. https://doi.org/10.1080/03605309408821072
G. Bayada and M. Chambat. Nonlinear variational formulation for a cavitation problem in lubrication. Journal of Mathematical Analysis and Applications, 90(2):286–298, 1982. https://doi.org/10.1016/0022-247X(82)90061-0
A. Bermúdez, M.C. Muñiz and P. Quintela. Existence and uniqueness for a free boundary problem in aluminum electrolysis. Journal of Mathematical Analysis and Applications, 191(3):497–527, 1995. https://doi.org/10.1006/jmaa.1995.1145
J. Carrillo and A. Lyaghfouri. On the dam problem with nonlinear Darcy’s laws and Dirichlet boundary conditions. Annali della Scuola Normale Superiore di Pisa Cl. Sci., 26(4):453–505, 1998.
S. Challal and A. Lyaghfouri. A filtration problem through a heterogeneous porous medium. Interfaces and Free Boundaries, 6(1):55–79, 2004. https://doi.org/10.4171/IFB/91
S. Challal and A. Lyaghfouri. On the continuity of the free boundary in problems of type div(a(x)∇u) = −(χ(u)h(x))x1. Nonlinear Analysis : Theory, Methods and Applications, 62(2):283–300, 2005. https://doi.org/10.1016/j.na.2005.02.115
S. Challal and A. Lyaghfouri. On a class of free boundary problems of type div(a(X)∇u) = −div(χ(u)H(X)). Differential and Integral Equations, 19(5):481–516, 2006.
S. Challal and A. Lyaghfouri. On the dam problem with two fluids governed by a nonlinear Darcy’s law. Advances in Differential Equations, 11(8):841–892, 2006.
S. Challal and A. Lyaghfouri. On the continuity of the free boundary in the problem ∆pu = −(h(x,y)χ(u))x. Applicable Analysis, 86(9):1177–1184, 2007. https://doi.org/10.1080/00036810701620601
S. Challal and A. Lyaghfouri. Hölder continuity of solutions to the A-Laplace equation involving measures. Communications in Pure and Applied Analysis, 8(5):1577–1583, 2009. https://doi.org/10.3934/cpaa.2009.8.1577
S. Challal and A. Lyaghfouri. Lipschitz continuity of solutions of a free boundary problem involving the p-Laplacian. Journal of Mathematical Analysis and Applications, 355(2):700–707, 2009. https://doi.org/10.1016/j.jmaa.2009.02.012
S. Challal and A. Lyaghfouri. The heterogeneous dam problem with leaky boundary condition. Communications in Pure and Applied Analysis, 10(1):93–125, 2011. https://doi.org/10.3934/cpaa.2011.10.93
S. Challal, A. Lyaghfouri and J.F. Rodrigues. On the A-Obstacle problem and the Hausdorff measure of its free boundary. Annali di Matematica Pura ed Applicata, 191(1):113–165, 2012. https://doi.org/10.1007/s10231-010-0177-7
S. Challal, A. Lyaghfouri, J.F. Rodrigues and R. Teymurazyan. On the regularity of the free boundary for a class of quasilinear obstacle problems. Interfaces and Free Boundaries, 16(3):359–394, 2014. https://doi.org/10.4171/IFB/323
M. Chipot. On the continuity of the free boundary in some class of twodimensional problems. Interfaces and Free Boundaries, 3(1):81–99, 2001. https://doi.org/10.4171/IFB/33
M. Chipot and A. Lyaghfouri. The dam problem with nonlinear Darcy’s law and leaky boundary conditions. Mathematical Methods in the Applied Sciences, 20(12):1045–1068, 1997.
M. Chipot and A. Lyaghfouri. The dam problem for linear Darcy’s law and nonlinear leaky boundary conditions. Advances in Differential Equations, 3(1):1– 50, 1998.
L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, AMS, 2010.
G.M. Lieberman. The natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations. Communications in Partial Differential Equations, 16(2-3):311–361, 1991. https://doi.org/10.1080/03605309108820761
A. Lyaghfouri. The inhomogeneous dam problem with linear Darcy’s law and Dirichlet boundary conditions. Mathematical Models and Methods in Applied Sciences, 8(6):1051–1077, 1996. https://doi.org/10.1142/S0218202596000432
A. Lyaghfouri. A unified formulation for the dam problem. Rivista di Matematica della Universita` di Parma, 6(1):113–148, 1998.
A. Lyaghfouri. A free boundary problem for a fluid flow in a heterogeneous porous medium. Annali dell’Universita di Ferrara, 49(1):209–262, 2003.
A. Lyaghfouri. The dam Problem. Handbook of Differential Equations, Stationary Partial Differential Equations, volume 3. North Holland, 2006.
A. Lyaghfouri. On the Lipschitz continuity of the solutions of a class of elliptic free boundary problems. Journal of Applied Analysis, 14(2):165–181, 2008.
A. Lyaghfouri and A. Saadi. Free boundary problems with Neuman boundary condition. Electronic Journal of Differential Equations, 2019(114):1–13, 2019.
J.F. Rodrigues and R. Teymurazyan. On the two obstacles problem in OrliczSobolev spaces and applications. Complex Variables and Elliptic Equations, 56(7-9):769–787, 2011. https://doi.org/10.1080/17476933.2010.505016
A. Saadi. Continuity of the free boundary in elliptic problems with Neuman boundary condition. Electronic Journal of Differential Equations, 2015(160):1– 16, 2015.
G. Bayada and M. Chambat. Nonlinear variational formulation for a cavitation problem in lubrication. Journal of Mathematical Analysis and Applications, 90(2):286–298, 1982. https://doi.org/10.1016/0022-247X(82)90061-0
A. Bermúdez, M.C. Muñiz and P. Quintela. Existence and uniqueness for a free boundary problem in aluminum electrolysis. Journal of Mathematical Analysis and Applications, 191(3):497–527, 1995. https://doi.org/10.1006/jmaa.1995.1145
J. Carrillo and A. Lyaghfouri. On the dam problem with nonlinear Darcy’s laws and Dirichlet boundary conditions. Annali della Scuola Normale Superiore di Pisa Cl. Sci., 26(4):453–505, 1998.
S. Challal and A. Lyaghfouri. A filtration problem through a heterogeneous porous medium. Interfaces and Free Boundaries, 6(1):55–79, 2004. https://doi.org/10.4171/IFB/91
S. Challal and A. Lyaghfouri. On the continuity of the free boundary in problems of type div(a(x)∇u) = −(χ(u)h(x))x1. Nonlinear Analysis : Theory, Methods and Applications, 62(2):283–300, 2005. https://doi.org/10.1016/j.na.2005.02.115
S. Challal and A. Lyaghfouri. On a class of free boundary problems of type div(a(X)∇u) = −div(χ(u)H(X)). Differential and Integral Equations, 19(5):481–516, 2006.
S. Challal and A. Lyaghfouri. On the dam problem with two fluids governed by a nonlinear Darcy’s law. Advances in Differential Equations, 11(8):841–892, 2006.
S. Challal and A. Lyaghfouri. On the continuity of the free boundary in the problem ∆pu = −(h(x,y)χ(u))x. Applicable Analysis, 86(9):1177–1184, 2007. https://doi.org/10.1080/00036810701620601
S. Challal and A. Lyaghfouri. Hölder continuity of solutions to the A-Laplace equation involving measures. Communications in Pure and Applied Analysis, 8(5):1577–1583, 2009. https://doi.org/10.3934/cpaa.2009.8.1577
S. Challal and A. Lyaghfouri. Lipschitz continuity of solutions of a free boundary problem involving the p-Laplacian. Journal of Mathematical Analysis and Applications, 355(2):700–707, 2009. https://doi.org/10.1016/j.jmaa.2009.02.012
S. Challal and A. Lyaghfouri. The heterogeneous dam problem with leaky boundary condition. Communications in Pure and Applied Analysis, 10(1):93–125, 2011. https://doi.org/10.3934/cpaa.2011.10.93
S. Challal, A. Lyaghfouri and J.F. Rodrigues. On the A-Obstacle problem and the Hausdorff measure of its free boundary. Annali di Matematica Pura ed Applicata, 191(1):113–165, 2012. https://doi.org/10.1007/s10231-010-0177-7
S. Challal, A. Lyaghfouri, J.F. Rodrigues and R. Teymurazyan. On the regularity of the free boundary for a class of quasilinear obstacle problems. Interfaces and Free Boundaries, 16(3):359–394, 2014. https://doi.org/10.4171/IFB/323
M. Chipot. On the continuity of the free boundary in some class of twodimensional problems. Interfaces and Free Boundaries, 3(1):81–99, 2001. https://doi.org/10.4171/IFB/33
M. Chipot and A. Lyaghfouri. The dam problem with nonlinear Darcy’s law and leaky boundary conditions. Mathematical Methods in the Applied Sciences, 20(12):1045–1068, 1997.
M. Chipot and A. Lyaghfouri. The dam problem for linear Darcy’s law and nonlinear leaky boundary conditions. Advances in Differential Equations, 3(1):1– 50, 1998.
L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, AMS, 2010.
G.M. Lieberman. The natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations. Communications in Partial Differential Equations, 16(2-3):311–361, 1991. https://doi.org/10.1080/03605309108820761
A. Lyaghfouri. The inhomogeneous dam problem with linear Darcy’s law and Dirichlet boundary conditions. Mathematical Models and Methods in Applied Sciences, 8(6):1051–1077, 1996. https://doi.org/10.1142/S0218202596000432
A. Lyaghfouri. A unified formulation for the dam problem. Rivista di Matematica della Universita` di Parma, 6(1):113–148, 1998.
A. Lyaghfouri. A free boundary problem for a fluid flow in a heterogeneous porous medium. Annali dell’Universita di Ferrara, 49(1):209–262, 2003.
A. Lyaghfouri. The dam Problem. Handbook of Differential Equations, Stationary Partial Differential Equations, volume 3. North Holland, 2006.
A. Lyaghfouri. On the Lipschitz continuity of the solutions of a class of elliptic free boundary problems. Journal of Applied Analysis, 14(2):165–181, 2008.
A. Lyaghfouri and A. Saadi. Free boundary problems with Neuman boundary condition. Electronic Journal of Differential Equations, 2019(114):1–13, 2019.
J.F. Rodrigues and R. Teymurazyan. On the two obstacles problem in OrliczSobolev spaces and applications. Complex Variables and Elliptic Equations, 56(7-9):769–787, 2011. https://doi.org/10.1080/17476933.2010.505016
A. Saadi. Continuity of the free boundary in elliptic problems with Neuman boundary condition. Electronic Journal of Differential Equations, 2015(160):1– 16, 2015.