Applications of Mathematics, Vol. 65, No. 1, pp. 67-87, 2020


A continuity result for a quasilinear elliptic free boundary problem

Abdeslem Lyaghfouri

Received August 4, 2019.   Published online January 31, 2020.

Abstract:  We investigate a two dimensional quasilinear free boundary problem, and show that the free boundary is a union of graphs of continuous functions.
Keywords:  quasilinear elliptic free boundary; continuity
Classification MSC:  35R35, 35J62


References:
[1] S. J. Alvarez, J. Carrillo: A free boundary problem in theory of lubrication. Commun. Partial Differ. Equations 19 (1994), 1743-1761. DOI 10.1080/03605309408821072 | MR 1301172 | Zbl 0809.35166
[2] J. Carrillo, A. Lyaghfouri: The dam problem for nonlinear Darcy's laws and Dirichlet boundary conditions. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 26 (1998), 453-505. MR 1635777 | Zbl 0928.76111
[3] S. Challal, A. Lyaghfouri: A filtration problem through a heterogeneous porous medium. Interfaces Free Bound. 6 (2004), 55-79. DOI 10.4171/IFB/91 | MR 2047073 | Zbl 1075.35115
[4] S. Challal, A. Lyaghfouri: On the continuity of the free boundary in problems of type $ div(a(x) \nabla u) = -(h(x)\chi(u))_{x_1}$. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 62 (2005), 283-300. DOI 10.1016/j.na.2005.02.115 | MR 2145607 | Zbl 1070.35136
[5] S. Challal, A. Lyaghfouri: On a class of free boundary problems of type $ div(a(X)\nabla u) =- div(\chi(u)H(x))$. Differ. Integral Equ. 19 (2006), 481-516. MR 2235138 | Zbl 1212.35508
[6] S. Challal, A. Lyaghfouri: On the dam problem with two fluids governed by a nonlinear Darcy's law. Adv. Differ. Equ. 11 (2006), 841-892. MR 2261733 | Zbl 1167.35550
[7] S. Challal, A. Lyaghfouri: On the continuity of the free boundary in the problem $\Delta_p u = -(h(x,y)\chi(u))_x$. Appl. Anal. 86 (2007), 1177-1184. DOI 10.1080/00036810701620601 | MR 2351639 | Zbl 1136.35105
[8] S. Challal, A. Lyaghfouri: Hölder continuity of solutions to the $A$-Laplace equation involving measures. Commun. Pure Appl. Anal. 8 (2009), 1577-1583. DOI 10.3934/cpaa.2009.8.1577 | MR 2505287 | Zbl 1179.35336
[9] S. Challal, A. Lyaghfouri: Lipschitz continuity of solutions of a free boundary problem involving the $p$-Laplacian. J. Math. Anal. Appl. 355 (2009), 700-707. DOI 10.1016/j.jmaa.2009.02.012 | MR 2521745 | Zbl 1167.35551
[10] S. Challal, A. Lyaghfouri: The heterogeneous dam problem with leaky boundary condition. Commun. Pure Appl. Anal. 10 (2011), 93-125. DOI 10.3934/cpaa.2011.10.93 | MR 2746529 | Zbl 1300.76028
[11] S. Challal, A. Lyaghfouri: Continuity of the free boundary in a problem involving the $A$-Laplacian. Available at https://arxiv.org/abs/1906.11791 (2019), 18 pages.
[12] S. Challal, A. Lyaghfouri: Lipschitz continuity of solutions to a free boundary problem involving the $A$-Laplacian. Available at https://arxiv.org/abs/1906.06511 (2019), 14 pages.
[13] S. Challal, A. Lyaghfouri, J. F. Rodrigues: On the $A$-obstacle problem and the Hausdorff measure of its free boundary. Ann. Mat. Pura Appl. (4) 191 (2012), 113-165. DOI 10.1007/s10231-010-0177-7 | MR 2886164 | Zbl 1235.35285
[14] M. Chipot: On the continuity of the free boundary in some class of two-dimensional problems. Interfaces Free Bound. 3 (2001), 81-99. DOI 10.4171/IFB/33 | MR 1805079 | Zbl 0973.35201
[15] M. Chipot, A. Lyaghfouri: The dam problem for non-linear Darcy’s laws and non-linear leaky boundary conditions. Math. Methods Appl. Sci. 20 (1997), 1045-1068. DOI 10.1002/(SICI)1099-1476(199708)20:12<1045::AID-MMA900>3.0.CO;2-3 | MR 1460692 | Zbl 0889.35125
[16] M. Chipot, A. Lyaghfouri: The dam problem for linear Darcy's law and nonlinear leaky boundary conditions. Adv. Differ. Equ. 3 (1998), 1-50. MR 1608073 | Zbl 0947.35177
[17] G. M. Lieberman: The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations. Commun. Partial Differ. Equations 16 (1991), 311-361. DOI 10.1080/03605309108820761 | MR 1104103 | Zbl 0742.35028
[18] A. Lyaghfouri: A free boundary problem for a fluid flow in a heterogeneous porous medium. Ann. Univ. Ferrara, Nuova Ser., Sez. VII (2003), 209-262. DOI 10.1007/BF02844918 | MR 2164998 | Zbl 1232.35196
[19] A. Lyaghfouri: The dam problem. Handbook of Differential Equations: Stationary Partial Differential Equations (M. Chipot, eds.). Handbook of Differential Equations 3, Elsevier, Amsterdam, 2006, 465-552. DOI 10.1016/S1874-5733(06)80010-0 | Zbl 1193.35257
[20] A. Lyaghfouri: On the Lipschitz continuity of the solutions of a class of elliptic free boundary problems. J. Appl. Anal. 14 (2008), 165-181. DOI 10.1515/JAA.2008.165 | MR 2501665 | Zbl 1175.35029

Affiliations:   Abdeslem Lyaghfouri, American University of Ras Al Khaimah, Department of Mathematics and Natural Sciences, Ras Al Khaimah, UAE, e-mail: abdeslem.lyaghfouri@aurak.ac.ae


 
PDF available at: