| Title:
             | 
A continuity result for a quasilinear elliptic free boundary problem (English) | 
| Author:
             | 
Lyaghfouri, Abdeslem | 
| Language:
             | 
English | 
| Journal:
             | 
Applications of Mathematics | 
| ISSN:
             | 
0862-7940 (print) | 
| ISSN:
             | 
1572-9109 (online) | 
| Volume:
             | 
65 | 
| Issue:
             | 
1 | 
| Year:
             | 
2020 | 
| Pages:
             | 
67-87 | 
| Summary lang:
             | 
English | 
| . | 
| Category:
             | 
math | 
| . | 
| Summary:
             | 
We investigate a two dimensional quasilinear free boundary problem, and show that the free boundary is a union of graphs of continuous functions. (English) | 
| Keyword:
             | 
quasilinear elliptic free boundary | 
| Keyword:
             | 
continuity | 
| MSC:
             | 
35J62 | 
| MSC:
             | 
35R35 | 
| idZBL:
             | 
07177872 | 
| idMR:
             | 
MR4064590 | 
| DOI:
             | 
10.21136/AM.2020.0190-19 | 
| . | 
| Date available:
             | 
2020-02-20T09:46:31Z | 
| Last updated:
             | 
2022-03-07 | 
| Stable URL:
             | 
http://hdl.handle.net/10338.dmlcz/147995 | 
| . | 
| Reference:
             | 
[1] Alvarez, S. J., Carrillo, J.: A free boundary problem in theory of lubrication.Commun. Partial Differ. Equations 19 (1994), 1743-1761. Zbl 0809.35166, MR 1301172, 10.1080/03605309408821072 | 
| Reference:
             | 
[2] Carrillo, J., Lyaghfouri, A.: 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. Zbl 0928.76111, MR 1635777 | 
| Reference:
             | 
[3] Challal, S., Lyaghfouri, A.: A filtration problem through a heterogeneous porous medium.Interfaces Free Bound. 6 (2004), 55-79. Zbl 1075.35115, MR 2047073, 10.4171/IFB/91 | 
| Reference:
             | 
[4] Challal, S., Lyaghfouri, A.: 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. Zbl 1070.35136, MR 2145607, 10.1016/j.na.2005.02.115 | 
| Reference:
             | 
[5] Challal, S., Lyaghfouri, A.: 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. Zbl 1212.35508, MR 2235138 | 
| Reference:
             | 
[6] Challal, S., Lyaghfouri, A.: On the dam problem with two fluids governed by a nonlinear Darcy's law.Adv. Differ. Equ. 11 (2006), 841-892. Zbl 1167.35550, MR 2261733 | 
| Reference:
             | 
[7] Challal, S., Lyaghfouri, A.: 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. Zbl 1136.35105, MR 2351639, 10.1080/00036810701620601 | 
| Reference:
             | 
[8] Challal, S., Lyaghfouri, A.: Hölder continuity of solutions to the $A$-Laplace equation involving measures.Commun. Pure Appl. Anal. 8 (2009), 1577-1583. Zbl 1179.35336, MR 2505287, 10.3934/cpaa.2009.8.1577 | 
| Reference:
             | 
[9] Challal, S., Lyaghfouri, A.: Lipschitz continuity of solutions of a free boundary problem involving the $p$-Laplacian.J. Math. Anal. Appl. 355 (2009), 700-707. Zbl 1167.35551, MR 2521745, 10.1016/j.jmaa.2009.02.012 | 
| Reference:
             | 
[10] Challal, S., Lyaghfouri, A.: The heterogeneous dam problem with leaky boundary condition.Commun. Pure Appl. Anal. 10 (2011), 93-125. Zbl 1300.76028, MR 2746529, 10.3934/cpaa.2011.10.93 | 
| Reference:
             | 
[11] Challal, S., Lyaghfouri, A.: Continuity of the free boundary in a problem involving the $A$-Laplacian.Available at https://arxiv.org/abs/1906.11791 (2019), 18 pages. MR 2521745 | 
| Reference:
             | 
[12] Challal, S., Lyaghfouri, A.: Lipschitz continuity of solutions to a free boundary problem involving the $A$-Laplacian.Available at https://arxiv.org/abs/1906.06511 (2019), 14 pages. MR 2521745 | 
| Reference:
             | 
[13] Challal, S., Lyaghfouri, A., Rodrigues, J. F.: On the $A$-obstacle problem and the Hausdorff measure of its free boundary.Ann. Mat. Pura Appl. (4) 191 (2012), 113-165. Zbl 1235.35285, MR 2886164, 10.1007/s10231-010-0177-7 | 
| Reference:
             | 
[14] Chipot, M.: On the continuity of the free boundary in some class of two-dimensional problems.Interfaces Free Bound. 3 (2001), 81-99. Zbl 0973.35201, MR 1805079, 10.4171/IFB/33 | 
| Reference:
             | 
[15] Chipot, M., Lyaghfouri, A.: The dam problem for non-linear Darcy’s laws and non-linear leaky boundary conditions.Math. Methods Appl. Sci. 20 (1997), 1045-1068. Zbl 0889.35125, MR 1460692, 10.1002/(SICI)1099-1476(199708)20:12<1045::AID-MMA900>3.0.CO;2-3 | 
| Reference:
             | 
[16] Chipot, M., Lyaghfouri, A.: The dam problem for linear Darcy's law and nonlinear leaky boundary conditions.Adv. Differ. Equ. 3 (1998), 1-50. Zbl 0947.35177, MR 1608073 | 
| Reference:
             | 
[17] Lieberman, G. M.: The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations.Commun. Partial Differ. Equations 16 (1991), 311-361. Zbl 0742.35028, MR 1104103, 10.1080/03605309108820761 | 
| Reference:
             | 
[18] Lyaghfouri, A.: A free boundary problem for a fluid flow in a heterogeneous porous medium.Ann. Univ. Ferrara, Nuova Ser., Sez. VII (2003), 209-262. Zbl 1232.35196, MR 2164998, 10.1007/BF02844918 | 
| Reference:
             | 
[19] Lyaghfouri, A.: The dam problem.Handbook of Differential Equations: Stationary Partial Differential Equations M. Chipot Handbook of Differential Equations 3, Elsevier, Amsterdam (2006), 465-552. Zbl 1193.35257, MR 2569337, 10.1016/S1874-5733(06)80010-0 | 
| Reference:
             | 
[20] Lyaghfouri, A.: On the Lipschitz continuity of the solutions of a class of elliptic free boundary problems.J. Appl. Anal. 14 (2008), 165-181. Zbl 1175.35029, MR 2501665, 10.1515/JAA.2008.165 | 
| . |