The dam problem for nonlinear Darcy's laws and Dirichlet boundary conditions

Carrillo, José; Lyaghfouri, Abdeslem

Carrillo, José ; Lyaghfouri, Abdeslem

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 26 (1998) no. 3, pp. 453-505.

MR 1635777 | Zbl 0928.76111

@article{ASNSP_1998_4_26_3_453_0,
     author = {Carrillo, Jos\'e and Lyaghfouri, Abdeslem},
     title = {The dam problem for nonlinear Darcy's laws and Dirichlet boundary conditions},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {453--505},
     publisher = {Scuola normale superiore},
     volume = {Ser. 4, 26},
     number = {3},
     year = {1998},
     zbl = {0928.76111},
     mrnumber = {1635777},
     language = {en},
     url = {archive.numdam.org/item/ASNSP_1998_4_26_3_453_0/}
}

Carrillo, José; Lyaghfouri, Abdeslem. The dam problem for nonlinear Darcy's laws and Dirichlet boundary conditions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 26 (1998) no. 3, pp. 453-505. http://archive.numdam.org/item/ASNSP_1998_4_26_3_453_0/

Bibliographie

[Ad] R.A. Adams, "Sobolev Spaces", Academic Press, New York, 1975. | MR 450957 | Zbl 0314.46030

[Al1] H.W. Alt, A free boundary problem associated with the flow of ground water, Arch. Rat. Mech. Anal. 64 (1977), 111-126. | MR 428901 | Zbl 0371.76079

[Al2] H.W. Alt, Strömunger durch inhomogene poröse Medien mit freiem Rand, J. Reine Angew. Math. 305 (1979), 89-115. | MR 518852 | Zbl 0392.76091

[Al3] H.W. AltThe fluid flow through porous media regularity of the free surface, Manuscripta Math. 21 (1977), 255-272. | MR 449170 | Zbl 0371.76080

[AC] A Alonso - J.C. Arrillo, A unified formulation for the boundary conditions in some convections-diffusion problems. Proc. European Conference on "Elliptic and Parabolic Problems", Pont-A-Mousson (June 1994) 325, 51-63. Pitman Research Notes in Mathematics. | Zbl 0831.35137

[AG] H.W. Alt - G. Gilardi, The behavior of the free boundary for the dam problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 9 (1981), 571-626. | Numdam | MR 693780 | Zbl 0521.76092

[AS] N. Ahmed - D.K. Sunada, Nonlinear flow in porous media, J. Hydraulics Div. Proc. Amer. Soc. Civil. Eng. 95 (1969), 1847-1857.

[Ba1] C. Baiocchi, Free boundary problems in the theory of fluid flow through porous media, Proceedings of the International Congress of Mathematicians -Vancouver (1974), 237-243. | MR 421331 | Zbl 0347.76067

[Ba2] C. Baiocchi, Free boundary problems in fluid flows through porous media and variational inequalities In: "Free Boundary Problems", Proceedings of a seminar held in Pavia Sept-Oct 1979, Vol. 1 (Roma 1980), pp. 175-191. | MR 630719 | Zbl 0481.76100

[BKS] H. Brezis - D. Kinderlehrer - G. Stampacchia, Sur une nouvelle formulation du problème de l'écoulement à travers une digue, C. R. Acad. Sci. Paris Sér. A 287 (1978), 711-714. | MR 515672 | Zbl 0391.76072

[Br] H. Brezis, "Analyse Fonctionnelle. Théorie et applications ", Masson, 1987. | MR 697382 | Zbl 00865216

[C1] J. Carrillo, "An Evolution Free Boundary Problem: Filtrations of a Compressible Fluid in a Porous Medium", Research Notes in Mathematics, Pitman, London, Vol. 89, 1983, pp. 97-110, | MR 730800 | Zbl 0541.76120

[C2] J. Carrillo, On the uniqueness of the solution of the evolution dam problem, Nonlinear Anal. Vol. 22, 5 (1994), 573-607. | MR 1266545 | Zbl 0810.76086

[CC1] J. Carrillo - M. Chipot, On the dam problem, J. Differential Equations 45 (1982), 234-271. | MR 665999 | Zbl 0504.35082

[CC2] J. Carrillo - M. Chipot, The dam problem with leaky boundary conditions, Appl. Math. Optim. 28 (1993), 57-85. | MR 1208798 | Zbl 0824.35141

[CF1] L.A. Caffarelli - A. Friedman, The dam problem with two layers, Arch. Ration. Mech. Anal. 68 (1978), 125-154. | MR 505510 | Zbl 0397.76087

[CF2] L.A. Caffarelli - A. Friedman, Asymptotic estimates for the dam problem with several layers, Indiana Univ. Math. J. 27 (1978), 551-580. | MR 500818 | Zbl 0386.76081

[CafG] L.A. Caffarelli - G. Gilardi, Monotonicity of the free boundary in the two dimensional dam problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1980), 523-537. | EuDML 83845 | Numdam | MR 597551 | Zbl 0513.76090

[CarG] J. Carrillo - G. Gilardi, La vitesse de propagation dans le problème de la digue, Ann. Fac. Sci. de Toulouse Math. 11 (1990), 7-28. | EuDML 73269 | Numdam | MR 1191718 | Zbl 0723.76094

[CL1] M. Chipot - A. Lyaghfouri, An existence theorem for an unbounded dam with leaky boundary conditions, Proc. European Conference on "Elliptic and Parabolic Problems", Pont-à-Mousson 325 (June 1994), 64-73. Pitman Research Notes in Mathematics. | MR 1416574 | Zbl 0834.35144

[CL2] M. Chipot - A. Lyaghfouri, The dam problem for nonlinear Darcy's law and leaky boundary conditions, Math. Methods Appl. Sci. 20 (1997), 1045-1068. | MR 1460692 | Zbl 0889.35125

[CL3] M. Chipot - A. Lyaghfouri, The dam problem for linear Darcy's law and nonlinear leaky boundary conditions, Advances in Differential Equations, 3 (1998), 1-50. | MR 1608073 | Zbl 0947.35177

[Dia] J.I. Diaz, "Non Linear Partial Differential Equations and Free Boundaries" Vol I: Elliptic Equations, Pitman Research Notes in Mathematics, London, 1985. | Zbl 0595.35100

[Dib] E. Dibenedetto, C1 local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), 827-850. | MR 709038 | Zbl 0539.35027

[DB] E. Dibenedetto - A. Friedman, Periodic behaviour for the evolutionary dam problem and related free boundary problems, Comm. Partial Differential Equations 11 (1986), 1297-1377. | MR 857169 | Zbl 0624.35084

[Do] Dongming Wei, An existence theorem for weak solution of a nonlinear dam problem, Appl. Anal. 34 (1989), 219-230. | MR 1387171 | Zbl 0663.35099

[Du] L. Dung, On a class of singular quasilinear elliptic equations with general structures and distributions data, Nonlinear Anal. 28 (1997), 1879-1902. | MR 1432639 | Zbl 0874.35039

[F] A. Friedman, "Variational Principles and Free-Boundary Problems", Robert E. Krieger Publishing Company, Malabar, Florida, 1988. | MR 1009785 | Zbl 0671.49001

[FH] A. Friedman - Shav-Yun Huang, The inhomogeneous dam problem with discontinuous permeability, Ann. Scuola. Norm. Sup. Pisa, Cl. Sci (4) 14 (1987), 49-77. | EuDML 83999 | Numdam | MR 937536 | Zbl 0644.35097

[FT] A. Friedman - A. Torelli, A free boundary problem connected with nonsteady filtration in porous media, Nonlinear Anal. 1 (1977), 503-545. | MR 605664 | Zbl 0373.76078

[G] G. Gilardi, A new approach to evolution free boundary problems, Comm. Partial Differential Equations 4 (1979) 1099-1123; 5 (1980), 983-984. | MR 544884 | Zbl 0426.35096

[GT] D. Gilbarg - N.S. Trudinger, "Elliptic Partial Differential Equations of Second Order", Springer-Verlag, New York, 1977. | MR 473443 | Zbl 0361.35003

[GZ1] R. Gariepy - W.P. Zeimer, Behavior at the boundary of solutions of quasilinear elliptic equations, Arch. Rational. Mech. Anal. 56 (1974), 372-384. | MR 355332 | Zbl 0297.35032

[GZ2] R. Gariepy - W.P. Zeimer, A regularity condition at the boundary for solutions of quasilinear elliptic equations, Arch. Rational. Mech. Anal. 67 (1978), 25-39. | MR 492836 | Zbl 0389.35023

[HKM] J. Heinonen - T. Kilpeläinen - O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations", Oxford Science Publications, 1993. | MR 1207810 | Zbl 0780.31001

[K1] T. Kilpeläinen, Hölder continuity of solutions to quasilinear elliptic equations involving measures, Potential Anal. 3 (1994), 265-272. | MR 1290667 | Zbl 0813.35016

[K2] T. Kilpeläinen, A Rado type theorem for p-harmonic functions in the plane, Electron. J. Differential Equations 9 (1994), approx. 4pp. (electronic). | EuDML 118527 | MR 1303907 | Zbl 0809.31005

[Le1] J.L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J. 32 (1983), 849-858. | MR 721568 | Zbl 0554.35048

[Le2] J.L. Lewis, Capacitary functions in convex rings, Arch. Rational. Mech. Anal. 66 (1977), 201-224. | MR 477094 | Zbl 0393.46028

[Lib] G.M. Liberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203-1219. | MR 969499 | Zbl 0675.35042

[Lio] J.L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires”, Dunod/ Gauthier-Villars, Paris, 1969. | Zbl 0189.40603

[Ly] A. Lyaghfouri, The inhomogeneous dam problem with linear Darcy's law and Dirichlet boundary conditions, Math. Models Methods Appl. Sci. 8 (1996), 1051-1077. | MR 1428145 | Zbl 0869.76088

[M] J.J. Manfredi, p-harmonicfunctions in the plane, Proc. Amer. Math. Soc. 103, (1988), 473-478. | MR 943069 | Zbl 0658.35041

[Ra] J.M. Rakotoson, Equivalence between the growth of ∫ B(x,r) |∇u|Pdy and T in the equation P [u] = T., J. Differential Equations 86 (1990), 102-122. | Zbl 0707.35033

[Ro] J.F. Rodrigues, On the dam problem with leaky boundary condition, Portugal. Math. 39 (1980), 399-411. | EuDML 115429 | MR 776251 | Zbl 0561.76105

[S] G. Stampacchia, On the filtration of a fluid through a porous medium with variable cross section, Russian Math. Surveys 29 (1974), 89-102. | MR 413739 | Zbl 0312.76058

[SV] R. Stavre - B. Vernescu, Incompressible fluid flow through a nonhomogeneous and anistropic dam, Nonlinear Anal. 9 (1985), 799-810. | MR 799885 | Zbl 0535.35080

[Tor1] A. Torelli, Existence and uniqueness of the solution of a non steady free boundary problem, Boll. Un. Mat. Ital. (5), 14-B (1977), 423-466. | MR 603295 | Zbl 0361.35031

[Tor2] A. Torelli, On a free boundary value problem connected with a non steady filtration phenomenon, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), 33-59. | EuDML 83745 | Numdam | MR 443564 | Zbl 0354.76069

[Tol] P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations (7) 8 (1983), 773-817. | MR 700735 | Zbl 0515.35024

[V] R.E. Volker, Nonlinear flow in porous media by finite element, J. Hydraulics Div. Proc. Amer. Soc. Civil. Eng. 95 (1969), 2093-2114.