Abstract
We introduce a notion of viscosity solutions for a general class of elliptic–parabolic phase transition problems. These include the Richards equation, which is a classical model in filtration theory. Existence and uniqueness results are proved via the comparison principle. In particular, we show existence and stability properties of maximal and minimal viscosity solutions for a general class of initial data. These results are new, even in the linear case, where we also show that viscosity solutions coincide with the regular weak solutions introduced in Alt and Luckhaus (Math Z 183:311–341, 1983).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alt H.W., Luckhaus S.: Quasilinear elliptic–parabolic differential equations. Math. Z. 183, 311–341 (1983). doi:10.1007/BF01176474
Armstrong, S.: Principal Half-Eigenvalues of Fully Nonlinear Homogeneous Elliptic Operators. Ph. D. thesis
Bertsch M., Hulshof J.: Regularity results for an elliptic–parabolic free boundary problem. Trans. Am. Math. Soc. 297, 337–350 (1986). doi:10.2307/2000472
Brändle C., Vázquez J.L.: Viscosity solutions for quasilinear degenerate parabolic equations of porous medium type. Indiana Univ. Math. J. 54, 817–860 (2005). doi:10.1512/iumj.2005.54.2565
Benilan P., Wittbold P.: On mild and weak solutions of elliptic–parabolic problems. Adv. Differ. Equ. 1, 1053–1073 (1996)
Caffarelli, L., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, Vol. 43. American Mathematical Society, Providence, 1995
Caffarelli, L., Salsa, S.: A Geometric Approach to Free Boundary Problems. Graduate Studies in Mathematics, Vol. 68. American Mathematical Society, Providence, 2005
Caffarelli, L., Vazquez, J.L.: Viscosity solutions for the porous medium equation. In: Differential Equations: La Pietra 1996 (Florence). Proc. Sympos. Pure Math., Vol. 65. American Mathematical Society, Providence, pp. 13–26, 1999
Carrillo J.: Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147, 269–361 (1999). doi:10.1007/s002050050152
Crandall M.G., Ishii H., Lions P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27, 1–67 (1992). doi:10.1090/S0273-0979-1992-00266-5
Douglas J. Jr., Dupont T., Serrin J.: Uniqueness and comparison theorems for nonlinear elliptic equations in divergence form. Arch. Ration. Mech. Anal. 42, 157–168 (1971)
DiBenedetto E., Gariepy R.: Local behavior of solutions of an elliptic–parabolic equation. Arch. Ration. Mech. Anal. 97, 1–17 (1987). doi:10.1007/BF00279843
Domencio, P.A., Schwartz, F.W.: Physical and Chemical Hydrogeology. Wiley, New-York, 1998
Evans, L.C.: Partial Differential Equations, 2nd edn. Graduate Studies in Mathematics, Vol. 19. American Mathematical Society, Providence, 2010
Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Applications of Mathematics (New York), Vol. 25. Springer, New York, 1993
Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Mathematische Lehrbücher und Monographien, II. Abteilung, Mathematische Monographien, Band 38. Akademie-Verlag, Berlin, 1974
Kim I.C.: Uniqueness and existence results on the Hele–Shaw and the Stefan problems. Arch. Ration. Mech. Anal. 168, 299–328 (2003). doi:10.1007/s00205-003-0251-z
Kim I.C.: A free boundary problem arising in flame propagation. J. Differ. Equ. 191, 470–489 (2003). doi:10.1016/S0022-0396(02)00195-X
Kim I.C.: A free boundary problem with curvature. Commun. Partial Differ. Equ. 30, 121–138 (2005). doi:10.1081/PDE-200044474
Kim I.C., Požár N.: Viscosity solutions for the two-phase Stefan problem. Commun. Partial Differ. Equ. 36, 42–66 (2011). doi:10.1080/03605302.2010.526980
Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, 1967
Merz W., Rybka P.: Strong solutions to the Richards equation in the unsaturated zone. J. Math. Anal. Appl. 371, 741–749 (2010). doi:10.1016/j.jmaa.2010.05.066
Mannucci P., Vazquez J.L.: Viscosity solutions for elliptic–parabolic problems. No DEA Nonlinear Differ. Equ. Appl. 14, 75–90 (2007). doi:10.1007/s00030-007-4044-1
Richards L.A.: Capillary conduction of liquids through porous mediums. Physics 1, 318–333 (1931)
Vázquez, J.L.: The Porous Medium Equation. Mathematical Theory. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, Oxford, 2007
van Duyn C.J., Peletier L.A.: Nonstationary filtration in partially saturated porous media. Arch. Ration. Mech. Anal. 78, 173–198 (1982). doi:10.1007/BF00250838
Wang L.: On the regularity theory of fully nonlinear parabolic equations, I. Commun. Pure Appl. Math. 45, 27–76 (1992). doi:10.1002/cpa.3160450103
Wang L.: On the regularity theory of fully nonlinear parabolic equations, II. Commun. Pure Appl. Math. 45, 141–178 (1992). doi:10.1002/cpa.3160450202
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F. Lin
Rights and permissions
About this article
Cite this article
Kim, I.C., Požár, N. Nonlinear Elliptic–Parabolic Problems. Arch Rational Mech Anal 210, 975–1020 (2013). https://doi.org/10.1007/s00205-013-0663-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-013-0663-3