Archive for Rational Mechanics and Analysis

, Volume 210, Issue 3, pp 975–1020 | Cite as

Nonlinear Elliptic–Parabolic Problems

  • Inwon C. Kim
  • Norbert PožárEmail author


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).


Weak Solution Viscosity Solution Elliptic Problem Comparison Principle Parabolic Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Alt H.W., Luckhaus S.: Quasilinear elliptic–parabolic differential equations. Math. Z. 183, 311–341 (1983). doi: 10.1007/BF01176474 MathSciNetCrossRefzbMATHGoogle Scholar
  2. Armstrong, S.: Principal Half-Eigenvalues of Fully Nonlinear Homogeneous Elliptic Operators. Ph. D. thesisGoogle Scholar
  3. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  5. Benilan P., Wittbold P.: On mild and weak solutions of elliptic–parabolic problems. Adv. Differ. Equ. 1, 1053–1073 (1996)MathSciNetzbMATHGoogle Scholar
  6. Caffarelli, L., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, Vol. 43. American Mathematical Society, Providence, 1995Google Scholar
  7. Caffarelli, L., Salsa, S.: A Geometric Approach to Free Boundary Problems. Graduate Studies in Mathematics, Vol. 68. American Mathematical Society, Providence, 2005Google Scholar
  8. 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, 1999Google Scholar
  9. Carrillo J.: Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147, 269–361 (1999). doi: 10.1007/s002050050152 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  13. Domencio, P.A., Schwartz, F.W.: Physical and Chemical Hydrogeology. Wiley, New-York, 1998Google Scholar
  14. Evans, L.C.: Partial Differential Equations, 2nd edn. Graduate Studies in Mathematics, Vol. 19. American Mathematical Society, Providence, 2010Google Scholar
  15. Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Applications of Mathematics (New York), Vol. 25. Springer, New York, 1993Google Scholar
  16. Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Mathematische Lehrbücher und Monographien, II. Abteilung, Mathematische Monographien, Band 38. Akademie-Verlag, Berlin, 1974Google Scholar
  17. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 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 CrossRefADSzbMATHGoogle Scholar
  19. Kim I.C.: A free boundary problem with curvature. Commun. Partial Differ. Equ. 30, 121–138 (2005). doi: 10.1081/PDE-200044474 CrossRefzbMATHGoogle Scholar
  20. 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 CrossRefzbMATHGoogle Scholar
  21. 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, 1967Google Scholar
  22. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  24. Richards L.A.: Capillary conduction of liquids through porous mediums. Physics 1, 318–333 (1931)ADSCrossRefzbMATHGoogle Scholar
  25. Vázquez, J.L.: The Porous Medium Equation. Mathematical Theory. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, Oxford, 2007Google Scholar
  26. 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 CrossRefzbMATHGoogle Scholar
  27. 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 CrossRefzbMATHGoogle Scholar
  28. 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 CrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA
  2. 2.Graduate School of Mathematical SciencesUniversity of TokyoTokyoJapan

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