Skip to main content
SpringerLink
Log in

Regularity in parabolic phase transition problems

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

We describe the main results obtained in a joint work with Athanasopoulos and Caffarelli on the regularity of viscosity solutions and of their free boundaries for a rather general class of parabolic phase transition problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alt, H.W. (1977). The fluid flow through porous media. Regularity of the free surface.Manuscripta Math.,21, 255–272.

    Article  MATH  MathSciNet  Google Scholar 

  2. Athanasopoulos, I. and Caffarelli, L.A. (1985). A theorem of real analysis and its application to free boundary problems.Comm. Pure Appl. Math.,38, 499–502.

    MathSciNet  Google Scholar 

  3. Alt, H.W., Caffarelli, L.A., and Friedman, A. (1984). Variational problems with two phases and their free boundaries,T.A.M.S.,282, 431–461.

    Article  MATH  MathSciNet  Google Scholar 

  4. Aronson, D., Caffarelli, L.A., and Vasquez, J.L. (1985). Interfaces with a corner point in one dimensional porous medium flow.Comm. Pure Appl. Math.,38, 374–404.

    Google Scholar 

  5. Athanasopoulos, I., Caffarelli, L.A., and Salsa, S. (1996). Caloric functions in Lipschitz domains and the regularity of solutions to phase transition problems,Annals of Math.,143, 413–434.

    Article  MATH  MathSciNet  Google Scholar 

  6. Athanasopoulos, I., Caffarelli, L.A., and Salsa, S. (1996). Regularity of the free boundary in parabolic phase transition problems,Acta Math.,176, 245–282.

    MATH  MathSciNet  Google Scholar 

  7. Athanasopoulos, I., Caffarelli, L.A., and Salsa, S. (1998). Phase Transition Problems of Parabolic Type. Flat Free Boundary are Smooth,Comm. Pure Appl. Math.,51, 75–110.

    Article  MathSciNet  Google Scholar 

  8. Athanasopoulos, I. and Salsa, S. (1991). An application of a parabolic comparison principle to free boundary problems.Indiana Univ. Math. J.,40, 29–32.

    Article  MATH  MathSciNet  Google Scholar 

  9. Caffarelli, L.A., (1987). A Harnack inequality approach to the regularity of free boundaries. Part I: Lipschitz free boundaries areC 1,α,Revista Math. Iberoamericana,3, 139–162.

    MATH  MathSciNet  Google Scholar 

  10. Caffarelli, L.A., (1989). A Harnack inequality approach to the regularity of free boundaries. Part II: Flat free boundaries are Lipschitz.Comm. Pure Appl. Math.,42, 55–78.

    MATH  MathSciNet  Google Scholar 

  11. Caffarelli, L.A. (1988). A Harnack inequality approach to the regularity of free boundaries. Part III: Existence theory, compactness, and dependence on x,Ann. Scuola Norm. Sup. Pisa,15, 583–602.

    MathSciNet  Google Scholar 

  12. Caffarelli, L.A. (1995). Uniform Lipschitz regularity of a singular perturbation problem.Diff. and Int. Eqs.,8, 1585–1590.

    MATH  MathSciNet  Google Scholar 

  13. Caffarelli, L.A. and Evans, L.C. (1983). Continuity of the temperature in the two phase Stefan problem,Arch. Rat. Mech. Anal.,81, 199–220.

    Article  MATH  MathSciNet  Google Scholar 

  14. Caffarelli, L.A. and Friedman, A. (1979). Continuity of the temperature in the Stefan problem,Indiana Univ. Math. J.,28, 53–70.

    Article  MATH  MathSciNet  Google Scholar 

  15. Caffarelli, L.A., Fabes, E.B., Mortola, S., and Salsa, S. (1981). Boundary behavior of non-negative solutions of elliptic operators in divergence form,Indiana Univ. Math. J.,30, 621–640.

    Article  MATH  MathSciNet  Google Scholar 

  16. Caffarelli, L.A. and Kenig, C. Gradient estimates for variable coefficient parabolic equations and singular perturbation problems, Preprint.

  17. DiBenedetto, E. (1982). Continuity of weak solutions to certain singular parabolic equations,Ann. Mat. Pura Appl.,121, 131–176.

    Article  MathSciNet  Google Scholar 

  18. Duvaut, G. (1973). Résolution d'un probleme de Stefan (fusion d'un bloc de glace à zéro degrés),C.R. Acad. Sci. Paris, Série I,276-A, 1461–1463.

    MathSciNet  Google Scholar 

  19. Friedman, A. (1968). The Stefan problem in several space variables,T.A.M.S.,133, 51–87.

    Article  MATH  Google Scholar 

  20. Friedman, A. (1970).Variational Principles and Free Boundary Problems, Wiley, New York.

    MATH  Google Scholar 

  21. Friedman, A. and Liu, Y. A free boundary problem arising in magneto hidrodynamics system,Ann. Scuola Norm. Sup. Pisa, cl. sci.,22, 375–448.

  22. Fabes, E.B., Garofalo, N., and Salsa, S. A Backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations,Ill Journal of Math.,30(4), 536–565.

  23. Fabes, E.B., Garofalo, N., and Salsa, A. (1984). Comparison theorems for temperatures in non-cylindrical domains,Atti Accad. Naz. Lincei, Ser. 8 Rend.,78, 1–12.

    MathSciNet  Google Scholar 

  24. Fabes, E.B. and Safonov, M.V. Behavior near the boundary of positive solutions of second order parabolic equations, to appear inProc. of El Escorial 96.

  25. Fabes, E.B., Safonov, M.V., and Yuan, Y. Behavior near the boundary of positive solutions of second order parabolic equations II. to appear inTrans. Amer. Math. Soc.

  26. Garofalo, N. (1984). Second order parabolic equations in non-variational form: Boundary Harnack principle and comparison theorems for non-negative solutions,Ann. Mat. Pura Appl.,138(4), 267–296.

    Article  MATH  MathSciNet  Google Scholar 

  27. Götz, I.G. and Zaltzman, B.B. (1991). Non-increase of mushy region in a nonhomogeneous Stefan problem,Quart. Appl. Math.,XLIX, 741–746.

    Google Scholar 

  28. Hanzawa, E.I. (1981). Classical solution of the Stefan problem,Tohoku Math.,33, 297–335.

    MATH  MathSciNet  Google Scholar 

  29. Ladyzenskaja, O.A., Solonnikov, V.A., and Ural'ceva, N.N. (1968). Linear and quasilinear equations of parabolic type,Trans. Math. Monographs,23, A.M.S., Providence, RI.

    Google Scholar 

  30. Meirmanov, A.M. (1981). On the classical solution of the multidimensional Stefan problem for quasilinear parabolic equations.Math. U.S.S.R-Sbornik,40, 157–178.

    Article  MATH  MathSciNet  Google Scholar 

  31. Nochetto, R.H. (1987). A class of non-degenerate two phase Stefan problems in several space variables,Comm. P.D.E.,12, 21–45.

    MATH  MathSciNet  Google Scholar 

  32. Rubenstein, L. (1979). The Stefan problem: comments on its present state,J. Inst. Maths. Applics.,124, 259–277.

    Google Scholar 

  33. Visintin, A. (1996).Models of Phase Transitions, Birkhäuser, Boston.

    MATH  Google Scholar 

  34. Widman, K.O. (1967). Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations,Math. Scand.,21, 7–27.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, 20133, Milan, Italy

    Sandro Salsa

Authors
  1. Sandro Salsa
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

To a great unforgettable Master.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Salsa, S. Regularity in parabolic phase transition problems. The Journal of Fourier Analysis and Applications 4, 549–564 (1998). https://doi.org/10.1007/BF02498224

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02498224

Keywords

Search

Navigation