Advertisement

Archive for Rational Mechanics and Analysis

, Volume 214, Issue 2, pp 545–573 | Cite as

A Quantitative Modulus of Continuity for the Two-Phase Stefan Problem

  • Paolo Baroni
  • Tuomo Kuusi
  • José Miguel Urbano
Article

Abstract

We derive the quantitative modulus of continuity
$$\omega(r)=\left[ p+\ln \left( \frac{r_0}{r}\right)\right]^{-\alpha (n, p)},$$
which we conjecture to be optimal for solutions of the p-degenerate two-phase Stefan problem. Even in the classical case p = 2, this represents a twofold improvement with respect to the early 1980’s state-of-the-art results by Caffarelli– Evans (Arch Rational Mech Anal 81(3):199–220, 1983) and DiBenedetto (Ann Mat Pura Appl 103(4):131–176, 1982), in the sense that we discard one logarithm iteration and obtain an explicit value for the exponent α(n, p).

Keywords

Weak Solution Stefan Problem Degenerate Parabolic Equation Rebalance Local Weak Solution 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Acerbi E., Mingione G.: Gradient estimates for a class of parabolic systems. Duke Math. J 136(2), 285–320 (2007)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Athanasopoulos I., Caffarelli L., Salsa S.: Caloric functions in Lipschitz domains and the regularity of solutions to phase transition problems. Ann. Math 143(3), 413–434 (1996)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Baroni P., Kuusi T., Urbano J.M.: A quantitative modulus of continuity for the two-phase Stefan problem (preprint).http://arxiv.org/abs/1401.2623
  4. 4.
    Baroni P., Kuusi T., Urbano J.M.: On the boundary regularity in phase transition problems (in preparation)Google Scholar
  5. 5.
    Caffarelli L., Evans L.C.: Continuity of the temperature in the two-phase Stefan problem. Arch. Rational Mech. Anal 81(3), 199–220 (1983)ADSCrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Caffarelli L., Friedman A.: Continuity of the temperature in the Stefan problem. Indiana Univ. Math. J 28(1), 53–70 (1979)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Dancer E.N., Hilhorst D., Mimura M., Peletier L.A.: Spatial segregation limit of a competition–diffusion system. Eur. J. Appl. Math 10, 97–115 (1999)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    DiBenedetto E.: Continuity of weak solutions to certain singular parabolic equations. Ann. Mat. Pura Appl. 4(103), 131–176 (1982)CrossRefMathSciNetGoogle Scholar
  9. 9.
    DiBenedetto E.: A boundary modulus of continuity for a class of singular parabolic equations. J. Differ. Equ 63(3), 418–447 (1986)ADSCrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    DiBenedetto E.: Degenerate Parabolic Equations Universitext. Springer-Verlag, New York (1993)CrossRefGoogle Scholar
  11. 11.
    DiBenedetto E., Friedman A.: Regularity of solutions of nonlinear degenerate parabolic systems. J. Reine Angew. Math. (Crelle’s J.) 349, 83–128 (1984)MATHMathSciNetGoogle Scholar
  12. 12.
    DiBenedetto E., Gianazza U., Vespri V.: Harnack estimates for quasi-linear degenerate parabolic differential equations. Acta Math 200(2), 181–209 (2008)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    DiBenedetto E., Gianazza U., Vespri V.: Harnack’s inequality for degenerate and singular parabolic equations. In: (eds) In: Springer Monographs in Mathematics., Springer, New York (2012)Google Scholar
  14. 14.
    DiBenedetto E., Vespri V.: On the singular equation \({\beta(u)_t=\triangle u}\). Arch. Rational Mech. Anal 132(3), 247–309 (1995)ADSCrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Friedman A.: The Stefan problem in several space variables. Trans. Am. Math. Soc 133, 51–87 (1968)CrossRefMATHGoogle Scholar
  16. 16.
    Friedman A.: Variational Principles and Free-Boundary Problems. Wiley, New York (1982)MATHGoogle Scholar
  17. 17.
    Gianazza U., Surnachev M., Vespri V.: A new proof of the Hölder continuity of solutions to p-Laplace type parabolic equations. Adv. Calc. Var 3(3), 263–278 (2010)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Igbida N., Urbano J.M.: Uniqueness for nonlinear degenerate problems. NoDEA Nonlinear Differ. Equ. Appl 10(3), 287–307 (2003)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Kamenomostskaya, S.L.,: On Stefan problem. Mat. Sbornik 53, 489–514 (1961) (Russian)Google Scholar
  20. 20.
    Kim I.C., Pozar N.: Viscosity solutions for the two phase Stefan problem. Comm. Partial Differ. Equ 36 42–66 (2011)Google Scholar
  21. 21.
    Kuusi T.: Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7(4), 673–716 (2008)MATHMathSciNetGoogle Scholar
  22. 22.
    Kuusi T., Mingione G., Nystrom K.: Sharp regularity for evolutionary obstacle problems, interpolative geometries and removable sets. J. Math. Pures Appl 101(2), 119–151 (2014)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Ladyzhenskaya O.A., Solonnikov V.A., Ural’tseva N.N.: Linear and quasi-linear equations of parabolic type. In: Translations of Mathematical Monographs, vol. 23. American Mathematical Society (1968)Google Scholar
  24. 24.
    Nochetto R.H.: A class of nondegenerate two-phase Stefan problems in several space variables. Comm. Partial Differ. Equ 12(1), 21–45 (1987)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Oleinik O.A.: A method of solution of the general Stefan problem. Soviet Math. Dokl 1, 1350–1354 (1960)MathSciNetGoogle Scholar
  26. 26.
    Peskir G., Shiryaev A.: Optimal stopping and free-boundary problems. In: (eds) In: Lectures in Mathematics., ETH Zürich, Birkhäuser Verlag (2006)Google Scholar
  27. 27.
    Rubin J.: Transport of reacting solutes in porous media: relation between mathematical nature of problem formulation and chemical nature of reactions. Water Resour. Res 19(5), 1231–1252 (1983)ADSCrossRefGoogle Scholar
  28. 28.
    Sacks P.E.: Continuity of solutions of a singular parabolic equation. Nonlinear Anal 7(4), 387–409 (1983)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Salsa S.: Two-phase Stefan problem. Recent results and open questions. Milan J. Math 80(2), 267–281 (2012)MATHMathSciNetGoogle Scholar
  30. 30.
    Stefan J.: Über die Theorie der Eisbildung. Monatshefte Mat. Phys 1, 1–6 (1890)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Urbano J.M.: A free boundary problem with convection for the p-Laplacian. Rend. Mat. Appl. (7) 17(1), 1–19 (1997)Google Scholar
  32. 32.
    Urbano J.M.: Continuous solutions for a degenerate free boundary problem. Ann. Mat. Pura Appl. (4) 178, 195–224 (2000)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Urbano J.M.: The method of intrinsic scaling, a systematic approach to regularity for degenerate and singular PDEs. In: Lecture Notes in Mathematics, vol. 1930. Springer-Verlag, Berlin (2008)Google Scholar
  34. 34.
    Ziemer W.P.: Interior and boundary continuity of weak solutions of degenerate parabolic equations. Trans. Am. Math. Soc 271(2), 733–748 (1982)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Paolo Baroni
    • 1
  • Tuomo Kuusi
    • 2
  • José Miguel Urbano
    • 3
  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden
  2. 2.Institute of MathematicsAalto UniversityAaltoFinland
  3. 3.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal

Personalised recommendations