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Journal of Mathematical Sciences

, Volume 213, Issue 4, pp 610–635 | Cite as

Hölder Continuity of Solutions to Nonlinear Parabolic Equations Degenerated on a Part of the Domain

  • M. D. Surnachev
Article
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We study the regularity of solutions to parabolic p-Laplace type equations degenerating uniformly with respect to a small parameter ε on a part of the domain. We prove ε-uniform estimates for the maximum of modulus, and Hölder estimates for the modulus of continuity of the solution. We also prove the Harnack inequality of a special form.

Keywords

Parabolic Equation Energy Estimate Harnack Inequality Nonnegative Solution Nonlinear Parabolic Equation 
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References

  1. 1.
    O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967)l English transl.: Am. Math. Soc., Providence RI (1968).Google Scholar
  2. 2.
    E. DiBenedetto, Degenerate Parabolic Equations, Springer, New York (1993).CrossRefMATHGoogle Scholar
  3. 3.
    E. DiBenedetto, U. Gianazza, and V. Vespri, “Harnack estimates for quasi-linear degenerate parabolic differential equations,” Acta Math. 200, No. 2, 181–209 (2008).CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    E. DiBenedetto, U. Gianazza, and V. Vespri, Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 9, No. 2, 385–422 (2010).Google Scholar
  5. 5.
    E. DiBenedetto, U. Gianazza, and V. Vespri, Harnack’s Inequality for Degenerate and Singular Parabolic Equations, Springer, Berlin (2012).CrossRefGoogle Scholar
  6. 6.
    Yu. A. Alkhutov and V. V. Zhikov, “On the Hölder property of solutions of degenerate elliptic equations” [in Russian], Dokl. Akad. Nauk, Ross. Akad. Nauk 378, No. 5, 583–588 (2001); Dokl. Math. 63, No. 3, 368–373 (2001).Google Scholar
  7. 7.
    Yu. A. Alkhutov and V. V. Zhikov, “A class of degenerate elliptic equations” [in Russian], Tr. Semin. I. G. Petrovskogo 23, 16-27 (2003); English transl.: J. Math. Sci., New York 120, No. 3, 1247–1254 (2004).Google Scholar
  8. 8.
    Yu. A. Alkhutov and E. A. Khrenova, “Harnack inequality for a class of second order degenerate elliptic equations” [in Russian], Tr. Mat. Inst. Steklova 278, 7-15 (2012); English transl.: Proc. Steklov Inst. Math. 278, 1-9 (2012).Google Scholar
  9. 9.
    Yu. A. Alkhutov and S. T. Guseinov, “Hölder continuity of solutions of an elliptic equation uniformly degenerating on part of the domain” [in Russian], Differ. Uravn. 45, No. 1, 54–59 (2009); English transl.: Differ. Equ. 45, No. 1, 53–58 (2009).Google Scholar
  10. 10.
    Yu. A. Alkhutov and V. Liskevich, “Gaussian upper bounds for fundamental solutions of a family of parabolic equations,” J. Evol. Equ. 12, No. 1, 165–179 (2012).CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Yu. A. Alkhutov and V. Liskevich, “Hölder continuity of solutions to parabolic equations uniformly degenerating on a part of the domain,” Adv. Differ. Equ. 17, No. 7-8, 747–766 (2012).MathSciNetMATHGoogle Scholar
  12. 12.
    E. Acerbi and N. Fusco, “A transmission problem in the calculus of variations,” Calc. Var. Partial Differ. Equ. 2, No. 1, 1–16 (1994).CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    J. Moser, “On a pointwise estimate for parabolic differential equations,” Comm. Pure Appl. Math. 24, 727-740 (1971).CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    M. D. Surnachev, “Improved estimates for parabolic p-Laplace type equations” [in Russian], Probl. Mat. Anal. 81, 81–106 (2015); English transl.: J. Math. Sci. 120, No. 4, 429–457 (2015).Google Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Keldysh Institute of Applied Mathematics RASMoscowRussia

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