Skip to main content
Boundary Value Problems
Download PDF

Hölder Regularity of Solutions to Second-Order Elliptic Equations in Nonsmooth Domains

Boundary Value Problems volume 2007, Article number: 057928 (2006) Cite this article

Abstract

We establish the global Hölder estimates for solutions to second-order elliptic equations, which vanish on the boundary, while the right-hand side is allowed to be unbounded. For nondivergence elliptic equations in domains satisfying an exterior cone condition, similar results were obtained by J. H. Michael, who in turn relied on the barrier techniques due to K. Miller. Our approach is based on special growth lemmas, and it works for both divergence and nondivergence, elliptic and parabolic equations, in domains satisfying a general "exterior measure" condition.

[1234567891011121314151617181920]

References

  1. Friedman A: Partial Differential Equations of Parabolic Type. Prentice-Hall, New Jersey; 1964:xiv+347.

    MATH  Google Scholar 

  2. Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order, Fundamental Principles of Mathematical Sciences. Volume 224. 2nd edition. Springer, Berlin; 1983:xiii+513.

    Book  Google Scholar 

  3. Krylov NV: Second-Order Nonlinear Elliptic and Parabolic Equations. Nauka, Moscow; 1985:376. English translation: Reidel, Dordrecht, 1987

    MATH  Google Scholar 

  4. Ladyzhenskaya OA, Solonnikov VA, Ural'tseva NN: Linear and Quasilinear Equations of Parabolic Type. Nauka, Moscow; 1967:xi+648. English translation: American Mathematical Society, Rhode Island, 1968

    MATH  Google Scholar 

  5. Ladyzhenskaya OA, Ural'tseva NN: Linear and Quasilinear Elliptic Equations. Nauka, Moscow; 1964. English translation: Academic Press, New York, 1968; 2nd Russian ed. 1973

    MATH  Google Scholar 

  6. Lieberman GM: Second Order Parabolic Differential Equations. World Scientific, New Jersey; 1996:xii+439.

    MATH  Book  Google Scholar 

  7. Littman W, Stampacchia G, Weinberger HF: Regular points for elliptic equations with discontinuous coefficients. Annali della Scuola Normale Superiore di Pisa (3) 1963, 17: 43–77.

    MATH  MathSciNet  Google Scholar 

  8. Gilbarg D, Serrin J: On isolated singularities of solutions of second order elliptic differential equations. Journal d'Analyse Mathématique 1955/1956, 4: 309–340.

    MathSciNet  Article  Google Scholar 

  9. Miller K: Barriers on cones for uniformly elliptic operators. Annali di Matematica Pura ed Applicata. Serie Quarta 1967,76(1):93–105. 10.1007/BF02412230

    MATH  MathSciNet  Article  Google Scholar 

  10. Michael JH: A general theory for linear elliptic partial differential equations. Journal of Differential Equations 1977,23(1):1–29. 10.1016/0022-0396(77)90134-6

    MATH  MathSciNet  Article  Google Scholar 

  11. Michael JH: Barriers for uniformly elliptic equations and the exterior cone condition. Journal of Mathematical Analysis and Applications 1981,79(1):203–217. 10.1016/0022-247X(81)90018-4

    MATH  MathSciNet  Article  Google Scholar 

  12. Gilbarg D, Hörmander L: Intermediate Schauder estimates. Archive for Rational Mechanics and Analysis 1980,74(4):297–318.

    MATH  MathSciNet  Article  Google Scholar 

  13. Landis EM: Second Order Equations of Elliptic and Parabolic Type. Nauka, Moscow; 1971:287. English translation: American Mathematical Society, Rhode Island, 1997

    MATH  Google Scholar 

  14. Krylov NV, Safonov M: A property of the solutions of parabolic equations with measurable coefficients. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 1980,44(1):161–175. English translation in Mathematics of the USSR-Izvestiya, 16 (1981), no. 1, 151–164

    MATH  MathSciNet  Google Scholar 

  15. Safonov M: Harnack's inequality for elliptic equations and Hölder property of their solutions. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta imeni V. A. Steklova Akademii Nauk SSSR (LOMI) 1980, 96: 272–287, 312. English translation in Journal of Soviet Mathematics 21 (1983), no. 5, 851–863

    MATH  MathSciNet  Google Scholar 

  16. Ferretti E, Safonov M: Growth theorems and Harnack inequality for second order parabolic equations. In Harmonic Analysis and Boundary Value Problems (Fayetteville, AR, 2000), Contemp. Math.. Volume 277. American Mathematical Society, Rhode Island; 2001:87–112.

    Chapter  Google Scholar 

  17. Kondrat'ev VA, Landis EM: Qualitative theory of second-order linear partial differential equations. In Partial Differential Equations, 3, Itogi Nauki i Tekhniki. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow; 1988:99–215. English translation: in Encyclopaedia of Mathematical Sciences, 32, Springer, New York, 1991, 141–192

    Google Scholar 

  18. de Giorgi E: Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari. Memorie dell'Accademia delle Scienze di Torino. Classe di Scienze Fisiche, Matematiche e Naturali. Serie III 1957, 3: 25–43.

    MathSciNet  Google Scholar 

  19. Moser J: On Harnack's theorem for elliptic differential equations. Communications on Pure and Applied Mathematics 1961, 14: 577–591. 10.1002/cpa.3160140329

    MATH  MathSciNet  Article  Google Scholar 

  20. Serrin J: Local behavior of solutions of quasi-linear equations. Acta Mathematica 1964,111(1):247–302. 10.1007/BF02391014

    MATH  MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Sungwon Cho

  2. School of Mathematics, University of Minnesota, 127 Vincent Hall, Minneapolis, MN, 55455, USA

    Mikhail Safonov

Authors
  1. Sungwon Cho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mikhail Safonov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sungwon Cho.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Cho, S., Safonov, M. Hölder Regularity of Solutions to Second-Order Elliptic Equations in Nonsmooth Domains. Bound Value Probl 2007, 057928 (2006). https://doi.org/10.1155/2007/57928

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2007/57928

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • Special Growth