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

  • Sungwon Cho1Email author and

    Affiliated with

    • Mikhail Safonov2

      Affiliated with

      Boundary Value Problems20062007:057928

      DOI: 10.1155/2007/57928

      Received: 16 March 2006

      Accepted: 28 May 2006

      Published: 3 December 2006

      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]

      Authors’ Affiliations

      (1)
      Department of Mathematics, Michigan State University
      (2)
      School of Mathematics, University of Minnesota

      References

      1. Friedman A: Partial Differential Equations of Parabolic Type. Prentice-Hall, New Jersey; 1964:xiv+347.MATH
      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.View Article
      3. Krylov NV: Second-Order Nonlinear Elliptic and Parabolic Equations. Nauka, Moscow; 1985:376. English translation: Reidel, Dordrecht, 1987MATH
      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, 1968MATH
      5. Ladyzhenskaya OA, Ural'tseva NN: Linear and Quasilinear Elliptic Equations. Nauka, Moscow; 1964. English translation: Academic Press, New York, 1968; 2nd Russian ed. 1973MATH
      6. Lieberman GM: Second Order Parabolic Differential Equations. World Scientific, New Jersey; 1996:xii+439.MATHView Article
      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.MATHMathSciNet
      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.MathSciNetView Article
      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/BF02412230MATHMathSciNetView Article
      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)90134-6MATHMathSciNetView Article
      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)90018-4MATHMathSciNetView Article
      12. Gilbarg D, Hörmander L: Intermediate Schauder estimates. Archive for Rational Mechanics and Analysis 1980,74(4):297–318.MATHMathSciNetView Article
      13. Landis EM: Second Order Equations of Elliptic and Parabolic Type. Nauka, Moscow; 1971:287. English translation: American Mathematical Society, Rhode Island, 1997MATH
      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, 151–164MATHMathSciNet
      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, 851–863MATHMathSciNet
      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.View Article
      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
      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
      19. Moser J: On Harnack's theorem for elliptic differential equations. Communications on Pure and Applied Mathematics 1961, 14: 577–591. 10.1002/cpa.3160140329MATHMathSciNetView Article
      20. Serrin J: Local behavior of solutions of quasi-linear equations. Acta Mathematica 1964,111(1):247–302. 10.1007/BF02391014MATHMathSciNetView Article

      Copyright

      © S. Cho and M. Safonov 2007

      This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.