Open Access

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

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.