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This article is part of the series Harnack's Estimates, Positivity and Local Behavior of Degenerate and Singular Parabolic Equations.

Open Access Research Article

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

Sungwon Cho1* and Mikhail Safonov2

Author Affiliations

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

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

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Boundary Value Problems 2007, 2007:057928  doi:10.1155/2007/57928

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2007/1/057928


Received:16 March 2006
Revisions received:25 April 2006
Accepted:28 May 2006
Published:3 December 2006

© 2007 Cho and Safonov

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.

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.

References

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

  2. Gilbarg, D, Trudinger, NS: Elliptic Partial Differential Equations of Second Order, Fundamental Principles of Mathematical Sciences,p. xiii+513. Springer, Berlin (1983)

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

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

  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

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

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

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

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

  10. Michael, JH: A general theory for linear elliptic partial differential equations. Journal of Differential Equations. 23(1), 1–29 (1977). Publisher Full Text OpenURL

  11. Michael, JH: Barriers for uniformly elliptic equations and the exterior cone condition. Journal of Mathematical Analysis and Applications. 79(1), 203–217 (1981). Publisher Full Text OpenURL

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

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

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

  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). 96, 272–287, 312 English translation in Journal of Soviet Mathematics 21 (1983), no. 5, 851–863 (1980)

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

  17. Kondrat'ev, VA, Landis, EM: Qualitative theory of second-order linear partial differential equations. Partial Differential Equations, 3, Itogi Nauki i Tekhniki, pp. 99–215. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (1988) 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. 3, 25–43 (1957)

  19. Moser, J: On Harnack's theorem for elliptic differential equations. Communications on Pure and Applied Mathematics. 14, 577–591 (1961). Publisher Full Text OpenURL

  20. Serrin, J: Local behavior of solutions of quasi-linear equations. Acta Mathematica. 111(1), 247–302 (1964). Publisher Full Text OpenURL