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Journal of Mathematical Sciences

, Volume 176, Issue 3, pp 281–360 | Cite as

On the regularity of domains satisfying a uniform hour–glass condition and a sharp version of the Hopf–Oleinik boundary point principle

  • R. Alvarado
  • D. Brigham
  • V. Maz’ya
  • M. Mitrea
  • E. Ziadé
Article

We prove that an open, proper, nonempty subset of \({\mathbb{R}}^n\) is a locally Lyapunov domain if and only if it satisfies a uniform hour-glass condition. The limiting cases are as follows: Lipschitz domains may be characterized by a uniform double cone condition, and domains of class Open image in new window may be characterized by a uniform two-sided ball condition. We discuss a sharp generalization of the Hopf–Oleinik boundary point principle for domains satisfying an interior pseudoball condition, for semi-elliptic operators with singular drift and obtain a sharp version of the Hopf strong maximum principle for second order, nondivergence form differential operators with singular drift. Bibliography: 66 titles. Illustrations: 7 figures.

Keywords

Maximum Principle Shape Function Boundary Point Elliptic Operator Nonempty Subset 
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References

  1. 1.
    D. S. Jerison and C. E. Kenig, “Boundary behavior of harmonic functions in nontangentially accessible domains,” Adv. Math. 46, No. 1, 80–147 (1982).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    P. W. Jones, “Quasiconformal mappings and extendability of functions in Sobolev spaces,” Acta Math. 47, 71–88 (1981).CrossRefGoogle Scholar
  3. 3.
    G. David and S. Semmes, “Singular Integrals and Rectifiable Sets in \({\mathbb{R}}^n\): Beyond Lipschitz Graphs,” Astérisque, No. 193 (1991).Google Scholar
  4. 4.
    S. Hofmann, M. Mitrea, and M. Taylor, “Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains,” Int. Math. Res. Not. No. 14, 2567–2865 (2010).MathSciNetGoogle Scholar
  5. 5.
    S. Hofmann, M. Mitrea, and M. Taylor, “Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains,” J. Geom. Anal. 17, No. 4, 593–647 (2007).MATHMathSciNetGoogle Scholar
  6. 6.
    N. M. Gunther, La Théorie du Potentiel et ses Applications aux Problèmes Fondamentaux de la Physique Mathématique, Gauthier-Villars, Paris (1934).Google Scholar
  7. 7.
    N. M. Gunther, Potential Theory, Ungar, New York (1967).Google Scholar
  8. 8.
    D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin etc. (1983).MATHGoogle Scholar
  9. 9.
    P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel etc. (2007).MATHGoogle Scholar
  10. 10.
    M. S. Zaremba, “Sur un problème mixte relatif à l’équation de Laplace,” Bull. Int. l’Acad. Sci. Cracovie, Série A, Class Math. Nat., 313–344 (1910).Google Scholar
  11. 11.
    E. Hopf, “A remark on linear elliptic differential equations of second order,” Proc. Am. Math. Soc. 3, 791–793 (1952).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    O. A. Oleinik, “On properties of solutions of certain boundary problems for equations of elliptic type” [in Russian], Mat. Sb., N. Ser. No. 30(72), 695–702 (1952).Google Scholar
  13. 13.
    A. V. Bitsadze, Boundary Value Problems for Second Order Elliptic Equations, North-Holland, Amsterdam (1968).MATHGoogle Scholar
  14. 14.
    M. E. Taylor, Partial Differential Equations. 1: Basic Theory, Springer, Berlin (1996).MATHGoogle Scholar
  15. 15.
    L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL (1992).MATHGoogle Scholar
  16. 16.
    W. Ziemer, Weakly Differentiable Functions, Springer, New York (1989).MATHCrossRefGoogle Scholar
  17. 17.
    S. Alexander, “Local and global convexity in complete Riemannian manifolds,” Pac. J. Math. 76, No. 2, 283–289 (1978).MATHGoogle Scholar
  18. 18.
    P. Grisvard, Elliptic Problems in Nonsmooth Domains Pitman, Boston etc. (1985).MATHGoogle Scholar
  19. 19.
    N. S. Nadirashvili, “Lemma on the interior derivative and uniqueness of the solution of the second boundary value problem for second order elliptic equations” [in Russian], Dokl. Akad. Nauk SSSR 261, No. 4, 804–808 (1981); English transl.: Sov. Math., Dokl. 24, 598–601 (1981).MATHGoogle Scholar
  20. 20.
    E. J. McShane, “Extension of range of functions,” Bull. Am. Math. Soc. 40, 837–842 (1934).CrossRefMathSciNetGoogle Scholar
  21. 21.
    H. Whitney, “Analytic extensions of functions defined on closed sets,” Trans. Am. Math. Soc. 36, 63–89 (1934).CrossRefMathSciNetGoogle Scholar
  22. 22.
    D. Gilbarg, “Some hydrodynamic applications of function theoretic properties of elliptic equations,” Math. Z. 72, No. 1, 165–174 (1959).MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    A. I. Nazarov, A Centennial of the Zaremba–Hopf–Oleinik Lemma, Preprint (2010); arXiv:1101.0164v1
  24. 24.
    R. Finn and D. Gilbarg, “Asymptotic behavior and uniqueness of plane subsonic flows,” Commun. Pure Appl. Math. 10, No. 1, 23–63 (1957).MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    C. Neumann, “Über die Methode des Arithmetischen Mittels,” Abhand. Königl. Sächs. Ges. Wiss., Leipzig, 10, 662–702 (1888).Google Scholar
  26. 26.
    A. Korn, Lehrbuch der Potentialtheorie. II. Allgemeine Theorie des logarithmischen Potentials und der Potentialfunctionen in der Ebene, F. Dümmler., Berlin (1901).MATHGoogle Scholar
  27. 27.
    L. Lichtenstein, “Neue Beiträge zur Theorie der linearen partiellen Differentialgleichungen zweiter Ordnung vom elliptischen Typus,” Math. Zeitschr. 20, 194–212 (1924).CrossRefMathSciNetGoogle Scholar
  28. 28.
    G. Giraud, “Generalisation des probl`emes sur les op´erations du type elliptique,” Bull. Sci. Math. 56, 316–352 (1932).Google Scholar
  29. 29.
    G. Giraud, “Problèmes de valeurs à la frontière rèlatifs à certaines données discontinues,” Bull. Soc. Math. Fr. 61, 1–54 (1933).MathSciNetGoogle Scholar
  30. 30.
    C. Miranda, Partial Differential Equations of Elliptic Type Springer, New York etc. (1970).MATHGoogle Scholar
  31. 31.
    M. Keldysch and M. Lavrentiev, “Sur l’unicité de la solution du problème de Neumann,” C. R. (Dokl.) Acad. Sci. URSS 16, No. 3, 141–142 (1937).Google Scholar
  32. 32.
    V. A. Kondrat’ev and E. M. Landis, “Qualitative theory of second order linear partial differential equations,” Partial Differential Equations III. Encycl. Math. Sci. 32, 87–192 (1991).Google Scholar
  33. 33.
    M. V. Safonov, Boundary Estimates for Positive Solutions to Second Order Elliptic Equations, Preprint (2008).Google Scholar
  34. 34.
    L. I. Kamynin, “A theorem on the internal derivative for a weakly degenerate second order elliptic equation” [in Russian], Mat. Sb., Nov. Ser. 126(168), No. 3, 307–326 (1985); English transl.: Math. USSR, Sb. 54, No. 2, 297–316 (1986).CrossRefGoogle Scholar
  35. 35.
    E. Hopf, “Elementare Bemerkung über die Lösung partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus,” Sitzungsberichte Akad. Berlin, 147–152 (1927).Google Scholar
  36. 36.
    D. Gilbarg, “Uniqueness of axially symmetric flows with free boundaries,” J. Ration. Mech. Anal. 1, 309–320 (1952).MATHMathSciNetGoogle Scholar
  37. 37.
    E. M. Landis, Second Order Equations of Elliptic Type and Parabolic Type Am. Math. Soc., Providence, RI (1998).MATHGoogle Scholar
  38. 38.
    L. C. Evans, Partial Differential Equations, Am. Math. Soc., Providence, RI (1998).MATHGoogle Scholar
  39. 39.
    L. E. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge Univ. Press, Cambridge (2000).MATHCrossRefGoogle Scholar
  40. 40.
    M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, NJ (1967).Google Scholar
  41. 41.
    A. D. Aleksandrov, “Investigations on the maximum principle. I” [in Russian], Izv. Vyssh. Uchebn. Zaved. Mat. No. 5(6), 126–157 (1958).Google Scholar
  42. 42.
    A. D. Aleksandrov, “Investigations on the maximum principle. II” [in Russian], Izv. Vyssh. Uchebn. Zaved. Mat. No. 3(10), 3–12 (1959).Google Scholar
  43. 43.
    A. D. Aleksandrov, “Investigations on the maximum principle. III” [in Russian], Izv. Vyssh. Uchebn. Zaved. Mat. No. 5(12), 16–32 (1959).Google Scholar
  44. 44.
    A. D. Aleksandrov, “Investigations on the maximum principle. IV” [in Russian], Izv. Vyssh. Uchebn. Zaved. Mat. No. 3(16), 3–15 (1960).Google Scholar
  45. 45.
    A. D. Aleksandrov, “Investigations on the maximum principle. V” [in Russian], Izv. Vyssh. Uchebn. Zaved. Mat. No. 5(18), 16–26 (1960).Google Scholar
  46. 46.
    A. D. Aleksandrov, “Investigations on the maximum principle. VI” [in Russian], Izv. Vyssh. Uchebn. Zaved. Mat. No. 1(20), 3–20 (1961).Google Scholar
  47. 47.
    L. I. Kamynin and B. N. Khimchenko, “On theorems of Giraud type for a second order elliptic operator weakly degenerate near the boundary” [in Russian], Dokl. Akad. Nauk SSSR 224, No. 4, 752–755 (1975); English transl.: Sov. Math., Dokl. 16, No. 5, 1287–1291 (1975).MATHGoogle Scholar
  48. 48.
    L. I. Kamynin and B. N. Khimchenko, “The maximum principle for an elliptic-parabolic equation of the second order” [in Russian], Sib. Mat. Zh. 13, 773–789 (1972); English transl.: Sib. Math. J. 13, 533–545 (1973).MATHCrossRefGoogle Scholar
  49. 49.
    L. I. Kamynin and B. N. Khimchenko, “Theorems of the Giraud type for second order equations with weakly degenerate nonnegative characteristic part” [in Russian], Sib. Mat. Zh. 18, No. 1, 103–121 (1977); English transl.: Sib. Math. J. 18, 76–91 (1977).MATHCrossRefGoogle Scholar
  50. 50.
    L. I. Kamynin and B. N. Khimchenko, “Investigations of the maximum principle,” Dokl. Akad. Nauk SSSR 240, No. 4, 774–777 (1978); English transl.: Sov. Math., Dokl. 19, 677–681 (1978).MATHGoogle Scholar
  51. 51.
    R. Výborný, “On certain extensions of the maximum principle,” In: Differential Equations and their Applications, Proc. Copnf. Prague Sept. 1962, pp. 223–228, Academic Press, New York (1963).Google Scholar
  52. 52.
    L. I. Kamynin and B. N. Khimchenko, “Development of Aleksandrov’s theory of the isotropic strict extremum principle” [in Russian], Differ. Uravn. 16, No. 2, 280–292 (1980); English transl.: Differ. Equations 16, No. 2, 181–189 (1980).MATHMathSciNetGoogle Scholar
  53. 53.
    C. Pucci, “Proprietà di massimo e minimo delle soluzioni di equazioni a derivate parziali del secondo ordine di tipo ellittico e parabolico. I,” Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat., (8) 23, No. 6, 370–375 (1957).Google Scholar
  54. 54.
    C. Pucci, “Proprietà di massimo e minimo delle soluzioni di equazioni a derivate parziali del secondo ordine di tipo ellittico e parabolico. II,” Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat., (8) 24, No. 1, 3–6 (1958).Google Scholar
  55. 55.
    G. M. Lieberman, “Regularized distance and its applications,” Pac. J. Math. 117, No. 2, 329–352 (1985).MATHMathSciNetGoogle Scholar
  56. 56.
    A. I. Nazarov and N. N. Ural’tseva, Qualitative Properties of Solutions to Elliptic and Parabolic Equations with Unbounded Lower Order Coefficients, Preprint (2009).Google Scholar
  57. 57.
    A. I. Nazarov and N. N. Ural’tseva, The Harnack Inequality and Related Properties for Solutions to Elliptic and Parabolic Equations with Divergence-Free Lower Order Coefficients [in Russian], Algebra Anal. 23, No. 1, 136–168 (2011); English transl.: St. Petersburg Math. J. 23, No. 1 (2012). [To appear]Google Scholar
  58. 58.
    M. V. Safonov, “Non-divergence elliptic equations of second order with unbounded drift,” In: Nonlinear Partial Differential Equations and Related Topic, pp. 211–232. Am. Math. Soc., Providence, RI (2010).Google Scholar
  59. 59.
    L. Nirenberg, “A strong maximum principle for parabolic equations,” Commun. Pure Appl. Math. 6, 167–177 (1953).MATHCrossRefMathSciNetGoogle Scholar
  60. 60.
    L. I. Kamynin, “A theorem on the internal derivative for a second order uniformly parabolic equation” [in Russian], Dokl. Akad. Nauk SSSR 299, No. 2, 280–283 (1988); English transl.: Sov. Math., Dokl. 37, No. 2, 373–376 (1988).MATHMathSciNetGoogle Scholar
  61. 61.
    L. I. Kamynin and B. N. Khimchenko, “The analogues of the Giraud theorem for a second order parabolic equation” [in Russian], Sib. Mat. Zh. 14, No. 1, 86–110 (1973); English transl.: Sib. Math. J. 14, 59–77 (1973).MATHCrossRefGoogle Scholar
  62. 62.
    L. I. Kamynin and B. N. Khimchenko, “An aspect of the development of the theory of the anisotropic strict A. D. Aleksandrov extremum principle” [in Russian], Differ. Uravn. 19, No. 3, 426–437 (1983); English transl.: Differ. Equations 19, No. 3, 318–327 (1983).MATHMathSciNetGoogle Scholar
  63. 63.
    A. Carbery, V. Maz’ya, M. Mitrea, and D. J. Rule, The Integrability of Negative Powers of the Solution of the Saint Venant Problem, Preprint (2010).Google Scholar
  64. 64.
    A. Ancona, “On strong barriers and an inequality of Hardy for domains in \({\mathbb{R}}^n\),” J. London Math. Soc. (2) 34, 247–290 (1986).CrossRefMathSciNetGoogle Scholar
  65. 65.
    R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, NJ (1970).MATHGoogle Scholar
  66. 66.
    C. S. Morawetz, J. B. Serrin, and Y. G. Sinai, Selected Works of Eberhard Hopf with Commentaries, Am. Math. Soc., Providence, RI (2002).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • R. Alvarado
    • 1
  • D. Brigham
    • 1
  • V. Maz’ya
    • 2
    • 3
  • M. Mitrea
    • 1
  • E. Ziadé
    • 1
  1. 1.University of Missouri at ColumbiaColumbiaUSA
  2. 2.University of LiverpoolLiverpoolUK
  3. 3.Linköping UniversityLinköpingSweden

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