Advertisement

Theoretical and Mathematical Physics

, Volume 157, Issue 3, pp 1655–1670 | Cite as

A strengthening of the interior Hölder continuity property for solutions of the Dirichlet problem for a second-order elliptic equation

  • A. K. GushchinEmail author
Article

Abstract

The classical solution of the Dirichlet problem with a continuous boundary function for a linear elliptic equation with Hölder continuous coefficients and right-hand side satisfies the interior Schauder estimates describing the possible increase of the solution smoothness characteristics as the boundary is approached, namely, of the solution derivatives and their difference ratios in the corresponding Hölder norm. We prove similar assertions for the generalized solution with some other smoothness characteristics. In contrast to the interior Schauder estimates for classical solutions, our established estimates for the differential characteristics imply the continuity of the generalized solution in a sense natural for the problem (in the sense of (n-1)-dimensional continuity) up to the boundary of the domain in question. We state the global properties in terms of the boundedness of the integrals of the square of the difference between the solution values at different points with respect to especially normalized measures in a certain class.

Keywords

elliptic equation smoothness of solution function space 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Grundlehren Math. Wiss., Vol. 224), Springer, Berlin (1983).zbMATHGoogle Scholar
  2. 2.
    V. P. Mikhailov, Differential Equations with Partial Derivitives [in Russian], Nauka, Moscow (1983); English transl. prev. ed.: Partial Differential Equations, Mir, Moscow (1978).Google Scholar
  3. 3.
    V. P. Mikhailov and A. K. Gushchin, Additional Chapters of Course “Equations of Mathematical Physics” [in Russian] (Lekts. Kursy NOTs, Vol. 7), Steklov Math. Inst., Russ. Acad. Sci., Moscow (2007).Google Scholar
  4. 4.
    V. P. Mikhajlov, Differential Equations, 12, 1320–1329 (1977).zbMATHGoogle Scholar
  5. 5.
    A. K. Gushchin, Math. USSR-Sb., 65, 19–66 (1990).zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    A. K. Gushchin and V. P. Mikhailov, Math. USSR-Sb., 81, 101–136 (1995).MathSciNetGoogle Scholar
  7. 7.
    A. K. Gushchin and V. P. Mikhailov, Dokl. Math., 48, 510–514 (1994).MathSciNetGoogle Scholar
  8. 8.
    A. K. Gushchin and V. P. Mikhailov, Sb. Math., 186, 197–219 (1995).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    A. K. Gushchin and V. P. Mikhailov, Dokl. Math., 54, 815–816 (1996).zbMATHGoogle Scholar
  10. 10.
    A. K. Gushchin, Dokl. Math., 62, 32–34 (2000).Google Scholar
  11. 11.
    A. K. Gushchin, Sb. Math., 193, 649–668 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    J. Necas, Czech. Math. J., 10(85), 283–298 (1960).MathSciNetGoogle Scholar
  13. 13.
    I. M. Petrushko, Math. USSR-Sb., 47, 43–72 (1984).zbMATHCrossRefGoogle Scholar
  14. 14.
    F. Riesz, Math. Z., 18, 87–95 (1923).CrossRefMathSciNetGoogle Scholar
  15. 15.
    J. E. Littlewood and R. Paley, J. London Math. Soc., 6, 230–233 (1931).zbMATHCrossRefGoogle Scholar
  16. 16.
    J. E. Littlewood and R. Paley, Proc. London Math. Soc., 42, 52–89 (1936).zbMATHCrossRefGoogle Scholar
  17. 17.
    J. E. Littlewood and R. Paley, Proc. London Math. Soc., 43, 105–126 (1937).zbMATHCrossRefGoogle Scholar
  18. 18.
    I. I. Privalov, Boundary Properties of Analytic Functions [in Russian] (2d ed.), Gostekhizdat, Moscow (1950).Google Scholar
  19. 19.
    E. M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton Math. Ser., Vol. 30), Princeton Univ. Press, Princeton, N. J. (1970).zbMATHGoogle Scholar
  20. 20.
    G. Cimmino, Rend. Circ. Mat. Palermo, 61, 177–221 (1938).zbMATHCrossRefGoogle Scholar
  21. 21.
    C. Miranda, Equazioni alle derivate parziali di tipo ellittico (Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 2), Springer, Berlin (1955).zbMATHGoogle Scholar
  22. 22.
    A. K. Gushchin and V. P. Mikhailov, “On boundary values of solutions of elliptic equations,” in: Generalized Functions and their Applications in Mathematical Physics [in Russian], Computing Center, USSR Acad. Sci., Moscow (1981), pp. 189–205.Google Scholar
  23. 23.
    O. I. Bogoyavlenskij et al., Proc. Steklov Inst. Math., No. 2, 65–105 (1988).Google Scholar
  24. 24.
    A. K. Gushchin, Sov. Math. Dokl., 38, 372–376 (1989).zbMATHMathSciNetGoogle Scholar
  25. 25.
    A. K. Gushchin and V. P. Mikhailov, Math. USSR-Sb., 73, 171–194 (1992).CrossRefMathSciNetGoogle Scholar
  26. 26.
    L. Carleson, Amer. J. Math., 80, 921–930 (1958).zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    L. Carleson, Ann. of Math., 76, 547–559 (1962).CrossRefMathSciNetGoogle Scholar
  28. 28.
    L. Hörmander, Math. Scand., 20, 65–78 (1967).zbMATHMathSciNetGoogle Scholar
  29. 29.
    E. De Giorgi, Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Natur., 3, 25–43 (1957).Google Scholar
  30. 30.
    J. Nash, Amer. J. Math., 80, 931–954 (1958).zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    J. Moser, Comm. Pure Appl. Math., 13, 457–468 (1960).zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1973); English transl. prev. ed.: Linear and Quasilinear Elliptic Equations, (Math. in Sci. and Engin., Vol. 46), Acad. Press, New York (1968).zbMATHGoogle Scholar
  33. 33.
    A. K. Gushchin, Siberian Math. J., 46, 826–840 (2005).CrossRefMathSciNetGoogle Scholar
  34. 34.
    A. K. Gushchin, Dokl. Math., 72, 665–668 (2005).zbMATHGoogle Scholar
  35. 35.
    A. K. Gushchin, Sb. Math., 189, 1009–1045 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    A. K. Gushchin, Dokl. Math., 57, 121–123 (1998).Google Scholar
  37. 37.
    A. K. Gushchin, Dokl. Math., 69, 329–331 (2004).zbMATHGoogle Scholar
  38. 38.
    A. K. Gushchin, Dokl. Math., 76, 486–489 (2007).CrossRefGoogle Scholar

Copyright information

© MAIK/Nauka 2008

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRASMoscowRussia

Personalised recommendations