Advertisement

Constructive Approximation

, Volume 36, Issue 1, pp 145–159 | Cite as

Parabolic Besov Regularity for the Heat Equation

  • Hugo Aimar
  • Ivana GómezEmail author
Article

Abstract

We obtain parabolic Besov smoothness improvement for temperatures on cylindrical regions based on Lipschitz domains. The results extend those for harmonic functions obtained by S. Dahlke and R. DeVore using the wavelet description of Besov regularity.

Keywords

Temperatures Besov regularity Wavelets Lipschitz cylindrical regions Nonlinear approximation 

Mathematics Subject Classification

35B65 35K05 46E35 

Notes

Acknowledgement

The research was supported by CONICET, UNL and ANPCyT (Argentina).

References

  1. 1.
    Aimar, H., Gómez, I.: Smoothness improvement for temperatures in terms of the Besov regularity of initial and Dirichlet data. Preprint Google Scholar
  2. 2.
    Aimar, H., Gómez, I., Iaffei, B.: Parabolic mean values and maximal estimates for gradients of temperatures. J. Funct. Anal. 255(8), 2008 (1939–1956) Google Scholar
  3. 3.
    Aimar, H., Gómez, I., Iaffei, B.: On Besov regularity of temperatures. J. Fourier Anal. Appl. 16(6), 1007–1020 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, vol. 223. Springer, Berlin (1976) zbMATHCrossRefGoogle Scholar
  5. 5.
    Besov, O.V., Il’in, V.P., Nikol’skiĭ, S.M.: Integral Representations of Functions and Imbedding Theorems. Scripta Series in Mathematics, vol. I+II. Winston, Washington (1978/1979). English transl.: Taibleson, M.H. (ed.) Google Scholar
  6. 6.
    Bownik, M.: The construction of r-regular wavelets for arbitrary dilations. J. Fourier Anal. Appl. 7(5), 489–506 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Dahlke, S.: Besov regularity for elliptic boundary value problems in polygonal domains. Appl. Math. Lett. 12(6), 31–36 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dahlke, S., DeVore, R.A.: Besov regularity for elliptic boundary value problems. Commun. Partial Differ. Equ. 22(1–2), 1–16 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    DeVore, R.A., Sharpley, R.C.: Maximal functions measuring smoothness. Mem. Am. Math. Soc. 47(293), viii+115 (1984) MathSciNetGoogle Scholar
  10. 10.
    Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93(1), 34–170 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Jakab, T., Mitrea, M.: Parabolic initial boundary value problems in nonsmooth cylinders with data in anisotropic Besov spaces. Math. Res. Lett. 13(5–6), 825–831 (2006) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Jerison, D., Kenig, C.E.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130(1), 161–219 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Leisner, C.: Nonlinear wavelet approximation in anisotropic Besov spaces. Indiana Univ. Math. J. 52(2), 437–455 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Meyer, Y.: Wavelets and Operators. Cambridge Studies in Advanced Mathematics, vol. 37. Cambridge University Press, Cambridge (1992). Translated from the 1990 French original by D.H. Salinger zbMATHGoogle Scholar
  15. 15.
    Peetre, J.: New Thoughts on Besov Spaces. Duke University Mathematics Series, vol. 1. Mathematics Department, Duke University, Durham (1976) zbMATHGoogle Scholar
  16. 16.
    Schmeisser, H.-J.: Anisotropic spaces. II. Equivalent norms for abstract spaces, function spaces with weights of Sobolev–Besov type. Math. Nachr. 79, 55–73 (1977) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Schmeisser, H.-J., Triebel, H.: Anisotropic spaces. I. Interpolation of abstract spaces and function spaces. Math. Nachr. 73, 107–123 (1976) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Instituto de Matemática Aplicada del Litoral (IMAL), Departamento de Matemática, Facultad de Ingeniería QuímicaCONICET-UNLSanta FeArgentina

Personalised recommendations