, Volume 15, Issue 3, pp 401–409 | Cite as

Measuring the level sets of anisotropic homogeneous functions

  • Hugo Aimar
  • Ivana GómezEmail author


In this note we investigate some basic properties of the level sets of functions which are homogeneous with respect to nonisotropic dilations. In particular we obtain a formula for the volume of the level sets in terms of the area on the level surfaces. We relate the results to some well known mean value formulas for solutions of PDE’s.


Homogeneous functions Volume of level sets Mean values formula Euler formula 

Mathematics Subject Classification (2000)

31C99 26B15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aimar H., Gómez I., Iaffei B.: Parabolic mean values and maximal estimates for gradients of temperatures. J. Funct. Anal. 255(8), 1939–1956 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Evans, L.C.: Partial differential equations, graduate studies in mathematics, vol. 19. American Mathematical Society, Providence, RI (1998)Google Scholar
  3. 3.
    Fulks W.: A mean value theorem for the heat equation. Proc. Amer. Math. Soc. 17, 6–11 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Jerison D., Kenig C.E.: The inhomogeneous dirichlet problem in lipschitz domains. J. Funct. Anal. 130(1), 161–219 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Lasserre J.B.: Integration and homogeneous functions. Proc. Amer. Math. Soc. 127(3), 813–818 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Pokrovskiĭ A.V.: Mean value theorems for solutions of linear partial differential equations. Mat. Zametki 64(2), 260–272 (1998)Google Scholar
  7. 7.
    Watson N.A.: A theory of subtemperatures in several variables. Proc. Lond. Math. Soc. 26(3), 385–417 (1973)zbMATHCrossRefGoogle Scholar
  8. 8.
    Zalcman L.: Mean values and differential equations. Israel J. Math. 14, 339–352 (1973)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Facultad de Ingeniería Química (UNL)Instituto de Matemática Aplicada del Litoral (CONICET, UNL)Santa FeArgentina

Personalised recommendations