Analysis and Mathematical Physics

, Volume 2, Issue 1, pp 79–88 | Cite as

On the logarithm of the minimizing integrand for certain variational problems in two dimensions

  • Murat Akman
  • John L. LewisEmail author
  • Andrew Vogel


Let f be a smooth convex homogeneous function of degreep, 1 < p < ∞, on \({\mathbb{C} \setminus \{0\}.}\) We show that if u is a minimizer for the functional whose integrand is \({f(\nabla v ), v}\) in a certain subclass of the Sobolev space W 1,p (Ω), and \({\nabla u \not = 0 }\) at \({z \in \Omega,}\) then in a neighborhood of z, \({ \log f (\nabla u ) }\) is a sub, super, or solution (depending on whether p > 2, p < 2, or p = 2) to L where
$$L \zeta=\sum_{k,j=1}^{2}\frac{\partial}{\partial x_k}\left( f_{\eta_k \eta_j}(\nabla u(z)) \frac{\partial \zeta }{ \partial x_j }\right),$$
we then indicate the importance of this fact in previous work of the authors when f(η) = |η| p and indicate possible future generalizations of this work in which this fact will play a fundamental role.


Calculus of variations Homogeneous integrands p-harmonic function p-harmonic measure Hausdorff dimension Dimension of a measure 

Mathematics Subject Classification (2000)

Primary 35J25 35J70 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bennewitz B., Lewis J.: On the dimension of p-harmonic measure. Ann. Acad. Sci. Fenn. Math. 30(2), 459–505 (2005)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Heinonen J., Kilpeläinen T., Martio O.: Nonlinear potential theory of degenerate elliptic equations. Oxford University Press, Oxford (1993)zbMATHGoogle Scholar
  3. 3.
    Lewis J.: Note on p harmonic measure. Comput. Methods Funct. Theory 6(1), 109–144 (2006)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Lewis J., Nyström K., Poggi Corradini P.: p harmonic measure in simply connected domains. Ann. Inst. Fourier Grenoble 61(2), 689–715 (2011)zbMATHCrossRefGoogle Scholar
  5. 5.
    Makarov N.: Distortion of boundary sets under conformal mapping. Proc. Lond. Math. Soc. 51, 369–384 (1985)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA
  2. 2.Department of MathematicsSyracuse UniversitySyracuseUSA

Personalised recommendations