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

Article

Abstract

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 W1,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.

Keywords

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

Mathematics Subject Classification (2000)

Primary 35J25 35J70 

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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

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

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