### Abstract

Let f be a smooth convex homogeneous function of degreep, 1 < p < ∞, on (Formula presented.) We show that if u is a minimizer for the functional whose integrand is (Formula presented.) in a certain subclass of the Sobolev space W^{1,p}(Ω), and (Formula presented.) then in a neighborhood of z, (Formula presented.) is a sub, super, or solution (depending on whether p > 2, p < 2, or p = 2) to L where (Formula presented.) 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.

Original language | English (US) |
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Pages (from-to) | 79-88 |

Number of pages | 10 |

Journal | Analysis and Mathematical Physics |

Volume | 2 |

Issue number | 1 |

DOIs | |

State | Published - Mar 25 2012 |

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Mathematical Physics

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## Cite this

Akman, M., Lewis, J. L., & Vogel, A. (2012). On the logarithm of the minimizing integrand for certain variational problems in two dimensions.

*Analysis and Mathematical Physics*,*2*(1), 79-88. https://doi.org/10.1007/s13324-012-0023-8