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

Murat Akman, John L. Lewis, Andrew Vogel

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

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 W1,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 languageEnglish (US)
Pages (from-to)79-88
Number of pages10
JournalAnalysis and Mathematical Physics
Volume2
Issue number1
DOIs
StatePublished - 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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