Continuity estimates for n-harmonic equations

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17 Scopus citations

Abstract

We investigate the nonhomogeneous n-harmonic equation div|∇u| n-2∇u = f for u in the Sobolev space 1,n(Ω), where f is a given function in the Zygmund class ℒ logα ℒ (Ω). In dimension n = 2 the solutions are continuous whenever f lies in the Hardy space ℋ1 (Ω), so in particular, if f ∈ log ℒ(Ω). We show in higher dimensions that within the Zygmund classes the condition α > n - 1 is both necessary and sufficient for the solutions to be continuous. We also investigate continuity of the map f → ∇u, from ℒ logα ℒ(Ω) into ℒn logβ ℒ(Ω), for -1 < β < nα/n-1 -1. These and other results of the present paper, though anticipated by simple examples, are in fact far from routine. Certainly, they are central in the p-harmonic theory. Indiana University Mathematics Journal

Original languageEnglish (US)
Pages (from-to)805-824
Number of pages20
JournalIndiana University Mathematics Journal
Volume56
Issue number2
DOIs
StatePublished - Jun 6 2007

Keywords

  • Maximal functions
  • Orlicz-Sobolev imbedding
  • p-harmonic equation

ASJC Scopus subject areas

  • Mathematics(all)

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