## 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 log^{α} ℒ (Ω). In dimension n = 2 the solutions are continuous whenever f lies in the Hardy space ℋ^{1} (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 log^{α} ℒ(Ω) into ℒ^{n} log^{β} 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 language | English (US) |
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Pages (from-to) | 805-824 |

Number of pages | 20 |

Journal | Indiana University Mathematics Journal |

Volume | 56 |

Issue number | 2 |

DOIs | |

State | Published - Jun 6 2007 |

## Keywords

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

## ASJC Scopus subject areas

- Mathematics(all)