### Abstract

The p-harmonic transforms are the most natural nonlinear counterparts of the Riesz transforms in ℝ^{n}. They originate from the study of the p-harmonic type equation div|∇u|^{p-2}∇u = divf, where f: Ω → ℝ^{n} is a given vector field in ℒ^{q} (Ω, ℝ^{n}) and u is an unknown function of Sobolev class W_{0}^{1,p}(Ω, ℝ^{n}), p + q = pq. The p-harmonic transform ℋ_{p}: ℒ^{q}(Ω, ℝ^{n}) → ℒ^{p}(Ω, ℝ^{n}) as-signs to f the gradient of the solution: ℋ_{p}f = ∇u ∈ ℒ^{p}(Ω, ℝ^{n}). More general PDE's and the corresponding nonlinear operators are also considered. We investigate the extension and continuity properties of the p-harmonic transform beyond its natural domain of definition. In particular, we identify the exponents λ > 1 for which the operator ℋ_{p}: ℒ ^{λq}(Ω, ℝ^{n}) → ℒ^{λp}(Ω, ℝ^{n}) is well defined and remains continuous. Rather surprisingly, the uniqueness of the solution ∇u ∈ ℒ^{λp}(Ω, ℝ^{n}) fails when λ exceeds certain critical value. In case p = n = dim Ω, there is no uniqueness in W^{1,λn}(ℝ^{n}) for any λ > 1.

Original language | English (US) |
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Pages (from-to) | 683-718 |

Number of pages | 36 |

Journal | Indiana University Mathematics Journal |

Volume | 53 |

Issue number | 3 |

DOIs | |

State | Published - Oct 18 2004 |

### Keywords

- Elliptic PDE's
- p-harmonic transform

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Indiana University Mathematics Journal*,

*53*(3), 683-718. https://doi.org/10.1512/iumj.2004.53.2462