## Abstract

Let ßcR^{n}, n≥3, and let p, 1 > p > ∞, p ≠ 2, be given. In this paper we study the dimension of p -harmonic measures that arise from nonnegative solutions to the p -Laplace equation, vanishing on a portion of ∂ω, in the setting of δ-Reifenberg flat domains. We prove, for p ≥ n, that there exists δ = δ(p, n) > 0 small such that if ω is a δ-Reifenberg flat domain with δ < δ, then p-harmonic measure is concentrated on a set of ω -finite H ^{n}∼ -measure. We prove, for p ≥ n, that for sufficiently flat Wolff snowflakes the Hausdorff dimension of p-harmonic measure is always less than n - 1. We also prove that if 2 < p < n, then there exist Wolff snowflakes such that the Hausdorff dimension of p-harmonic measure is less than n - 1, while if 1 < p < 2, then there exist Wolff snowflakes such that the Hausdorff dimension of p-harmonic measure is larger than n - 1. Furthermore, perturbing off the case p = 2, we derive estimates for the Hausdorff dimension of p-harmonic measure when p is near 2.

Original language | English (US) |
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Pages (from-to) | 2197-2256 |

Number of pages | 60 |

Journal | Journal of the European Mathematical Society |

Volume | 15 |

Issue number | 6 |

DOIs | |

State | Published - 2013 |

## Keywords

- Hausdorff dimension
- P-harmonic function
- P-harmonic measure
- Reifenberg flat domain
- Wolff snowflake

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics