### Abstract

In this paper we study the Hausdorff dimension of a elliptic measure μ_{f} in space associated to a positive weak solution to a certain quasilinear elliptic PDE in an open subset and vanishing on a portion of the boundary of that open set. We show that this measure is concentrated on a set of σ-finite n−1 dimensional Hausdorff measure for p>n and the same result holds for p=n with an assumption on the boundary. We also construct an example of a domain in space for which the corresponding measure has Hausdorff dimension ≤n−1−δ for p≥n for some δ which depends on various constants including p. The first result generalizes the authors previous work in [3] when the PDE is the p-Laplacian and the second result generalizes the well known theorem of Wolff in [24] when p=2 and n=2.

Original language | English (US) |
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Pages (from-to) | 512-557 |

Number of pages | 46 |

Journal | Advances in Mathematics |

Volume | 309 |

DOIs | |

State | Published - Mar 17 2017 |

### Keywords

- Hausdorff Dimension of a Borel measure
- Hausdorff dimension
- Hausdorff measure
- Quasilinear elliptic PDEs
- The four-corner Cantor set

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Advances in Mathematics*,

*309*, 512-557. https://doi.org/10.1016/j.aim.2017.01.013