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首页> 外文期刊>Journal of the European Mathematical Society: JEMS >On the dimension of p-harmonic measure in space
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On the dimension of p-harmonic measure in space

机译：关于空间中p调和测度的维数

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Let ?? R~n, n ≥ 3, and let p, 1 < p < ∞, p 6≠ 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 ??, 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 ? is a δ-Reifenberg flat domain with δ < δ, then p-harmonic measure is concentrated on a set of σ-finite H~(n-1)-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.
机译：让?? R〜n，n≥3，设p，1 <∞，p 6≠2。在本文中，我们研究了在δ-Reifenberg平坦域的情况下，对p-Laplace方程的非负解，在Δε的一部分上消失的p调和测量的维数。我们证明，对于p≥n，存在δ=δ（p，n）/> 0小，使得是一个δ<δ的δ-Reifenberg平坦域，则p调和测度集中在一组σ有限H〜（n-1）测度上。对于p≥n，我们证明对于足够平坦的Wolff雪花，p调和量度的Hausdorff维数始终小于n-1。我们还证明，如果2

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