The dimension of harmonic measure on some AD-regular flat sets of fractional dimension

2023/01/10 by Xavier Tolsa, Tolsa, Xavier · 1 citation
Computer Science · Mathematics · #31B15 #31B20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2301.04084

openalex publication_date 2023/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

In this paper it is shown that if E⊂\mathbb Rⁿ⁺¹ is an s-AD regular compact set, with s∈ [n-\frac12,n), and E is contained in a hyperplane or, more generally, in an n-dimensional C¹ manifold, then the Hausdorff dimension of the harmonic measure for the domain \mathbb Rⁿ⁺¹∖ E is strictly smaller than s, i.e., than the Hausdorff dimension of E.

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