The Hausdorff dimension of elliptic measure—a counterexample to the Oksendahl conjecture in 𝑅²

Sweezy, Caroline

doi:10.1090/S0002-9939-1992-1161401-5

The Hausdorff dimension of elliptic measure—a counterexample to the Oksendahl conjecture in $\textbf {R}^ 2$
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by Caroline Sweezy PDF

Proc. Amer. Math. Soc. 116 (1992), 361-368 Request permission

Abstract:

Two counterexamples to the Oksendahl conjecture in ${\mathbb {R}^2}$ for elliptic measure are constructed. It is shown that there exists a strictly elliptic divergence form operator in a specially constructed quasi-disk such that the associated elliptic measure has as its support a set of Hausdorff dimension arbitrarily close to 2. The method is the construction of a quasi-conformal map from a quasi-disk whose boundary has high Hausdorff dimension to the unit disk. The $L$-operator is the pull-back of $\Delta$ on the unit disk.

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