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The Hausdorff dimension of elliptic measure--a counterexample to the Oksendahl conjecture in $ {\bf R}\sp 2$


Author: Caroline Sweezy
Journal: Proc. Amer. Math. Soc. 116 (1992), 361-368
MSC: Primary 42B25
DOI: https://doi.org/10.1090/S0002-9939-1992-1161401-5
MathSciNet review: 1161401
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1161401-5
Keywords: Elliptic measure, Hausdorff dimension, quasi-circle, quasi-conformal map
Article copyright: © Copyright 1992 American Mathematical Society

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