Riesz capacity: Hausdorff measure and extremal ratios
Riesz capacity measures the size of a set in $\mathbb{R}^n$ in terms of a pairwise interaction kernel $|x-y|^{-p}$ with exponent $p In the first part of the dissertation, the decay rate of Riesz capacity as the exponent $p$ increases to $n$ is shown to yield the Hausdorff measure of the set. The result applies to strongly rectifiable sets, and so in particular to submanifolds of Euclidean space. For strictly self-similar fractals, a one-sided decay estimate is found. How does the capacity change when the exponent $p$ increases? A longstanding conjecture by P\'olya and Szeg\H{o} claims that the ball maximizes the ratio of $q$-capacity over $p$-capacity when $q>p>0$. In the second part of the dissertation, we investigate the capacity ratio when $p
Riesz capacity, Hausdorff measure, extremal ratios
University of Illinois - Urbana-Champaign
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