月曜解析セミナー： Hausdorff Dimension of measures associated to positive weak solution of certain PDEs

2014年 　 2月 24日 14時 45分 ～ 2014年 　 2月 24日 16時 15分

Murat Akman (University of Kentucky)

In the first part of my talk I will present a recent result on the study of Hausdorff dimension of a measure associated to p-harmonic function defined in an open set $\Omega\subset\mathbb{R}^{n}$ and vanishing on a portion $\Gamma$ of $\partial\Omega$. If $p>n$ we show this measure is concentrated on a set of $\sigma-$finite $n-1$ dimensional Hausdorff measure, while if $p=n$ the same conclusion holds provided that $\Gamma$ is locally uniformly fat in the sense of $n-$capacity. This is a joint work with John Lewis and Andrew Vogel.

In the second part I will discuss Hausdorff dimension of a measure related to a positive weak solution of a certain partial differential equation, generalized p-laplace equation, in a simply connected domain. This work generalizes work of Lewis and coauthors when the measure is p-harmonic and also for p = 2, the well known theorem of Makarov regarding the Hausdorff dimension of harmonic measure relative to a point in a simply connected domain.