Mathematical Sciences: Problems in Potential Theory

9400687吴该奖项支持对复杂分析的经典理论和从几个新观点出发的势理论所产生的问题的数学研究。本文研究了拟共形映射上加倍测度的零集的性质及相关问题。特别讨论了单位圆上拟对称同胚的零集在bilipschitz同胚下是否不变的问题。研究一致椭圆型偏微分算子的调和测度。特别强调的是通过真正的n维方法在正正交上建立算子，以便相应的谐波措施的支持具有小的豪斯多夫维数。我们将努力研究在满足容量密度条件的域上估计条件布朗运动预期寿命的实用方法。这些领域的一个新的特征预计将导致。复变分析和势理论研究的重点是满足一定偏微分方程的多变量实函数和复函数问题。该理论是高度几何化的，解决许多重要问题的方法来自于大量的数学理论，如概率论、微分几何、微分方程和泛函分析。＊＊＊

9400687 Wu This award supports mathematical research on problems arising from the classical theory of complex analysis and potential theory approached from several new points of view. The work concerns investigations into the nature of null sets for doubling measures and related problems on quasiconformal mapping. In particular the question of whether null sets of quasisymmetric homeomophisms on the unit circle are invariant under bilipschitz homeomorphisms. Work will also continue on harmonic measures for uniformly elliptic partial differential operators. Particular emphasis will be placed on building operators on the positive orthant by genuinely n-dimensional methods so that the support of the corresponding harmonic measures have small Hausdorff dimensions. Effort will be made to investigate practical ways of estimating expected lifetime for conditional Brownian motion for domains satisfying a capacity density condition. A new characterization of such domains is expected to result. Research in complex analysis and potential theory focuses on problems related to real and complex functions of several variables which satisfy certain partial differential equations. The theory is highly geometric and methods for attacking the many important problems are drawn from a multitude of mathematical theories such as probability, differential geometry, differential equations and functional analysis. ***