Mathematical Sciences: Applications of Harmonic and Parabolic Measure to Problems in Function Theory and PartialDifferential Equations

这个项目涉及数学研究， 用偏微分测度论的某些进展 方程和经典函数论。 问题 后一领域有几个来源。 一个是利特尔伍德 一个预言球面平均值的猜想 多项式和有理函数的导数。 的 猜想已被证明是尖锐的有理函数 但不适用于多项式。 工作将在获得 锐常数在理性的情况下，并制定出 另一个是多项式次数的正确幂。 这些 结果预计将来自更好的估计， 调和测度 这些应遵循最近的工作， 支持调和测度的集合的维数。 椭圆型偏微分方程的工作将是 关注的是衡量解决方案的增长， 用集合的Riesz容度表示的双调和方程 梯度消失的地方。 较低的尖锐结果 尺寸已经获得。 在抛物线和 热方程的绝对连续性问题 关于Lebesgue测度的抛物测度将是 被占用了事实并非总是如此， 为二维域建立。 在这一背景下， 的边界上的充分必要条件 绝对的连续性已经建立。 本工作 希望将理论扩展到更高维度 解的表示论的应用 抛物方程的解

This project involves mathematical research combining certain advances in measure theory with partial differential equations and classical function theory. Problems in the latter area stem from several sources. One is the Littlewood conjecture which predicts the mean value of the spherical derivature of polynomials and rational functions. The conjecture has been shown to be sharp for rational functions but not for polynomials. Work will be done in obtaining sharp constants in the rational case and working out the correct power of the polynomial's degree in the other. These results are expected to derive from better estimates for harmonic measure. These should follow from recent work on the dimension of sets which can support harmonic measure. Work on elliptic partial differential equations will be concerned with measuring the growth of solutions to the biharmonic equation in terms of the Riesz capacity of set where the gradient vanishes. Sharp results for lower dimensions have been obtained. In the area of parabolic and heat equations, the question of absolute continuity of the parabolic measure with respect to Lebesgue measure will be taken up. That this is not always the case has already been established for two-dimensional domains. In this context, good necessary and sufficient conditions on the boundary for absolute continuity have been established. The present work looks to extend the theory to higher dimensions. Applications to the representation theory for solutions of parabolic equations will follow.