Mathematical Sciences: Symmetry Problems in Complex Analysisand Potential Theory

数学科学：复分析中的对称问题和势论

• 批准号: 8618755
• 负责人: Dmitry Khavinson
• 金额: $ 2.2万
• 依托单位: University of Arkansas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-06-01 至 1989-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8618755&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Symmetry; Problems; Complex

A two-year award is recommended in support of mathematical research focusing on approximation in the complex domain and associated higher dimensional generalizations. The principal investigator plans to use techniques from functional analysis, complex function theory and potential theory to address fundamental questions in two areas. The first concerns the correspondence between geometric properties of bounded sets and basic approximations related to the sets. Specifically, the principal investigator has established a lower bound for the distance between the classical conjugate of the identity function and the algebra of rational functions on a set. It is twice the area divided by the perimeter. Equality holds for discs and annuli. Work will be done investigating other possible cases of equality. Interestingly, this result includes the classical isoperimetric inequality. The solution to this geometric question involves questions about the Neumann problem for the Laplace operator and ordinary differential equations in the complex domain. All the concepts for approximating the conjugate identity in the plane extend to higher dimensions. This leads to the second line of study, generalizing classical Schwarz functions to dimension greater than two, replacing the conjugate function by one which becomes constant under the Laplacian (the distance function). Rational functions go over to harmonic functions (the closure of the kernel of the Laplacian). One has an easy upper bound on the approximating distance in terms of the volume of the set in question. The conjectured lower bound is volume divided by boundary area. Results in this context will be harder to achieve. However a viable form of the Schwarz function has been proposed for higher dimension. Taken together with additional work on the interplay between geometry and basic concepts of potential theory, one expects progress toward a better understanding of the nature of the lower bound sought. This work is expected to have broad application to mathematical analysis and the geometry of function spaces.

建议获得为期两年的奖励，以支持数学研究，重点是复杂域中的近似值和相关的更高维度概括。 主要研究者计划利用功能分析，复杂功能理论和潜在理论中的技术来解决两个领域的基本问题。 第一个涉及有限集的几何特性与与集合相关的基本近似值之间的对应关系。 具体而言，主要研究者已经为身份函数的经典结合物与集合上的理性函数代数之间的距离建立了一个下限。 它是区域除以周长的两倍。 平等适用于光盘和Annuli。 将完成调查其他可能的平等案例的工作。 有趣的是，该结果包括经典的等级不平等。 对这个几何问题的解决方案涉及有关laplace操作员和复杂域中普通微分方程的Neumann问题的问题。 近似平面中共轭身份的所有概念扩展到更高的维度。 这导致了第二道研究，将经典的Schwarz函数概括为大于两个的尺寸，将共轭函数替换为在laplacian（距离函数）下变为恒定的函数。 理性函数转到谐波函数（Laplacian的内核的关闭）。 就相关集合的体积而言，一个近似距离上的上限容易上限。 猜想的下界的体积除以边界区域。 在这种情况下的结果将很难实现。 然而，已经提出了一种可行的施瓦茨函数形式，以实现更高的维度。 结合了关于几何学与潜在理论基本概念之间相互作用的其他工作，人们期望进步以更好地理解下限型的性质。 预计这项工作将在数学分析和功能空间的几何形状上具有广泛的应用。