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.

建议颁发两年期奖项，以支持专注于复杂域近似和相关高维概括的数学研究。 首席研究员计划使用泛函分析、复杂函数理论和势理论等技术来解决两个领域的基本问题。 第一个涉及有界集合的几何性质和与集合相关的基本近似之间的对应关系。 具体来说，主要研究者已经为集合上恒等函数的经典共轭与有理函数的代数之间的距离建立了下界。 它是面积除以周长的两倍。 圆盘和环面相等。 将开展调查其他可能的平等案例的工作。 有趣的是，这个结果包括经典的等周不等式。 这个几何问题的求解涉及拉普拉斯算子的诺依曼问题和复域中的常微分方程。 所有用于近似平面中共轭恒等式的概念都扩展到更高的维度。 这导致了第二条研究路线，将经典施瓦茨函数推广到大于二的维度，用在拉普拉斯算子（距离函数）下恒定的一代替共轭函数。 有理函数转向调和函数（拉普拉斯核的闭合）。 就所讨论的集合的体积而言，近似距离有一个简单的上限。 推测的下限是体积除以边界面积。 在这种情况下，取得成果将更加困难。 然而，已经针对更高维度提出了施瓦茨函数的可行形式。 结合几何学和势论基本概念之间相互作用的额外工作，人们期望在更好地理解所寻求的下限的本质方面取得进展。 这项工作预计将广泛应用于数学分析和函数空间的几何。