Complex Analysis and Potential Theory

摘要Khavinson利用偏微分方程和复分析技术，计划研究非线性偏微分方程组、逼近理论和位势理论中的几个问题。他将研究复杂p-Laplace方程的解的辐角原理及其与解析函数极值问题的关系，解析函数最佳平均逼近的定性性质，反射原理和“天线问题”，以及双层位势的反问题。这些问题中的大多数虽然起源于相当经典的数学，但需要大量的现代工具来解决，尽管它们的表述非常基础，但对所有解决它们的尝试都是有弹性的。所有这些都在数学和应用的其他分支中有重要的影响，例如，“天线问题”涉及为超越障碍的柱面波寻找可能的最佳“接收”天线，对双层势的研究将允许人们通过涉及双层积分与标准程序的直接公式来表征可以解决最重要的边值问题--狄利克雷问题--的一类平面曲线和数据。Khavinson将与三名研究生合作参与这些项目，这将对阿肯色州的研究生课程和人力资源开发产生强大的积极影响。

ABSTRACT Khavinson Using techniques of partial differential equations and complex analysis, Khavinson plans to study several problems in nonlinear pde's, approximation theory and potential theory. He will study the argument principle for solutions of the complex p-Laplace equation and its relation to extremal problems for analytic functions, qualitative properties of the best approximations in the mean by analytic functions, the reflection principle and the "antenna problem", and an inverse problem for double layer potentials. Most of these problems although stemming from rather classical mathematics will require the arsenal of the modern tools for their solution and, though quite basic in their formulation, have been resilient to all attempts to solve them. All of them have important ramifications in other branches of mathematics and applications, e.g. the "antenna problem" deals with the search for a best possible "receiving" antenna for cylindrical waves beyond an obstacle, research on double layer potentials would allow one to characterize the class of plane curves and data for which one can solve the most important boundary value problem - the Dirichlet problem - by direct formulae involving the integrals of the double layer vs. standard procedures. Khavinson will work in these projects in collaboration with three graduate students which should have a strong positive impact on the graduate program and human-resource development in Arkansas.