Complex Analysis and Potential Theory

Khavinson计划利用偏微分方程和复分析技术，研究非线性偏微分方程、近似理论和势理论中的几个问题。他将研究复p-拉普拉斯方程解的实参原理及其与解析函数极值问题的关系，解析函数在均值处最佳逼近的定性性质，反射原理和“天线问题”，以及双层势的逆问题。这些问题中的大多数虽然源于相当经典的数学，但需要大量的现代工具来解决，尽管它们的公式相当基本，但它们对所有解决它们的尝试都具有弹性。它们在数学和应用的其他分支中都有重要的分支，例如，“天线问题”处理的是在障碍物之外寻找圆柱形波的最佳“接收”天线；对双层势的研究将允许人们描述一类平面曲线和数据，人们可以通过涉及双层积分与标准程序的直接公式来解决最重要的边值问题-狄利克雷问题。卡文森将与三名研究生合作开展这些项目，这将对阿肯色州的研究生项目和人力资源发展产生积极的影响。

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.