Mathematical Sciences: Harmonic Analysis and Elliptic Partial Differential Equations

该奖项支持专注于分析和偏微分方程式三个领域的数学研究。第一部分讨论一般二阶椭圆型算子的Neumann问题的可解性。这里的基本特征是假设系数缺乏光滑性，这使得研究更接近于物理世界中发生的问题的表示。关于狄利克莱特问题的这类问题已经成功地完成了工作。在目前的上下文中，人们感兴趣的是指定边界通量而不是解的边界值。第二个领域将集中于具有不光滑边界的区域上的高阶椭圆型方程边值问题的可解性。也就是说，具有角点和不可微边的边界。这个问题还涉及到在更具体的应用中出现的问题。在常系数Dirichlet问题和在区域上寻找Neumann问题可解的尖锐条件方面取得了进展。在Besov或Soblev空间中对具有边界数据的齐次问题的解进行分类方面仍有工作要做。第三个问题是关于球面上的Laplace Beltrami算子的第一个本征值的凸下界的存在性。给出了本征值作为盖子大小的函数的数值估计。更精确的解析估计将为次调和函数的增长提供估计，并对自由边界问题有重要的影响。

This award supports mathematical research focusing on three areas of analysis and partial differential equations. The first concerns the solvability of the Neumann problem for general second order elliptic operators with bounded measurable coefficients. The essential feature here is the lack of smoothness assumed on the coefficients which brings the research closer to representations of problems occurring in the physical world. Successful work has already been done on such questions for the Dirichlet problem. In the present context one is interested in specifying the boundary flux rather than the boundary values of the solution. The second area will concentrate on the solvability of boundary value problems for higher order elliptic equations in domains with less than smooth boundaries. That is, boundaries with corners and nondifferentiable edges. This problem also relates to problems arising in more concrete applications. Progress has been made on the Dirichlet problem with constant coefficients and on finding sharp conditions on the domain which allow for solvability of the Neumann problem. Work remains to be done in classifying solutions of homogeneous problems with boundary data in the Besov or Sobolev spaces. The third problem concerns the existence of a convex lower bound for the first eigenvalue of the Laplace Beltrami operator on the sphere minus a polar cap. Numerical estimates have been obtained on the value of the eigenvalue as a function of the size of the cap. Sharper, analytic estimates will provide estimates for the growth of subharmonic functions as well as have important consequences for free boundary problems.