Mathematical Sciences: Harmonic Analysis and Elliptic Partial Differential Equations

该奖项支持数学研究，重点是三个 分析和偏微分方程。 第一 关于一般Neumann问题的可解性 二阶有界可测椭圆算子 系数 这里的基本特征是缺乏 平滑假设的系数，这带来了研究 更接近于物理问题的表现， 世界 在这些问题上已经做了成功的工作 关于Dirichlet问题 在目前的情况下，一个是 感兴趣的是指定边界通量，而不是 解的边界值。 第二个领域将集中在以下问题的可解性： 高阶椭圆型方程边值问题 边界不太平滑的区域。 也就是说， 有角和不可微的边。 这个问题也 涉及更具体应用中出现的问题。 关于常系数Dirichlet问题的研究取得了一些进展 系数和寻找尖锐的条件域， 考虑到诺依曼问题的可解性。 还有一些工作要 在对齐次问题的解决方案进行分类时， Besov或Sobolev空间中的边界数据。 第三个问题是关于凸下 拉普拉斯Beltrami算子第一特征值的界 在没有极冠的球体上。 数字估计已经 得到的特征值的值作为大小的函数 盖子的 更清晰的分析估计将提供估计 对于次谐波函数的增长， 自由边界问题。

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.