Mathematical Sciences: Problems in Analysis Related to Symmetrization

Baernstein DMS-9501293 贝恩斯坦将继续他的工作理论的对称化。 特别感兴趣的将是使用对称化技术来解决各种极值问题。 例子包括找到精确值的布洛赫和朗道常数在复变函数理论，确定最低临界屈曲载荷对一个固定板的给定面积，并证明等周不等式中的大小边界的一组黎曼流形上的比较与球的相同体积的空间中的常曲率。 对称化是一个过程，在这个过程中，集合或函数被改变成其他集合或函数，它们具有相同的“大小”，但具有更多的对称性。 例如，欧几里得空间中的一组给定测度可以用一个以原点为中心、具有相同测度的球来代替。 一个函数可以通过用一个其半行原像是原函数的半行原像的对称化的函数来替换它。

Baernstein DMS-9501293 Baernstein will continue his work on the theory of symmetrization. Of particular interest will be the use of symmetrization techniques to solve extremal problems of various sorts. Examples include finding the exact values of the Bloch and Landau constants in complex function theory, determining the lowest critical buckling load on a clamped plate of given area, and proving isoperimetric inequalities in which the size of the boundary of a set on a Riemannian manifold is compared with that of a ball of the same volume in a space of constant curvature. Symmetrization is a process in which sets or functions are changed into other sets or functions of the same "size," but with more symmetry. For example, a set of given measure in Euclidean space might be replaced with a ball, centered at the origin, with the same measure. A function might then be symmetrized by replacing it with a function whose pre-image of half lines are the symmetrization of the pre-image of half lines of the original function.