Mathematical Sciences: Problems in Analysis Related to Symmetrization

巴恩斯坦DMS-9501293巴恩斯坦将继续他在对称化理论方面的工作。特别令人感兴趣的将是使用对称化技术来解决各种极端问题。例如求复变函数论中Bloch常数和Landau常数的精确值，确定给定面积的固定板上的最低临界屈曲载荷，以及证明等周不等式，其中黎曼流形上的集合的边界的大小与常曲率空间中相同体积的球的边界的大小相比较。对称化是一个过程，在这个过程中，集合或函数被改变为其他具有相同“大小”的集合或函数，但具有更大的对称性。例如，欧几里得空间中的一组给定的度量可以用一个以原点为中心的球替换，并具有相同的度量。然后，可以通过用其半线的前像是原始函数的半线的前像的对称化的函数来替换该函数来对其进行对称化。

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.