Mathematical Sciences: Problems in Analysis Related to Symmetrization

Baernstein将继续他对对称理论的研究。特别感兴趣的将是使用对称技术来解决各种极端问题。例子包括找到复变函数理论中布洛赫常数和朗道常数的精确值，确定给定面积的夹紧板的最低临界屈曲载荷，以及证明等周不等式，其中黎曼流形上集合的边界大小与恒定曲率空间中相同体积的球的边界大小进行比较。对称是一个过程，在这个过程中，集合或函数被改变成具有相同“大小”但更对称的其他集合或函数。例如，在欧几里得空间中，一组给定的度量可以用一个以原点为中心的具有相同度量的球来代替。然后可以用一个函数来代替一个函数，这个函数的半线前像是原函数的半线前像的对称化。

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.