Mathematical Sciences: Problems in Analysis Related to Symmetrization

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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.