Mathematical Sciences: Geometric Extremal Problems in the Plane

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Matheson DMS-9500835 Matheson will investigate the existence and properties of extremal functions for a certain class of nonlinear functionals defined over the unit ball of the Dirichlet space. The functionals in question remain bounded when applied to holomorphic functions for which the area of the range is bounded by pi. For functionals satisfying appropriate convexity properties it is known that the maximum over functions with range in a fixed domain is attained for the covering plane and is equal to the value of a certain harmonic majorant at the origin. It is known for some functionals that extremal domains are simply connected and hence the maximum coincides with the maximum over the ball of the Dirichlet space. Matheson will investigate conditions on functionals leading to the above conclusion. It is expected that symmetrization methods will play an important role in thie investigation. 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.