Mathematical Sciences: Geometric Extremal Problems in the Plane

Matheson将研究在Dirichlet空间的单位球上定义的一类非线性泛函的极值函数的存在性和性质。当应用于全纯函数时，所讨论的泛函保持有界，对于全纯函数，值域的面积以π为界。对于满足适当凸性的泛函，已知覆盖平面上具有定域范围的泛函的最大值等于原点处的调和主量。已知某些泛函的极值域是单连通的，因此最大值与狄利克雷空间球上的最大值重合。Matheson将研究导致上述结论的泛函的条件。预计对称方法将在调查中发挥重要作用。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。

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.