Mathematical Sciences: Symmetry Problems in Complex Analysisand Potential Theory

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A two-year award is recommended in support of mathematical research focusing on approximation in the complex domain and associated higher dimensional generalizations. The principal investigator plans to use techniques from functional analysis, complex function theory and potential theory to address fundamental questions in two areas. The first concerns the correspondence between geometric properties of bounded sets and basic approximations related to the sets. Specifically, the principal investigator has established a lower bound for the distance between the classical conjugate of the identity function and the algebra of rational functions on a set. It is twice the area divided by the perimeter. Equality holds for discs and annuli. Work will be done investigating other possible cases of equality. Interestingly, this result includes the classical isoperimetric inequality. The solution to this geometric question involves questions about the Neumann problem for the Laplace operator and ordinary differential equations in the complex domain. All the concepts for approximating the conjugate identity in the plane extend to higher dimensions. This leads to the second line of study, generalizing classical Schwarz functions to dimension greater than two, replacing the conjugate function by one which becomes constant under the Laplacian (the distance function). Rational functions go over to harmonic functions (the closure of the kernel of the Laplacian). One has an easy upper bound on the approximating distance in terms of the volume of the set in question. The conjectured lower bound is volume divided by boundary area. Results in this context will be harder to achieve. However a viable form of the Schwarz function has been proposed for higher dimension. Taken together with additional work on the interplay between geometry and basic concepts of potential theory, one expects progress toward a better understanding of the nature of the lower bound sought. This work is expected to have broad application to mathematical analysis and the geometry of function spaces.