Mathematical Sciences: Applications of Harmonic and Parabolic Measure to Problems in Function Theory and PartialDifferential Equations

点击翻译按钮获取中文摘要

This project involves mathematical research combining certain advances in measure theory with partial differential equations and classical function theory. Problems in the latter area stem from several sources. One is the Littlewood conjecture which predicts the mean value of the spherical derivature of polynomials and rational functions. The conjecture has been shown to be sharp for rational functions but not for polynomials. Work will be done in obtaining sharp constants in the rational case and working out the correct power of the polynomial's degree in the other. These results are expected to derive from better estimates for harmonic measure. These should follow from recent work on the dimension of sets which can support harmonic measure. Work on elliptic partial differential equations will be concerned with measuring the growth of solutions to the biharmonic equation in terms of the Riesz capacity of set where the gradient vanishes. Sharp results for lower dimensions have been obtained. In the area of parabolic and heat equations, the question of absolute continuity of the parabolic measure with respect to Lebesgue measure will be taken up. That this is not always the case has already been established for two-dimensional domains. In this context, good necessary and sufficient conditions on the boundary for absolute continuity have been established. The present work looks to extend the theory to higher dimensions. Applications to the representation theory for solutions of parabolic equations will follow.