Mathematical Sciences: Extremal Sobolev Inequalities and Applications

主要研究人员将在傅立叶分析、非线性偏微分方程和谱分析问题的背景下继续她对极值函数和极值Sobolev不等式的研究。在过去的三年里，她将Soblev不等式应用于2-球面上的高斯曲率和n-球面上的标量曲率以及紧致流形上的等谱问题。在这个项目中，主要研究人员将利用极值Sobolev不等式研究以下四个问题：Szego投影定理，n-球面上共形拉普拉斯的对数行列式，关于曲率的进一步工作和紧致2-流形的统一化定理。

The principal investigator will continue her study of extremal functions and extremal Sobolev inequalities in the context of problems in Fourier analysis, nonlinear partial differential equations and spectral analysis. In the past three years she has applied Sobolev inequalities to problems involving prescribing the Gaussian curvature on the 2-sphere and the scalar curvature on the n-sphere and to isospectral problems on compact manifolds. In this project the principal investigator will study the following four problems using extremal Sobolev inequalities: Szego's projection theorem, the logarithm determinant formula for the conformal Laplacian on the n-sphere, further work on prescribing curvature and the Uniformization Theorem for compact 2-manifolds.