Mathematical Sciences: Extremal Sobolev Inequalities and Applications

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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.