Special Applied Mathematics Seminar - Changfeng Gui-New Sharp Inequalities in Analysis and Geometry

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Title - New Sharp Inequalities in Analysis and Geometry

Speaker - Changfeng Gui - University of Texas at San Antonio

Abstract: The classical Moser-Trudinger inequality is a borderline case of Soblolev inequalities and plays an important role in geometric analysis. Aubin in 1979 showed that the best constant in the Moser-Trudinger inequality can be improved by reducing to one half if the functions are restricted to the complement of a three dimensional subspace of the Sobolev space H1, while Onofri in 1982 discovered an elegant optimal form of Moser-Trudinger inequality on sphere. In this talk, I will present new sharp inequalities which are variants of Aubin and Onofri inequalities on the sphere with or without constraints.
The main ingredient leading to the above inequalities is a novel geometric inequality: Sphere Covering Inequality (SCI), discovered jointly with Amir Moradifam from UC Riverside. SCI states that the total area of two distinct surfaces with Gaussian curvature less than 1, which are also conformal to the Euclidean unit disk with the same conformal factor on the boundary, must be at least 4π. In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. SCI has also been successfully applied to solve other important questions in PDE.
Efforts have also been made to show similar inequalities in higher dimensions. Among the preliminary results, we have improved Beckner’s inequality for axially symmetric functions when the dimension n = 4, 6, 8. Many questions remain open.

Location: Virtual

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