Mathematical Sciences: Moser-Trudinger Inequality and Applications

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Chang 9401465 This award continues support for mathematical research on problems involving sharp forms of the classical Sobolev embedding theorem. The result is stated in terms of fundamental comparisons called inequalities of Moser-Trudinger type. These inequalities have been shown to be valuable in a number of problems involving nonlinear partial differential equations, spectral analysis and conformal geometry. Work will be done on the ratio of determinant of the Laplacian operator of two conformally related metrics, on the extremal metric of zeta functional determinants on four-manifolds and on the geometric meaning of the constants in the sharp form of the Moser-Trudinger inequalities. The study of Sobolev embedding is part of a broad program in geometric analysis aimed at understanding various curvature metrics which can exist on surfaces (manifolds) and how the surfaces can be constructed from a given description of their curvature. The inequalities arise in the study of the Laplace operator on the surfaces and the resulting constants which occur in the relationship between integrals of solutions and integrals of their gradients. ***