Fourth Order Equations Involoving Critical Sobolev Exponentsand Systems of PDE's From Geometry

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Abstract Gursky 9623048 The research to be supported by this grant is concerned with the study of fourth order differential equations and systems of equations from differential geometry. The equations under consideration are Euler equations for functionals that are connected to certain sharp Sobolev inequalities. The systems of equations arise in differential geometry when studying the problem of prescribing curvature and the existence of critical points of a functional introduced in relativity. Historically, the most fruitful applications of pure mathematics to very real world problems are often the result of the interaction of two different fields within the subject. The work to be supported here attempts to use methods of mathematical analysis to study certain problems in the field of differential geometry. At least one of these problems is relevant to physics, but it is often the case that the mathematics precedes the application by many years, so that it is difficult to assess the immediate use of the problems posed here: except to say that they are very basic questions that hopefully will increase the depth of our understanding of a subject that has seen many applications in the past.