Mathematical Sciences: Methods for Solutions of Some Nonlinear Eliptic Equations

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Professor Chen will work on the problem of prescribing curvature on manifolds which have the topological type of a sphere. In addition, he will attempt to apply variational methods to reprove the classical uniformization theorem and prove an extension of the theorem to surfaces with conical singularities. Variational methods will also be used to classify solutions of some nonlinear partial differential equations of differential geometry and to study solutions of nonlinear elliptic boundary value problems for domains in Euclidean space. The research supported by this award is frequently referred to as global analysis. It attempts to find and describe global solutions to nonlinear partial differential equations on a surface or manifold. These equations describe local properties of the surface such as curvature. The research, therefore, seeks to relate local information to global information.