Second Order Nonlinear Elliptic Equations and Systems

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9706887 Li This research is on geometric problems in three related areas, all based on solving nonlinear second order partial differential equations of various types. The first concerns an analogue of the Yamabe problem on manifolds with boundaries: To study, for a given compact Riemannian manifold with boundary (M,g), the existence and compactness of metrics conformal to g that have constant scalar curvature and constant boundary mean curvature. This is equivalent to the study of the existence and compactness of positive solutions to a certain semilinear elliptic equation with critical exponent satisfying certain nonlinear Neuman boundary conditions. The second concerns the problem of prescribing scalar curvature on the n-sphere. The problem is better understood in dimension n=2,3,4. The goal here is to search for good existence criterion in dimension n = 5 that allow the prescribed curvature functions K to be Morse functions. The third concerns the study of degenerate Monge-Ampere equations, especially those arising from geometry. Partial differential equations arise naturally from physics, geometry, and many other fields and form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to create qualitative and quantitative information about the solutions. This may include answers to questions about existence, uniqueness, smoothness, and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.