Mathematical Sciences: On Nonlinear Elliptic Equations

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9401815 Li This award supports mathematical research on geometric problems in three related areas all based on solving various types of partial differential equations. The first concerns the problems of prescribing scalar curvature on the n-sphere with certain hypothesis on the critical points. The problem has been solved in dimensions two and three, this project will consider, with flatness assumptions on the critical points, the existence of a prescribed curvature(s) for a given infinitely differentiable function. The second area involves the Weyl problem with nonnegative Gauss curvature. Roughly this problem focuses on whether or not a sphere, with a given Riemannian metric an everywhere positive Gauss curvature can be isometrically embedded in three-dimensional Euclidean space with the standard flat metric. While this problem has been solved, only partial results exist for the case where the curvature is only nonnegative. To find necessary and sufficient conditions for smooth isometric embedding when the Gauss curvature is only nonnegative is the goal of this effort. Finally, work will be done in the study of the compressible fluid model of uniformly rotating white dwarf stars. Equilibrium solutions have been established for small angular velocity and the nonexistence of equilibrium solutions for large angular velocity has also been shown. What now must be done is to establish the existence of a global branch of equilibrium solutions - which amounts to solving a certain free boundary problem. Partial differential equations 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 provide 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. ***