Nonlinear Elliptic Equations and Systems

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The first part of the proposed work concerns Monge-Ampere equations, including extensions of classical results of Jorgens, Calabi and Pogorelov which state that entire solutions to such equations are quadratic polynomials. Dirichlet problem for Monge-Ampere equations in exterior domains of Euclidean space will also be investigated. Related problems concerning the affine Bernstein problem will be studied. The second part of the proposed work concerns best Sobolev inequality on Riemannian manifolds and the Yamabe problem on manifolds with boundary.This includes efforts in establishing a new form of best Sobolev inequalities on Riemannian manifolds as well as existence and compactness results concerning the Yamabe problem on manifolds with boundary. Sharp pointwise estimates to blow up solutions play important roles in such studies.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.