On regularity and singularity of solutions of some nonlinear elliptic equations

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This project concerns the analysis and applications of nonlinear partial differential equations, with emphasis on integro-differential equations. Integro-differential equations are equations which involve both integrals and derivatives. They arise from models such as in diffusion with long range interactions in physics, future options in mathematical finance, and population dynamics in social science. They also appear intrinsically in geometry. One main goal of the proposal is to study fine analysis of a particular family of integro-differential equations, which will be applied to understand a number of scientific phenomena in geometry and physics. Another main goal is to develop general mathematical theories on integro-differential equations for their wide usage in future. Other than its applications, the analysis itself for those equations is of independent interest. It not only extends the current theories of partial differential equations, but also gives new insights and creates new methods of establishing them. This proposal focuses on regularity and singularity of solutions of nonlinear elliptic partial differential equations. The PI proposes to develop a unified approach to study existence and compactness of solutions to a family of prescribed fractional order curvature problems in conformal geometry. The investigation of their singular solutions will not only develop further the fractional singular Yamabe problem, but also lead naturally to boundary reaction-diffusion equations which appear as models of dislocations in crystals and soft thin films in micromagnetism. The study on regularity of fully nonlinear integro-differential equations, which usually arise from stochastic control problems with purely jump Levy process, will enrich the existing general regularity theory. The proposed research on Monge-Ampere equations is motivated by Monge-Ampere metrics on affine manifolds with singularities. Its goal is to address the regularity and analyze the behavior of solutions of such equations with singularities, and to understand their connections to differential geometry.