Mathematical Sciences: Existence, Singularity & Regularity of Solutions of Scalar Equations & Systems in Connection with Differential Geometry & Mathematical Physic

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The focus of this project is the analysis of solution of partial differential equations, systems as well as scalar equations. The equations of interest arise in mathematical physics and differential geometry. Emphasis will be placed on the study of the Yamabe problem in a complete Riemannian manifold, a problem of considerable interest which can be formulated in terms of semi-linear elliptic equations. The problem is then one of finding positive solutions to the equations and analyzing their asymptotic behavior. This work motivates in a natural way efforts to understand a larger class of solutions by means of critical points of the Sobolev quotient. The solutions are believed to be infinitely differentiable in the complement of sets whose Hausdorff dimension is precisely estimated by means of the nonlinear term in the differential equation. Work will also be done on harmonic maps, particularly in studying the regularity of minimizing harmonic maps into singular objects like varifolds. A recent new proof of the regularity of minimizing harmonic maps between two compact Riemannian manifolds suggests that an expansion is possible to target domains which are Lipschitz graphs. Most of the attention is paid to the sets of regularity for harmonic maps. But there are now some general results concerning the singular sets as well. They may have reasonable structures, such as curves or unions of smooth curves. Further work will be done to determine what classifications of singular sets are possible. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them and validate methods for expressing solutions.