Mathematical Sciences: Singularity and Regularity of Solutions of Scalar Equations and Systems in Connection with Differential Geometry

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The source of problems to be analyzed in this project derive from differential geometry and mathematical physics. These areas of science provide important prototypes of non-linear equations in which fundamental and difficult analysis must be done to understand the properties of solutions. Work will concentrate on the study of problems related to singularity and regularity of solutions of partial differential equations such as those encountered in the Yamabe equation in geometry and in the theory of liquid crystals. Other work will consider the boundary regularity of very degenerate non-linear elliptic systems which arise in efforts to decide when it is possible to find harmonic maps between two (non necessarily smooth) domains when the boundary values are prescribed. The work will focus on several specific goals. Among them will be the study of singular solutions of nonlinear elliptic equations in an effort to obtain estimates on the dimension of the singular set as well as to understand the location of such sets. In cases where the singular set is isolated, its location appears to be rigid. In studies of liquid crystal models, work will be done examining the transition between the nematic and smectic states of crystal configurations, while more geometric work will seek to provide information as to how one can determine whether or not a compact Riemannian manifold is conformally equivalent to a sphere.