Work sponsored by this Presidential Young Investigator Award will focus primarily on questions arising in the applications of mathematics to problems of elsticity and liquid crystals, to geometric problems related to the study of curvature on manifolds with boundary. The problems to be addressed are presented in terms of various partial differential equations arising either from physical considerations - minimization of functionals or variational inequalities, or are motivated by geometrical concepts. The latter include the problem of finding a metric on a manifold with prescribed Ricci tensor. This problem is studied locally or globally through the analysis of systems of elliptic equations. In the Hodge theory, the problem reduces to that of solving the boundary regularity of solutions of degenerate elliptic systems. For a single equation, the problem has recently been settled. Focus of the elsticity (plasticity) studies will be on the Singonolli problem, seeking the best possible regularity of vector valued functions which solve obstacle problems. In liquid crystals many physical properties such as line singularities, phase transition, and moving defects, may be explained within a comprehensible theory. Related work will be done on studies of the corresponding dynamical equations. In an isotropic case, these problems reduce to the analysis of heat flow of harmonic maps and to harmonic maps from Minkowski space to singular target manifolds.
