In this project the principal investigator will examine several problems involving nonlinear elliptic systems of partial differential equations in two and three space dimensions. The specific systems to be considered come from the theory of nonlinear elasticity, and the principal investigator will study the properties of solutions that are the energy minimizers of the associated variational problems. Her primary goal is to determine various regularity properties of solutions and to examine their qualitative behavior. The main tools to be used are previous results of the principal investigator concerning maximum principles and a'priori estimates. The solutions of many problems that describe physical reality are what mathematicians call "variational" solutions, that is, the correct physical solution is the mathematical solution of a variational problem. In ordinary language, a variational solution is one that maximizes or minimizes a fundamental quantity like the total energy of a system. For example, the bubbles in your glass of beer are spherical, because the spherical shape minimizes the surface energy. In this project the principal investigator will study various mathematical properties of solutions of variational problems that model two- and three-dimensional nonlinear elastic phenomena.
