In this project, the principal investigator pursues a research program in nonlinear partial differential equations, the calculus of variations and singular perturbation theory. The research topics come from three applied settings: Ginzburg-Landau type models for superconductivity, von Karman type models for thin film blistering, and a model in micromagnetics. In the area of superconductivity, the P.I. will focus on the response of samples to large magnetic fields, with particular attention paid to the bifurcation from the normal state to a superconducting state. In the area of thin film blisters, the P.I. will investigate the nature of instabilities of the blistered region through the analysis of various dynamical models for blister growth and thin film growth. Finally, in the area of micromagnetics, the P.I. will analytically explore a model thought to capture a new kind of magnetic wall structure associated with a geometric constriction within the sample.
This project concerns the behavior of various materials when subjected to outside fields or when forced to take on specific shapes. The energy of these systems is generally described through a function, often called an `order parameter,' whose values indicate what state is taken on by the material under a given set of circumstances (such as geometry, applied fields, etc.). Through this type of study, one hopes to gain an understanding of what shapes are optimal for a given sample in order to enhance or diminish various physical effects. For example, in the case of a superconductor, one hopes to learn which shapes are most conducive to producing a supercurrent that conducts without losses due to resistance. The relevant mathematical tools come from the calculus of variations and from the theory of nonlinear partial differential equations, as well as from methods of asymptotic analysis as applied to the previou