Regularity Problems for Multiple Integrals and Interfacial Coarsening for Energy-Driven Models

Michigan State University, East Lansing MI

Investigators

Abstract

Proposal DMS-0401048 PI: Xiaodong Yan, Michigan State University Title: Regularity properties for multiple integrals and interfacial coarsening for energy-driven models The proposal addresses problems in three main areas. In part one, we are concerned with regularity property of minimizers of uniformly convex functionals. The objective is to generalize method used in previous work of Sverak and PI to construct counterexamples in lower target dimensions, as well as to understand regularity property of minimizers under further assumptions on the integrand. The second part deals with regularity property of minimizers for both compressible and incompressible models from nonlinear elasticity. The main feature here is the singular behavior of the integrand in the compressible case and nonlinear nonconvex constraint in the incompressible case. Currently there is no general method to deal with regularity problems for such integral. We hope that the simple model problems studied in this project can help to gain insight into more general cases. The third part addresses interfacial coarsening phenomenon observed in many energy-driven models from materials science. In particular, the aim is to provide rigorous analysis on the value of the coarsening exponent for different models. The analysis is closely related to deep and interesting issues in calculus of variations, e.g.gamma limit of singularly perturbed variational problems. The well developed tools from those areas provide a better vision on the last project. The questions to be studied are motivated both by classical theory of optimization and contemporary issues in nonlinear elasticity and materials science. An outstanding open question in continuum mechanics is to determine what singularities might exist in surfaces which optimize a collection of nonlinear constraints. Knowledge of this type can help in the design of exotic materials. The study of coarsening is important for both bulk and thin film materials. Such materials are of great practical interest. For example, superconducting thin film devices are presently being evaluated for possible use in high-performance microwave filters and other applications.