Variational methods for materials science, mechanics, and imaging

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

Leoni DMS-1007989 The central theme of this project is the application of modern calculus of variations to materials science, mechanics, and imaging. The principal investigator emphasizes the following specific topics: - Material Voids in Elastic Solids with Anisotropic Surface Energies. Here he studies how the qualitative features of the equilibrium shape of the void depend on the smoothness and convexity (or lack thereof) of the interfacial energy. - Evolution by Surface Diffusion. The principal investigator studies the short-time existence and regularity of solutions of a model of second-phase particles and material voids in a linearly elastic solid. - Non-Local Operators for Image Recovery. Here he studies regularity and qualitative features of solutions of a non-local model for image and signal processing proposed by Gilboa and Osher. He also takes up problems in water waves and micromagnetics. The central theme of this project is the application of modern calculus of variations to materials science, mechanics, and imaging. When an elastic material is subject to stresses, it can stretch and develop voids within the material; the shape of the voids affects other properties of the material. The principal investigator studies how the shape depends on the energy at the interface between the material and the void. The movement of interfaces by means of surface diffusion is not well understood when the interfacial energy is highly anisotropic. The investigator studies this problem, focusing on the short-time behavior; rigorous results here improve our understanding of various behaviors of materials. He also considers problems in image processing, water waves, and micromagnetics. The results and methods developed in the project are incorporated in new graduate courses, and some are included in a series of books that the investigator is writing in collaboration with I. Fonseca. Graduate students are involved in the project.