The central theme of this project is the application of modern Calculus of Variations to materials science and hydrodynamics. The investigator undertakes the following specific projects: -Heteroepitaxial thin films. The investigator studies from the variational point of view a mathematical model for the epitaxial deposition of a film onto a relatively thick substrate in the case of a mismatch between the lattice parameters of the two crystalline solids. -Strain gradient plasticity. The investigator derives, in a mathematically rigorous way, a strain gradient theory of plasticity starting from a variational formulation of the equilibrium problem for a finite number of dislocations in a plane domain. -Water waves. An alternative approach for the existence of regular water waves and of Stokes wave of greatest height is undertaken. Other projects include image segmentation problems and micromagnetics.
The central theme of this project is the application of modern techniques in mathematics to various problems in materials science, mechanics, and hydrodynamics. When a thin film of a crystalline solid is deposited on a substrate, a common industrial process, difficulties can arise if the crystal lattice structures of the film and the substrate do not match. The investigator studies this problem. A fundamental question in materials is the effect of dislocations in the material microstructure on larger-scale behavior. Classical plasticity theory does not address this issue well. The investigator develops a theory of strain gradient plasticity that incorporates dislocations and their interactions. He also considers problems in water waves, image processing, and micromagnetics. The results and methods developed in the project will be incorporated in new graduate courses, and some will be included in a series of books that the investigator is writing in collaboration with I. Fonseca,
