Energy-driven pattern formation is ubiquitous in nature; the character and dynamics of such patterns is selected as local minimizers and gradient flows, respectively, of non-convex, and often, non-local energies with multiple spatio-temporal scales. Analysis of such patterns sheds valuable insight upon their origins, and from the viewpoint of applications, is necessary for their control.

In this talk, after introducing a selection of interdisciplinary problems from liquid crystals, micromagnetics, and phase transitions in heterogeneous media, we focus on a model problem that seeks to homogenize the Allen-Cahn equation in heterogeneous media. The combined effect of oscillation and concentration results in a formidable, and long-standing open question in analysis and PDEs. We will discuss our recent resolution of this problem using a confluence of tools from Nonlinear PDE, the Calculus of Variations, Geometric Measure Theory and Differential Geometry.

Collaborators on the various projects discussed in this talk are Rustum Choksi (McGill), Vivek Dabade (IISc), Irene Fonseca (CMU), Dmitry Golovaty (U. Akron),  Richard D. James (U. Minnesota), Jessica Lin (McGill), Michael R. Novack (U.T. Austin), Peter J. Sternberg (Indiana U.).

https://ncsu.zoom.us/j/92032810898?pwd=VllOS0ZtQXYrU1UvRDdPL3Q1d3BRZz09