Interplay Between Interfacial Patterns, Singular Perturbations and Heterogeneous Medium

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YipDMS-1009102 The investigator and his colleagues study the interactionbetween the defect structures in materials (such as phase andgrain boundaries) and the background heterogeneous medium. Theheterogeneity can come from intrinsic atomic structures,externally imposed forcing, or even numerical approximations. The emphasis is on understanding the structure and propagation ofinterfaces. Due to the highly nonlinear interaction between thematerials interface and background inhomogeneity, many newfeatures arise, most notably boundaries between patterns insteadof pure phases, and pinning and de-pinning transitions. In termsof mathematical disciplines, the project is related to theconcept of homogenization of singularly perturbed problems inwhich multiple scales coexist. To tackle nonlinear problems asin the current setting, new techniques and interpretations beyondtraditional approaches are required. The project investigatesthe structure and propagation of boundaries between patterns, andthe connection between pulsating waves and singularperturbations. Both the discrete and continuum regimes areanalyzed. The project is highly interdisciplinary and iswell-motivated by problems coming from applied science, inparticular materials science. It brings together rigorousapplied analysis, numerical simulations and modeling in a varietyof physical phenomena. It emphasizes the connections betweenmultiple scales, as practical materials often are composed ofstructures that have greatly different length scales. Resultscan lead to better understanding of how applied forcing andchanges of external environments can affect the behavior ofmaterials responses. This in turn can lead to the manufacturingof stronger and better materials.