Tunable Materials: Geometry, Nonlocality, Chirality

This project aims at building a comprehensive mathematical theory for tunable materials, namely artificially designed composites (metamaterials) capable to adapt their mechanical, magnetic, or electrical responses according to the external environment, and considered the future for optical-data processing and quantum information. The proposal is located at the interface between Calculus of Variations, PDEs, and Applied Mathematics. It will leave a lasting footprint both on state-of-the-art research in mathematics and on the modeling of innovative materials, possibly having also an impact on frontier research in materials science, biology, and physics. First, we will open a completely new research line in the study of a special class of metamaterials, namely those exhibiting a strong difference (high-contrast) in the properties of their components, and we will investigate phase transitions in this setting. Applications for this analysis are in structural engineering, in the development of smart laminated structures with innovative insulation features capable to counteract climate-change consequences. Second, we will move radically beyond the current state-of-the art on evolutionary phase-transition problems to address two open questions in the modeling of nonlocal Cahn-Hilliard equations: (i) Do such equations provide a diffuse counterpart of sharp-interface models? (ii) How does nonlocality interact with microstructure formation? Question (i) is motivated by applications in the modeling of tumor-growth and polymers, whereas (ii) has connections with quantum-dots modeling. Third, we will push the mathematical understanding of magnetic skyrmions several steps further. These are spin textures emerging in magnetic systems lacking inversion symmetry. We will lay the basis for the analysis of two open questions related to the creation of innovative memory-storage devices, and to the development of modern sensors and actuators. In particular, we will develop a mathematical theory for high-contrast chiral multilayers and for skyrmions in magnetoelastic media. These topics are at the forefront of nowadays spintronics.