Quantitative Analysis on High-Contrast Composite Materials

Composite materials, created by closely bonding two or more distinct substances, are common in both natural and engineered systems. Many of these materials display significant spatial variation in properties such as electrical conductivity and dielectric permeability, and are known as high-contrast composites. Examples include filled polymers, porous media, and biological tissues, with applications in bioengineering, medical imaging, and electronics. The Principal Investigator (PI) focuses on quantitatively analyzing field concentration -- a ubiquitous phenomenon in material science that arises when material properties change drastically over a very small length scale. Gaining a deeper understanding of this effect is crucial for accurately determining the effective properties of such materials and for designing them more efficiently to enhance performance. The PI will also perform undergraduate and graduate mentorship. Mathematically, high-contrast composites are modeled by partial differential equations (PDEs) with highly oscillatory coefficients. These equations are challenging to analyze, as classical analytical techniques often fail to apply. This project aims to develop new mathematical methods to address several open problems in this area. The first focus is on the buildup of the electric field between insulators, with particular attention to the asymptotic behavior of solutions and equations involving the p-Laplacian and the Lamé system. The second objective is to study composites composed of perfect or mixed conductors, emphasizing nonlinear governing equations and systems. Finally, the project will investigate models in which the conductors are imperfectly bonded. These involve more complicated transmission conditions and boundary conditions, including Robin-type boundary conditions. Such models are also of practical importance in biology, as they serve as approximations of membrane structures in biological systems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.