The proposed project pertains to one of the most compelling questions in contemporary science and engineering: how do irregular geometry and material inhomogeneity influence the behavior of a physical system? The main objective of the present proposal is to obtain precise mathematical description of this influence for systems and processes governed by fundamental elliptic partial differential equations, with the direct application of results to several eminent problems in Materials Science and Engineering. Indeed, most classical mathematical models provide a nice, smooth, approximation of objects and processes. However, every real object inadvertently possesses irregularities (a sharp edge, a defect, an abrupt change of the material). They often have a decisive effect on the occurring phenomena, but cannot be captured in a classical, smooth and homogeneous, set-up (e.g., fractal structure of lungs, propagation of cracks in solids, electronic states in a disordered crystal). Mathematically, all these features, which translate into complex singularities in the geometry and the arising equations, are still very poorly understood.
The present project will be focused on the following two major goals. i. Establish the mathematical foundations of the localization of vibrations in irregular or disordered systems. This part is based on the recent breakthrough work of the PI and M. Filoche. It introduced a fundamentally novel mathematical approach to wave localization, governing both Anderson localization by disorder and weak localization induced by geometric or material irregularities in the same mathematical framework. ii. Describe exact relations between the geometry of the domains, the structure of the operators, and the properties of the solutions to the elliptic boundary problems. This part is aimed at sharp estimates translating microscopic features of the materials (seen via coefficients of the equation and the geometry) into the global properties of numerous phenomena in elasticity, electrostatics, thermodynamics modeled by elliptic PDEs. The project is a synergy of cutting-edge ideas and techniques of modern Harmonic Analysis and Geometric Measure Theory on one side, and state-of-the-art numerical experiments and engineering studies on the other. Its completion will have an impact in several areas of both pure and applied Materials Research.
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