Investigation of the fine microscopic features appearing in multifunctional materials and their interactions on the overall macroscopic properties is of strategic importance in the design of materials for engineering applications. From the theoretical point of view, the relevance of these problems lies in their association with systems of singular Partial Differential Equations and with Boundary Value Problems characterized by highly oscillating solutions. Traditionally, mathematical analysis has been mainly focused on the description of the asymptotic properties of the equations and the corresponding solutions as well as their implication on the physical model. Taking advantage of the variational structure of the problem, a series of mathematical tools and techniques has been created over the years using the language of functional analysis, geometric measure theory and PDEs. In my research, energy-minimization approaches are utilized, such as relaxation and Gamma-convergence, to characterize the homogenized engineering properties of soft and elastic crystals and shape-memory metal alloys, with a special emphasis on pattern formation and interaction with topological defects. Lately, tools from probability theory have been employed to model criticality and self-similarity occurring during a dynamical process related to the solid-to-solid phase-transformation of elastic crystals.
|キーワード||PDEs, Calculus of Variations, Gamma-convergence, Materials Science|