Mathematical Sciences: Calculus of Variations, Material Microstructure, and Stochastic Evolution Problems

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Three distinct but loosely related research projects are planned: (a) A study of the Lavrentiev phenomenon for isoperimetrically constrained problems and its relationship with cavitation effects in nonlinear elasticity, (b) A study of materials with layered microstructure and materials with negative capillarity, and (c) Stochastic hyperbolic equations and stochastic control in nematic liquid crystals. Each part represents a further development of the mathematical tools developed in the investigator's previous work, as well as the introduction of new research topics. The three projects are motivated by problems from continuum mechanics. The objective is to understand the physics of an assortment of novel materials and to develop mathematics which will help to explain, predict, and control their mechanical behavior.