Variational Models for Self-Assembly and Pattern Formation: Analysis and Computation

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Self-assembly, a process whereby a disordered system of components forms an organized, structured pattern solely as a consequence of component interactions, is both ubiquitous in nature and important for the synthesis of many designer materials. Many systems which exhibit self-assembly and pattern formation may be viewed as energy-driven; therefore, one can successfully capture and analyze the phenomena via minimization of some, often postulated, free energy defined over all possible configurations. The resulting variational problems involve highly non-trivial energy landscapes and are mathematically and computationally very challenging. In this proposal we will study three such variational models (and connections between them) from the point of view of mathematical analysis and computation. A common theme will be the focus on nonlocal problems and global techniques, i.e. those whose vision extends beyond the local study of critical points and the analysis of their stability.