Singular Structures Arising from Variational Problems in Materials Science

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Proposal DMS-0401328PI: Peter Sternberg (Indiana University)Title: Singular structures arising from variational problems in materials scienceABSTRACT This proposal focuses on a study of singular structures that arise inthe solution to various variational problems modeling materials. Thesestructures include Ginzburg-Landau vortices in models for superconductors,blister patterns in models for thin film delamination and segregationpatterns in models for diblock copolymers. Mathematically, all of thesemodels involve singular perturbations of nonconvex variational problems. Thetools that will be brought to bear include ideas from Gamma convergence anda careful examination of the second variation of the energy. This proposal involves an investigation of material behavior based on amathematical study of the energyassociated with various physical configurations. The materials to be studiedinclude superconductors and related devices, thin films which have been laidon top of a substrate for manufacturing purposes, and diblockcopolymers--soft "designer" materials formed by two self-segregating speciesof molecules. Based on the principle that materials naturally configurethemselves so as to expend the least amount of energy possible, one bringsto bear mathematical tools from the calculus of variations so as to betterunderstand the behavior of these materials.