Proceedings of the International Congress of Mathematicians
Madrid, August 22–30, 2006
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pp: 359–383
DOI: 10.4171/022-1/15
Many physical systems can be modelled by nonconvex variational problems regularized by higher-order terms. Examples include martensitic phase transformation, micromagnetics, and the Ginzburg–Landau model of nucleation. We are interested in the singular limit, when the coefficient of the higher-order term tends to zero. Our attention is on the internal structure of walls, and the character of microstructure when it forms. We also study the pathways of thermally-activated transitions, modeled via the minimization of action rather than energy. Our viewpoint is variational, focusing on matching upper and lower bounds.
Keywords: Action minimization, Aviles–Giga problem, calculus of variations, cross-tie wall, martensitic transformation, micromagnetics, microstructure