| A fundamental question in the understanding of pattern formation is to describe and predict spatial variation of patterns. Natural patterns often show near-periodicity at small scales, combined with large and sometimes abrupt variations at larger scales. The challenge is to develop a theory of such metapatterns - patterns of patterns - that allows to construct, compare, and classify these structures, and relate their large-scale properties to those of the microstructure. We propose to characterize metapatterns in variational systems in the limit of complete scale separation, by using a combination of singular limit analysis, variational methods, and geometric measure theory. The method will be generic and in principle applicable to a wide scale of variational pattern-forming systems. The limit of complete scale separation is characterized by the shrinking of the patterns at the smallest scale, while keeping the larger scale constant. Thus at every point in space a full local pattern - called a texture - shrinks to infinitesimal size, and the limiting functional is defined on such texture-valued maps. The large-scale structures observed in metapatterns can then be studied via the properties of the limit functional. For the specific class of variational pattern-forming systems this project will borrow from, generalize, and extend a number of existing approaches. The resulting theory will be rigorous, impose no restriction on smoothness or geometry, and provide insight in the limiting behaviour via the limiting functional. We achieve this by exploiting the restriction to variational systems, which provides powerful tools such as Gamma-convergence, pattern-valued Young measures, and metric-space-valued functions of bounded variation. |
