Striped patterns and the eikonal equation

In this talk I discuss work with Marco Veneroni on curvature-penalizing energies that arise in models of pattern formation. These nonlocal energies contain competing terms, penalizing both rapid variation and large-scale aggregation, and this competition results in structures with a preferred length scale.

I will concentrate on a model that produces striped patterns that may be curved and may show defects. One of the central challenges in the field of pattern formation is to rigorously connect the properties of the microscopic model on one hand with large-scale features of the striped pattern (such as curvature and defects) on the other. For the model at hand we have proved a result of this type, which connects the value of the microscopic energy to the presence or absense of certain large-scale features.

An interesting consequence of this result is the appearance of a new formulation of the eikonal equation, in terms of projection matrices. This formulation avoids unphysical singularities that arise in the usual vectorial formulation of the eikonal equation and raises many questions about the properties of this equation.