Geometrically constrained walls in two dimension.

We address the effect of extreme geometry on a non-convex variational problem motivated by recent investigations of magnetic domain walls trapped by sharp thin necks. We prove the existence of local minimizers representing geometrically constrained walls under suitable symmetry assumptions on the domains and provide an asymptotic characterization of the wall profile. The asymptotic behavior, which depends critically on the scaling of length and width of the neck, turns out to be qualitatively different from the higher-dimensional case and a richer variety of regimes is shown to exist.