Geometrically Constrained Walls

We address the effect of extreme geometry on a non-convex variational problem. The analysis is motivated by recent investigations of magnetic domain walls trapped by sharp thin necks. To capture the essential issues in the simplest possible setting, we focus on a scalar variational problem with a symmetric double well potential, whose spatial domain is a dumbell with a sharp neck. Our main results are (a) the existence of local minimizers representing geometrically constrained walls, and (b) an asymptotic characterization of the wall profile. Our analysis uses methods similar to Γ-convergence; in particular, the wall profile minimizes a certain “reduced problem” – the limit of the original problem, suitably rescaled near the neck. The structure of the wall depends critically on the choice of scaling, specifically the ratio between length and width of the neck. ∗This research was supported in part by NSF grants DMS 0101439 and 0313744. RVK also acknowledges helpful discussions with Francois Murat, facilitated by funding from Laboratoire Jacques-Louis Lions at Université de Paris VI in Fall 2001. †This research was supported in part by NSF grant DMS-0405343. Address from September 2005: Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK. An early version of this work was part of VVS’s PhD thesis at NYU.