"Geometrically constrained walls"

Abstract:
We address the effect of extreme geometry on a non-convex variational
problem, focusing 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 the local minimizers
representing geometrically constrained walls, and (b) an asymptotic
characterization of the structure of such wall. Our analysis uses methods 
similar to $\Gamma$-convergence; in particular, the asymptotic wall 
structure 
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, i.e. on the ratio between length 
and width of the neck.