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Description

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.