Geometrically constrained walls

RV Kohn, V Slastikov

Research output: Contribution to journalArticle (Academic Journal)peer-review


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.
Translated title of the contributionGeometrically constrained walls
Original languageEnglish
Pages (from-to)33 - 57
Number of pages25
JournalCalculus of Variable and Partial Differential Equations
Volume28 (1)
Publication statusPublished - Jan 2007

Bibliographical note

Publisher: Springer Berlin/Heidelberg


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