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Geometrically constrained walls
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Kohn, Robert V. and Slastikov, Valeriy V. (2007) Geometrically constrained walls. Calculus of Variations and Partial Differential Equations, Volume 28 (Number 1). pp. 33-57. doi:10.1007/s00526-006-0030-4 ISSN 0944-2669.
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Official URL: http://dx.doi.org/10.1007/s00526-006-0030-4
Abstract
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 Gamma-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.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Calculus of Variations and Partial Differential Equations | ||||
Publisher: | Springer | ||||
ISSN: | 0944-2669 | ||||
Official Date: | January 2007 | ||||
Dates: |
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Volume: | Volume 28 | ||||
Number: | Number 1 | ||||
Number of Pages: | 25 | ||||
Page Range: | pp. 33-57 | ||||
DOI: | 10.1007/s00526-006-0030-4 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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