Singular perturbation problems involving curvature

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

Consider an anisotropic area functional, giving rise to a variational principle for the shape of crystal surfaces. Sometimes such a functional is regularised with an additional curvature term to avoid difficulties coming from a lack of convexity. We study the asymptotic behaviour of the resulting functional as the strength of the regularisation tends to 0. We consider two cases. The first corresponds to a cubic crystal structure. The expected shapes of the crystal surfaces are polyhedra with faces parallel to the coordinate planes, and for the regularised functionals, we discover a limiting energy depending on the lengths of the edges. In the second case, we have a uniaxial anisotropy. We calculate the limiting energy for surfaces of revolution and give a lower bound for topological spheres.
LanguageEnglish
Title of host publicationDifferential Geometry and Continuum Mechanics
EditorsGui-Qiang Chen, Michael Grinfeld, R. J. Knops
PublisherSpringer
Pages49-75
ISBN (Print)978-3-319-18572-9
StatusPublished - 2015

Publication series

NameSpringer Proceedings in Mathematics & Statistics
PublisherSpringer
Volume137

Fingerprint

Singular Perturbation Problems
Crystal
Limiting
Curvature
Cartesian plane
Surface of revolution
Crystal Structure
Energy
Variational Principle
Polyhedron
Convexity
Anisotropy
Regularization
Asymptotic Behavior
Face
Tend
Lower bound
Calculate
Term

Cite this

Moser, R. (2015). Singular perturbation problems involving curvature. In G-Q. Chen, M. Grinfeld, & R. J. Knops (Eds.), Differential Geometry and Continuum Mechanics (pp. 49-75). (Springer Proceedings in Mathematics & Statistics; Vol. 137). Springer.

Singular perturbation problems involving curvature. / Moser, Roger.

Differential Geometry and Continuum Mechanics. ed. / Gui-Qiang Chen; Michael Grinfeld; R. J. Knops. Springer, 2015. p. 49-75 (Springer Proceedings in Mathematics & Statistics; Vol. 137).

Research output: Chapter in Book/Report/Conference proceedingChapter

Moser, R 2015, Singular perturbation problems involving curvature. in G-Q Chen, M Grinfeld & RJ Knops (eds), Differential Geometry and Continuum Mechanics. Springer Proceedings in Mathematics & Statistics, vol. 137, Springer, pp. 49-75.
Moser R. Singular perturbation problems involving curvature. In Chen G-Q, Grinfeld M, Knops RJ, editors, Differential Geometry and Continuum Mechanics. Springer. 2015. p. 49-75. (Springer Proceedings in Mathematics & Statistics).
Moser, Roger. / Singular perturbation problems involving curvature. Differential Geometry and Continuum Mechanics. editor / Gui-Qiang Chen ; Michael Grinfeld ; R. J. Knops. Springer, 2015. pp. 49-75 (Springer Proceedings in Mathematics & Statistics).
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