TY - JOUR

T1 - Towards a variational theory of phase transitions involving curvature

AU - Moser, Roger

PY - 2012/8

Y1 - 2012/8

N2 - An anisotropic area functional is often used as a model for the free
energy of a crystal surface. For models of faceting, the anisotropy is
typically such that the functional becomes nonconvex, and then it may be
appropriate to regularize it with an additional term involving curvature.
When the weight of the curvature term tends to 0, this gives rise to a
singular perturbation problem.
The structure of this problem is comparable to the theory of phase
transitions studied first by Modica and Mortola. Their ideas are also use-
ful in this context, but they have to be combined with adequate geometric
tools. In particular, a variant of the theory of curvature varifolds, intro-
duced by Hutchinson, is used in this paper. This allows an analysis of the
asymptotic behaviour of the energy functionals.

AB - An anisotropic area functional is often used as a model for the free
energy of a crystal surface. For models of faceting, the anisotropy is
typically such that the functional becomes nonconvex, and then it may be
appropriate to regularize it with an additional term involving curvature.
When the weight of the curvature term tends to 0, this gives rise to a
singular perturbation problem.
The structure of this problem is comparable to the theory of phase
transitions studied first by Modica and Mortola. Their ideas are also use-
ful in this context, but they have to be combined with adequate geometric
tools. In particular, a variant of the theory of curvature varifolds, intro-
duced by Hutchinson, is used in this paper. This allows an analysis of the
asymptotic behaviour of the energy functionals.

UR - http://www.scopus.com/inward/record.url?scp=84864951990&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1017/S0308210510000995

U2 - 10.1017/S0308210510000995

DO - 10.1017/S0308210510000995

M3 - Article

SN - 0308-2105

VL - 142

SP - 839

EP - 865

JO - Proceedings of the Royal Society of Edinburgh Section A - Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A - Mathematics

IS - 4

ER -