# Anisotropic geometric functionals and gradient flows

Giovanni Bellettini; Luca Mugnai

Banach Center Publications (2009)

- Volume: 86, Issue: 1, page 21-43
- ISSN: 0137-6934

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topGiovanni Bellettini, and Luca Mugnai. "Anisotropic geometric functionals and gradient flows." Banach Center Publications 86.1 (2009): 21-43. <http://eudml.org/doc/282060>.

@article{GiovanniBellettini2009,

abstract = {We survey some recent results on the gradient flow of an anisotropic surface energy, the integrand of which is one-homogeneous in the normal vector. We discuss the reasons for assuming convexity of the anisotropy, and we review some known results in the smooth, mixed and crystalline case. In particular, we recall the notion of calibrability and the related facet-breaking phenomenon. Minimal barriers as weak solutions to the gradient flow in case of nonsmooth anisotropies are proposed. Furthermore, we discuss some relations between cylindrical anisotropies, the prescribed curvature problem and the capillarity problem. We conclude the paper by examining some higher order geometric functionals. In particular we discuss the anisotropic Willmore functional and compute its first variation in the smooth case.},

author = {Giovanni Bellettini, Luca Mugnai},

journal = {Banach Center Publications},

keywords = {gradient flow; anisotropy; minimal barrier; calibrability; cylindrical anisotropies; capillarity problem; Willmore functional},

language = {eng},

number = {1},

pages = {21-43},

title = {Anisotropic geometric functionals and gradient flows},

url = {http://eudml.org/doc/282060},

volume = {86},

year = {2009},

}

TY - JOUR

AU - Giovanni Bellettini

AU - Luca Mugnai

TI - Anisotropic geometric functionals and gradient flows

JO - Banach Center Publications

PY - 2009

VL - 86

IS - 1

SP - 21

EP - 43

AB - We survey some recent results on the gradient flow of an anisotropic surface energy, the integrand of which is one-homogeneous in the normal vector. We discuss the reasons for assuming convexity of the anisotropy, and we review some known results in the smooth, mixed and crystalline case. In particular, we recall the notion of calibrability and the related facet-breaking phenomenon. Minimal barriers as weak solutions to the gradient flow in case of nonsmooth anisotropies are proposed. Furthermore, we discuss some relations between cylindrical anisotropies, the prescribed curvature problem and the capillarity problem. We conclude the paper by examining some higher order geometric functionals. In particular we discuss the anisotropic Willmore functional and compute its first variation in the smooth case.

LA - eng

KW - gradient flow; anisotropy; minimal barrier; calibrability; cylindrical anisotropies; capillarity problem; Willmore functional

UR - http://eudml.org/doc/282060

ER -

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