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Anisotropic geometric functionals and gradient flows

100%

Bellettini G., Mugnai L.

Banach Center Publications

2009

tom 86

nr 1

21-43

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.

Anisotropic mean curvature on facets and relations with capillarity

81%

Amato S., Bellettini G., Tealdi L.

Geometric Flows

2015

tom 1

nr 1

Given an anisotropy ɸ on R3, we discuss the relations between the ɸ-calibrability of a facet F ⊂ ∂E of a solid crystal E, and the capillary problem on a capillary tube with base F. When F is parallel to a facet ̃︀ BFɸ of the unit ball of ɸ, ɸ-calibrability is equivalent to show the existence of a ɸ-subunitary vector field in F, with suitable normal trace on @F, and with constant divergence equal to the ɸ-mean curvature of F. Assuming E convex at F, ̃︀ BFɸ a disk, and F (strictly) ɸ-calibrable, such a vector field is obtained by solving the capillary problem on F in absence of gravity and with zero contact angle. We show some examples of facets for which it is possible, even without the strict ɸ-calibrability assumption, to build one of these vector fields. The construction provides, at least for convex facets of class C1,1, the solution of the total variation flow starting at 1F.

Ograniczanie wyników

1 Banach Center Publications

1 Geometric Flows

2 Bellettini G.

1 Amato S.

1 Mugnai L.

1 Tealdi L.

1 2015

1 2009

Wyniki wyszukiwania

Anisotropic geometric functionals and gradient flows

Anisotropic mean curvature on facets and relations with capillarity