Anisotropic mean curvature on facets and relations with capillarity

Stefano Amato; Giovanni Bellettini; Lucia Tealdi

Geometric Flows (2015)

  • Volume: 1, Issue: 1
  • ISSN: 2353-3382

Abstract

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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.

How to cite

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Stefano Amato, Giovanni Bellettini, and Lucia Tealdi. "Anisotropic mean curvature on facets and relations with capillarity." Geometric Flows 1.1 (2015): null. <http://eudml.org/doc/275884>.

@article{StefanoAmato2015,
abstract = {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.},
author = {Stefano Amato, Giovanni Bellettini, Lucia Tealdi},
journal = {Geometric Flows},
language = {eng},
number = {1},
pages = {null},
title = {Anisotropic mean curvature on facets and relations with capillarity},
url = {http://eudml.org/doc/275884},
volume = {1},
year = {2015},
}

TY - JOUR
AU - Stefano Amato
AU - Giovanni Bellettini
AU - Lucia Tealdi
TI - Anisotropic mean curvature on facets and relations with capillarity
JO - Geometric Flows
PY - 2015
VL - 1
IS - 1
SP - null
AB - 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.
LA - eng
UR - http://eudml.org/doc/275884
ER -

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