Some regularity results for minimal crystals

L. Ambrosio; M. Novaga; E. Paolini

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 8, page 69-103
  • ISSN: 1292-8119

Abstract

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We introduce an intrinsic notion of perimeter for subsets of a general Minkowski space (i.e. a finite dimensional Banach space in which the norm is not required to be even). We prove that this notion of perimeter is equivalent to the usual definition of surface energy for crystals and we study the regularity properties of the minimizers and the quasi-minimizers of perimeter. In the two-dimensional case we obtain optimal regularity results: apart from a singular set (which is -negligible and is empty when the unit ball is neither a triangle nor a quadrilateral), we find that quasi-minimizers can be locally parameterized by means of a bi-lipschitz curve, while sets with prescribed bounded curvature are, locally, lipschitz graphs.

How to cite

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Ambrosio, L., Novaga, M., and Paolini, E.. "Some regularity results for minimal crystals." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 69-103. <http://eudml.org/doc/90666>.

@article{Ambrosio2010,
abstract = { We introduce an intrinsic notion of perimeter for subsets of a general Minkowski space (i.e. a finite dimensional Banach space in which the norm is not required to be even). We prove that this notion of perimeter is equivalent to the usual definition of surface energy for crystals and we study the regularity properties of the minimizers and the quasi-minimizers of perimeter. In the two-dimensional case we obtain optimal regularity results: apart from a singular set (which is $\{\mathcal H\}^1$-negligible and is empty when the unit ball is neither a triangle nor a quadrilateral), we find that quasi-minimizers can be locally parameterized by means of a bi-lipschitz curve, while sets with prescribed bounded curvature are, locally, lipschitz graphs. },
author = {Ambrosio, L., Novaga, M., Paolini, E.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Quasi-minimal sets; Wulff shape; crystalline norm.; quasi-minimal sets; crystalline norm; surface energy},
language = {eng},
month = {3},
pages = {69-103},
publisher = {EDP Sciences},
title = {Some regularity results for minimal crystals},
url = {http://eudml.org/doc/90666},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Ambrosio, L.
AU - Novaga, M.
AU - Paolini, E.
TI - Some regularity results for minimal crystals
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 69
EP - 103
AB - We introduce an intrinsic notion of perimeter for subsets of a general Minkowski space (i.e. a finite dimensional Banach space in which the norm is not required to be even). We prove that this notion of perimeter is equivalent to the usual definition of surface energy for crystals and we study the regularity properties of the minimizers and the quasi-minimizers of perimeter. In the two-dimensional case we obtain optimal regularity results: apart from a singular set (which is ${\mathcal H}^1$-negligible and is empty when the unit ball is neither a triangle nor a quadrilateral), we find that quasi-minimizers can be locally parameterized by means of a bi-lipschitz curve, while sets with prescribed bounded curvature are, locally, lipschitz graphs.
LA - eng
KW - Quasi-minimal sets; Wulff shape; crystalline norm.; quasi-minimal sets; crystalline norm; surface energy
UR - http://eudml.org/doc/90666
ER -

References

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