# 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

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topAmbrosio, 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 -

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