Geodesic
Some regularity results for minimal crystals
ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 69-103

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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 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.

DOI : 10.1051/cocv:2002018
Classification : 49J45, 49Q20
Keywords: quasi-minimal sets, Wulff shape, crystalline norm 
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     author = {Ambrosio, L. and Novaga, M. and Paolini, E.},
     title = {Some regularity results for minimal crystals},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {69--103},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     doi = {10.1051/cocv:2002018},
     mrnumber = {1932945},
     zbl = {1066.49021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2002018/}
}
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Ambrosio, L.; Novaga, M.; Paolini, E. Some regularity results for minimal crystals. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 69-103. doi: 10.1051/cocv:2002018

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