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Rectifiability and parameterization of intrinsic regular surfaces in the Heisenberg group

Bernd KirchheimFrancesco Serra Cassano — 2004

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We construct an intrinsic regular surface in the first Heisenberg group 1 3 equipped wiht its Carnot-Carathéodory metric which has euclidean Hausdorff dimension  2 . 5 . Moreover we prove that each intrinsic regular surface in this setting is a 2 -dimensional topological manifold admitting a 1 2 -Hölder continuous parameterization.

Sufficient conditions for the validity of the Cauchy-Born rule close to SO ( n )

Sergio ContiGeorg DolzmannBernd KirchheimStefan Müller — 2006

Journal of the European Mathematical Society

The Cauchy–Born rule provides a crucial link between continuum theories of elasticity and the atomistic nature of matter. In its strongest form it says that application of affine displacement boundary conditions to a monatomic crystal will lead to an affine deformation of the whole crystal lattice. We give a general condition in arbitrary dimensions which ensures the validity of the Cauchy–Born rule for boundary deformations which are close to rigid motions. This generalizes results of Friesecke...

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