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Curl bounds grad on SO(3)

Ingo Münch, Patrizio Neff (2008)

ESAIM: Control, Optimisation and Calculus of Variations

Let F p ∈ GL ( 3 ) be the plastic deformation from the multiplicative decomposition in elasto-plasticity. We show that the geometric dislocation density tensor of Gurtin in the form Curl [ F p ] · ( F p ) T applied to rotations controls the gradient in the sense that pointwise ∀ R ∈ C 1 ( ℝ 3 , SO ( 3 ) ) : ∥ Curl [ R ] · R T ∥ 𝕄 3 × 3 2 ≥ 1 2 ∥ D R ∥ ℝ 27 2 . This result complements rigidity results [Friesecke, James and Müller, Comme Pure Appl. Math. 55 (2002) 1461–1506; John, Comme Pure Appl. Math. 14 (1961) 391–413; Reshetnyak, Siberian Math. J. 8 (1967) 631–653)] as well as an associated linearized theorem...

Curl bounds Grad on SO(3)

Patrizio Neff, Ingo Münch (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Let F p ∈ GL ( 3 ) be the plastic deformation from the multiplicative decomposition in elasto-plasticity. We show that the geometric dislocation density tensor of Gurtin in the form Curl [ F p ] · ( F p ) T applied to rotations controls the gradient in the sense that pointwise ∀ R ∈ C 1 ( ℝ 3 , SO ( 3 ) ) : ∥ Curl [ R ] · R T ∥ 𝕄 3 × 3 2 ≥ 1 2 ∥ D R ∥ ℝ 27 2 . This result complements rigidity results [Friesecke, James and Müller, Comme Pure Appl. Math.55 (2002) 1461–1506; John, Comme Pure Appl. Math.14 (1961) 391–413; Reshetnyak, Siberian Math. J.8 (1967) 631–653)] as well as an associated linearized theorem saying...

Fastest curves and toroidal black holes

E. Woolgar (1997)

Banach Center Publications

We discuss an apparent paradox (and conjectured resolution) of Jacobson and Venkataramani concerning 'temporarily toroidal' black hole horizons, in light of a recent connectivity theorem for spaces of complete causal curves. We do this in a self-contained manner by first reviewing the 'fastest curve argument' which proves this connectivity theorem, and we note that active topological censorship can be derived as a corollary of this argument. We argue that the apparent paradox arises only when one...

Generalized PN manifolds and separation of variables

Fernand Pelletier, Patrick Cabau (2008)

Banach Center Publications

The notion of generalized PN manifold is a framework which allows one to get properties of first integrals of the associated bihamiltonian system: conditions of existence of a bi-abelian subalgebra obtained from the momentum map and characterization of such an algebra linked with the problem of separation of variables.

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