scholarly journals L p -versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with p-integrable exterior derivative

2021 ◽  
Vol 359 (6) ◽  
pp. 749-755
Author(s):  
Peter Lewintan ◽  
Patrizio Neff
Keyword(s):  
Exterior Derivative ◽  
Tensor Fields ◽  
Arbitrary Dimensions

scholarly journals $$L^p$$-trace-free version of the generalized Korn inequality for incompatible tensor fields in arbitrary dimensions

2021 ◽  
Vol 72 (3) ◽  
Author(s):  
Peter Lewintan ◽  
Patrizio Neff
Keyword(s):  
Matrix Representation ◽  
Three Dimensional ◽  
Type Inequality ◽  
Cross Product ◽  
Dimensional Case ◽  
Tensor Fields ◽  
Korn Inequality ◽  
Free Part ◽  
Arbitrary Dimensions ◽  
Square Matrix

AbstractFor $$n\ge 3$$ n ≥ 3 and $$1<p<\infty $$ 1 < p < ∞ , we prove an $$L^p$$ L p -version of the generalized trace-free Korn-type inequality for incompatible, p-integrable tensor fields $$P:\Omega \rightarrow \mathbb {R}^{n\times n}$$ P : Ω → R n × n having p-integrable generalized $${\text {Curl}}_{n}$$ Curl n and generalized vanishing tangential trace $$P\,\tau _l=0$$ P τ l = 0 on $$\partial \Omega $$ ∂ Ω , denoting by $$\{\tau _l\}_{l=1,\ldots , n-1}$$ { τ l } l = 1 , … , n - 1 a moving tangent frame on $$\partial \Omega $$ ∂ Ω . More precisely, there exists a constant $$c=c(n,p,\Omega )$$ c = c ( n , p , Ω ) such that $$\begin{aligned} \Vert P \Vert _{L^p(\Omega ,\mathbb {R}^{n\times n})}\le c\,\left( \Vert {\text {dev}}_n {\text {sym}}P \Vert _{L^p(\Omega ,\mathbb {R}^{n \times n})}+ \Vert {\text {Curl}}_{n} P \Vert _{L^p\left( \Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}}\right) }\right) , \end{aligned}$$ ‖ P ‖ L p ( Ω , R n × n ) ≤ c ‖ dev n sym P ‖ L p ( Ω , R n × n ) + ‖ Curl n P ‖ L p Ω , R n × n ( n - 1 ) 2 , where the generalized $${\text {Curl}}_{n}$$ Curl n is given by $$({\text {Curl}}_{n} P)_{ijk} :=\partial _i P_{kj}-\partial _j P_{ki}$$ ( Curl n P ) ijk : = ∂ i P kj - ∂ j P ki and "Equation missing" denotes the deviatoric (trace-free) part of the square matrix X. The improvement towards the three-dimensional case comes from a novel matrix representation of the generalized cross product.

Z-Graded Extensions of Poisson Brackets

1997 ◽  
Vol 09 (01) ◽  
pp. 1-27 ◽  
Author(s):  
Janusz Grabowski
Keyword(s):  
Poisson Bracket ◽  
Jacobi Identity ◽  
Differential Forms ◽  
Poisson Manifold ◽  
Exterior Derivative ◽  
Tensor Fields ◽  
Lie Bracket ◽  
Nijenhuis Bracket ◽  
Hamiltonian Map ◽  
Vector Valued

A Z-graded Lie bracket { , }P on the exterior algebra Ω(M) of differential forms, which is an extension of the Poisson bracket of functions on a Poisson manifold (M,P), is found. This bracket is simultaneously graded skew-symmetric and satisfies the graded Jacobi identity. It is a kind of an 'integral' of the Koszul–Schouten bracket [ , ]P of differential forms in the sense that the exterior derivative is a bracket homomorphism: [dμ, dν]P=d{μ, ν}P. A naturally defined generalized Hamiltonian map is proved to be a homomorphism between { , }P and the Frölicher–Nijenhuis bracket of vector valued forms. Also relations of this graded Poisson bracket to the Schouten–Nijenhuis bracket and an extension of { , }P to a graded bracket on certain multivector fields, being an 'integral' of the Schouten–Nijenhuis bracket, are studied. All these constructions are generalized to tensor fields associated with an arbitrary Lie algebroid.

Spacetime symmetries

2018 ◽  
Author(s):  
Michael Kachelriess
Keyword(s):  
Riemannian Manifold ◽  
Conservation Laws ◽  
Vector Fields ◽  
Killing Vector ◽  
Geodesic Equation ◽  
Spacetime Symmetries ◽  
Tensor Fields ◽  
Killing Vector Fields ◽  
Covariant Derivatives ◽  
Killing Equation

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the Killing equation, and how the existence of Killing vector fields is connected to global conservation laws.

scholarly journals Spherically Symmetric Tensor Fields and Their Application in Nonlinear Theory of Dislocations

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 830
Author(s):  
Evgeniya V. Goloveshkina ◽  
Leonid M. Zubov
Keyword(s):  
Nonlinear Theory ◽  
Hollow Sphere ◽  
Tensor Field ◽  
Exact Analytical Solution ◽  
Symmetric Tensor ◽  
Symmetric Distribution ◽  
Spherically Symmetric ◽  
Tensor Fields ◽  
Spherically Symmetric Distribution ◽  
Nonlinear Elastic

The concept of a spherically symmetric second-rank tensor field is formulated. A general representation of such a tensor field is derived. Results related to tensor analysis of spherically symmetric fields and their geometric properties are presented. Using these results, a formulation of the spherically symmetric problem of the nonlinear theory of dislocations is given. For an isotropic nonlinear elastic material with an arbitrary spherically symmetric distribution of dislocations, this problem is reduced to a nonlinear boundary value problem for a system of ordinary differential equations. In the case of an incompressible isotropic material and a spherically symmetric distribution of screw dislocations in the radial direction, an exact analytical solution is found for the equilibrium of a hollow sphere loaded from the outside and from the inside by hydrostatic pressures. This solution is suitable for any models of an isotropic incompressible body, i. e., universal in the specified class of materials. Based on the obtained solution, numerical calculations on the effect of dislocations on the stress state of an elastic hollow sphere at large deformations are carried out.

scholarly journals Gauge × gauge = gravity on homogeneous spaces using tensor convolutions

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
L. Borsten ◽  
I. Jubb ◽  
V. Makwana ◽  
S. Nagy
Keyword(s):  
Homogeneous Spaces ◽  
Gauge Fields ◽  
Convolution Product ◽  
Tensor Fields ◽  
Einstein Universe ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Static Einstein Universe ◽  
Definition Of ◽  
Gauge Gravity

Abstract A definition of a convolution of tensor fields on group manifolds is given, which is then generalised to generic homogeneous spaces. This is applied to the product of gauge fields in the context of ‘gravity = gauge × gauge’. In particular, it is shown that the linear Becchi-Rouet-Stora-Tyutin (BRST) gauge transformations of two Yang-Mills gauge fields generate the linear BRST diffeomorphism transformations of the graviton. This facilitates the definition of the ‘gauge × gauge’ convolution product on, for example, the static Einstein universe, and more generally for ultrastatic spacetimes with compact spatial slices.

Sign in / Sign up

Export Citation Format

Share Document