Nečas–Lions lemma revisited: An L p ‐version of the generalized Korn inequality for incompatible tensor fields

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.

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Estimates for the constant in two nonlinear Korn inequalities

Mathematics and Mechanics of Solids ◽

10.1177/1081286520957705 ◽

2020 ◽

pp. 108128652095770

Author(s):

Maria Malin ◽

Cristinel Mardare

Keyword(s):

Euclidean Space ◽

Open Subset ◽

Rigid Motion ◽

Tensor Fields ◽

Metric Tensor ◽

Square Roots ◽

Korn Inequality ◽

New Inequalities

A nonlinear Korn inequality estimates the distance between two immersions from an open subset of [Formula: see text] into the Euclidean space [Formula: see text], [Formula: see text], in terms of the distance between specific tensor fields that determine the two immersions up to a rigid motion in [Formula: see text]. We establish new inequalities of this type in two cases: when k = n, in which case the tensor fields are the square roots of the metric tensor fields induced by the two immersions, and when k = 3 and n = 2, in which case the tensor fields are defined in terms of the fundamental forms induced by the immersions. These inequalities have the property that their constants depend only on the open subset over which the immersions are defined and on three scalar parameters defining the regularity of the immersions, instead of constants depending on one of the immersions, considered as fixed, as up to now.

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On Horizontal and Complete Lifts of $(1,1)\;$ Tensor Fields F Satisfying ${f(2K+S,S)=0}$

International Journal of Mathematics and Soft Computing ◽

10.26708/ijmsc.2016.1.6.12 ◽

2016 ◽

Vol 6 (1) ◽

pp. 143

Author(s):

Abhishek Singh ◽

Ramesh Kumar Pandey ◽

Sachin Khare

Keyword(s):

Tensor Fields

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Handling ray transform of symmetric tensor fields and the Radon transform of differential forms on incomplete data

Daghestan Electronic Mathematical Reports ◽

10.31029/demr.1.2 ◽

2014 ◽

pp. 56-70

Author(s):

Ziyaudin Medzhidov ◽

Keyword(s):

Radon Transform ◽

Incomplete Data ◽

Differential Forms ◽

Symmetric Tensor ◽

Tensor Fields ◽

Ray Transform

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Spacetime symmetries

10.1093/oso/9780198802877.003.0006 ◽

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.

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

Symmetry ◽

10.3390/sym13050830 ◽

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.

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Gauge × gauge = gravity on homogeneous spaces using tensor convolutions

Journal of High Energy Physics ◽

10.1007/jhep06(2021)117 ◽

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.

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