Abstract—
Tensor fields of arbitrary rank in multidimensional space are naturally extended to the concepts of divergence, gradient, curl, deformer operators, as well as their second-order superpositions. Two options for generalizing the rotor as an external product are presented. Differential operators of the second order that do not change the rank of the tensor to which they are applied are considered in detail. Square matrices are introduced, consisting of differential operators \({\text{Di}}{{{\text{v}}}_{{(l)}}}{\text{Gra}}{{{\text{d}}}_{{(k)}}}\), \({\text{Gra}}{{{\text{d}}}_{{(k)}}}{\text{Di}}{{{\text{v}}}_{{(l)}}}\), and their relationship is established. An explicit expression is written for the repeated operator rotor. All introduced generalized operators in particular cases agree in their properties with the corresponding classical operators in vector and tensor analysis.
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REFERENCES
L. P. Lebedev and I. I. Vorovich, Functional Analysis in Mechanics (Springer, New York, 2003).
V. V. Vasiliev, “Singular solutions in the problems of mechanics and mathematical physics,” Mech. Solids 53 (4), 397–410 (2018).
P. K. Rashevskii, Riemannian Geometry and Tensor Calculus (URSS, Moscow, 2003).
B. E. Pobedrya, Lections on Tensor Calculus (Mosk. Gos. Univ., Moscow, 1986) [in Russian].
Yu. I. Dimitrienko, The Mechanics of a Continuous Medium, Vol. 1: The Tensor Analysis (MGTU Im. Baumana, Moscow, 2011) [in Russian].
M. U. Nikabadze, Several Problems of Tensor Calculus (TsPI MGU, Moscow, 2007) [in Russian].
D. V. Georgievskii and M. V. Shamolin, “Levi-Civita symbols, generalized vector products, and new integrable cases in mechanics of multidimensional bodies,” J. Math. Sci. 187 (3), 280–299 (2012).
D. V. Georgievskii, “Compatibility equations in systems based on generalized Cauchy kinematic relations,” Mech. Solids 49 (1), 99–103 (2014).
D.V. Georgievskii, “High-rank deformators and the Kröner incompatibility tensors with two-dimensional structure of indices,” Dokl. Phys. 64 (6), 256–257 (2019).
P. G. Ciarlet, P. Ciarlet Jr., G. Geymonat, and F. Krasucki, “Characterization of the kernel of the operator CURL CURL,” C.R. Acad. Sci. Paris Ser. I. 344, 305–308 (2007).
X. Zeng, “Cylindrically symmetric ground state solutions for curl-curl equations with critical exponent,” Z. Angew. Math. Phys. 68 (6), 135 (2017).
J. Mederski, “The Bresis–Nirenberg problem for the curl-curl operator,” J. Funct. Anal. 274 (5), 1345–1380 (2018).
Z. Zhang, “Comparison results for eigenvalues of curl curl operator and Stokes operator,” Z. Angew. Math. Phys. 69 (4), 104 (2018).
Funding
The work was carried out within the framework of state assignment AAAA-A20-120011690136-2 with the support of the Russian Foundation for Basic Research (grant nos. 18-29-10085mk, 19-01-00016a).
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Translated by M. K. Katuev
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Georgievskii, D.V. Second Order Linear Differential Operators over High Rank Tensor Fields. Mech. Solids 55, 808–812 (2020). https://doi.org/10.3103/S0025654420060060
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DOI: https://doi.org/10.3103/S0025654420060060