Geodesic
On the Noether theory of two-dimensional singular operators and applications to boundary-value problems for systems of fourth-order elliptic equations
Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 26 (2020) no. 1, pp. 7-13 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

It is known that the general theory of multidimensional singular integral operators over the entire space $E_m$ was constructed by S. G. Mikhlin. It is shown that in the two-dimensional case, if the operator symbol does not turn into zero, then the Fredholm theory holds. As for operators over a bounded domain, in this case the boundary of the domain significantly affects the solvability of such operator equations. In this paper we consider two-dimensional singular operators with continuous coefficients over a bounded domain. Such operators are widely used in many problems of the theory of partial differential equations. In this regard, it would be interesting to find criteria of Noetherity of such operators as explicit conditions for its coefficients. Depending on the $2m + 1$ connected components, necessary and sufficient conditions of Noetherity for such operators are obtained and a formula for the evaluation of the index is given. The results are applied to the Dirichlet problem for general fourth-order elliptic systems.
Keywords: singular integral operator, index, Noetherity of an operator, elliptic system.
Mots-clés : symbol 
@article{VSGU_2020_26_1_a0,
     author = {G. Dzhangibekov and J. M. Odinabekov},
     title = {On the {Noether} theory of two-dimensional singular operators and applications to boundary-value problems for systems of fourth-order elliptic equations},
     journal = {Vestnik Samarskogo universiteta. Estestvennonau\v{c}na\^a seri\^a},
     pages = {7--13},
     year = {2020},
     volume = {26},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGU_2020_26_1_a0/}
}
TY  - JOUR
AU  - G. Dzhangibekov
AU  - J. M. Odinabekov
TI  - On the Noether theory of two-dimensional singular operators and applications to boundary-value problems for systems of fourth-order elliptic equations
JO  - Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ
PY  - 2020
SP  - 7
EP  - 13
VL  - 26
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VSGU_2020_26_1_a0/
LA  - ru
ID  - VSGU_2020_26_1_a0
ER  - 
%0 Journal Article
%A G. Dzhangibekov
%A J. M. Odinabekov
%T On the Noether theory of two-dimensional singular operators and applications to boundary-value problems for systems of fourth-order elliptic equations
%J Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ
%D 2020
%P 7-13
%V 26
%N 1
%U http://geodesic.mathdoc.fr/item/VSGU_2020_26_1_a0/
%G ru
%F VSGU_2020_26_1_a0
G. Dzhangibekov; J. M. Odinabekov. On the Noether theory of two-dimensional singular operators and applications to boundary-value problems for systems of fourth-order elliptic equations. Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 26 (2020) no. 1, pp. 7-13. http://geodesic.mathdoc.fr/item/VSGU_2020_26_1_a0/

[1] S. G. Mikhlin, Multidimensional singular integrals and integral equations, Fizmatgiz, M., 1962, 254 pp. (In Russ.)

[2] E. M. Stein, “Note on singular integrals”, Proc. Amer. Math. Soc., 8 (1957), 250–254 | DOI | MR | Zbl

[3] I. N. Vekua, Generalized analytic functions, Nauka, M., 1959, 652 pp. (In Russ.)

[4] B. V. Boyarsky, Studies on elliptic equations in the plane and boundary problems of function theory, Doctoral of Physical and Mathematical Sciences thesis, M., 1960 (In Russ.)

[5] A. D. Juraev, Method of singular integral equations, Nauka, M., 1987, 415 pp. (In Russ.)

[6] K. Kh. Boymatov, G. Dzhangibekov, “On a singular integral operator”, Russian Mathematical Surveys, 43:3 (1988), 199–200 | DOI | MR

[7] G. Dzhangibekov, “On a class of two-dimensional singular integral operators and its applications to boundary value problems for elliptic systems of equations in the plane”, Doklady Mathematics, 47:3 (1993), 498–503 | MR | Zbl

[8] G. Jangibekov, G. Kh. Khujanazarova, “On the Noether property and the index for some two-dimensional singular integral operators in a connected domain”, Doklady Mathematics, 396:4 (2004), 449–454 (In Russ.) | MR | Zbl

[9] G. Dzhangibekov, D. M. Odinabekov, G. Kh. Khudzhanazarov, “The Noetherian Conditions and the Index of Some Class of Singular Integral Operators over a Bounded Simply Connected Domain”, Moscow University Mathematics Bulletin, 74:2 (2019), 49–54 | DOI | MR | Zbl

[10] R. V. Duduchava, “On multidimensional singular integral operators II: The case of compact manifolds”, Journal of Operator Theory, 11:1 (1984), 41–76 https://www.researchgate.net/publication/266063688_On_multidimensional_singular_integral_operators_I_The_half-space_case | MR | Zbl

[11] G. Jangibekov, G. Kh. Khujanazarova, “On the Dirichlet problem for elliptic systems of two equations of the fourth order on a plane”, Doklady Mathematics, 398:2 (2004), 151–155 (In Russ.) | MR

[12] G. Dzhangibekov, D. M. Odinabekov, “On the theory of two-dimensional singular integral operators with even characteristic over a bounded domains”, Contemporary problems of mathematics and mechanics, Proceedings of the conference dedicated to the 80th anniversary of academician V.A. Sadovnichy, MGU, M., 2019, 50–53 (In Russ.) | DOI