Geodesic
Constructive solution of ellipticity problem for the first order differential systems
Modelirovanie i analiz informacionnyh sistem, Tome 24 (2017) no. 5, pp. 655-670.

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We built first order elliptic systems with any possible number of unknown functions and the maximum possible number of unknowns, i.e, in general. These systems provide the basis for studying the properties of any first order elliptic systems. The study of the Cauchy-Riemann system and its generalizations led to the identification of a class of elliptic systems of first-order of a special structure. An integral representation of solutions is of great importance in the study of these systems. Only by means of a constructive method of integral representations we can solve a number of problems in the theory of elliptic systems related mainly to the boundary properties of solutions. The obtained integral representation could be applied to solve a number of problems that are hard to solve, if you rely only on the non-constructive methods. Some analogues of the theorems of Liouville, Weierstrass, Cauchy, Gauss, Morera, an analogue of Green's formula are established, as well as an analogue of the maximum principle. The used matrix operators allow the new structural arrangement of the maximum number of linearly independent vector fields on spheres of any possible dimension. Also the built operators allow to obtain a constructive solution of the extended problem "of the sum of squares" known in algebra.
Keywords: elliptic systems, differential equations. 
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V. E. Balabaev. Constructive solution of ellipticity problem for the first order differential systems. Modelirovanie i analiz informacionnyh sistem, Tome 24 (2017) no. 5, pp. 655-670. http://geodesic.mathdoc.fr/item/MAIS_2017_24_5_a9/

[1] Shafarevich I. R., “Foundations of algebraic geometry”, Russian Math. Surveys, 24:6 (1969), 178 pp. | DOI | MR | MR | Zbl

[2] Yuzvinski S., “Ortogonal pairings of Euclidean spaces”, Michigan Math. J., 28:2 (1981), 131–145 | DOI | MR

[3] Adams J. F., “Vector fields on spheres”, Ann. of Math., 75:3 (1962), 603–632 | DOI | MR | Zbl

[4] Solomyak M.Z., “O linejnyh ehllipticheskih sistemah pervogo poryadka”, Dokl. AN SSSR, 150:1 (1963), 48–51 (in Russian) | Zbl

[5] Bicadze A.V., Kraevye zadachi dlya ehllipticheskih uravnenij vtorogo poryadka, Nauka, Moskva, 1966 (in Russian) | MR

[6] Vinogradov V.S., “Investigation of boundary value problems for elliptic systems of first order”, Math. Notes, 14:2 (1973), 724–731 | DOI | MR

[7] Balabaev V.E., “On one system of equations in octaves in eight dimensional Euclidean space”, Fundam. Prikl. Mat., 1:2 (1995), 517–521 (in Russian) | MR | Zbl

[8] Dezin A.A., Mnogomernyi analiz i diskretnye modeli, Nauka, Moskva, 1990, 240 pp. (in Russian) | MR