Geodesic
On differential operators and differential equations on torus
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 22 (2018) no. 4, pp. 607-619

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In this paper, we consider periodic boundary value problems for a differential equation whose coefficients are trigonometric polynomials. The spaces of generalized functions are constructed, in which the problems considered have solutions, in particular, the solvability space of a periodic analogue of the Mizohata equation is constructed. A periodic analogue and a generalization of the construction of a nonstandard analysis are constructed, containing not only functions, but also functional spaces. As an illustration of the statement that not all constructions on a torus lead to simplification compared to a plane, a periodic analogue of the concept of a hypoelliptic differential operator is considered, where number-theoretic properties are significant. In particular, it turns out that if a polynomial with integer coefficients is irreducible in the rational field, then the corresponding differential operator is hypoelliptic on the torus.
Keywords: differential operator on torus, linear differential equation on torus, nonstandard analysis
Mots-clés : Mizohata equation, hypoellipticity. 
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     author = {V. P. Burskii},
     title = {On differential operators and differential equations on torus},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {607--619},
     publisher = {mathdoc},
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     number = {4},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2018_22_4_a0/}
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V. P. Burskii. On differential operators and differential equations on torus. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 22 (2018) no. 4, pp. 607-619. http://geodesic.mathdoc.fr/item/VSGTU_2018_22_4_a0/