On differential operators an differential equations on torus

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.

differential operator on torus, linear differential equation on torus, Mizohata equation, nonstandard analysis, hypoellipticity