On Linear Differential Equations on the Torus and Non-Standard Analysis

Abstract

In this paper, we consider periodic boundary value problems for differential equations whose coefficients are trigonometric polynomials. We construct the spaces of generalized functions, where such problems have solutions. In particular, the solvability space of a periodic analogue of the Mizohata equation is constructed. We build also a periodic analogue and a generalization of the construction of the nonstandard analysis, where infinitely small are not only functions, but also functional spaces. To show that not all constructions on the torus lead to a simplification in compare with the plane, we consider a periodic analogue of the hypoelliptic differential operator and show that its 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.