On differential operators and linear differential equations on torus

TL;DR

This paper explores differential operators and equations on the torus, constructing solution spaces for periodic boundary value problems with trigonometric polynomial coefficients, and analyzing their properties including hypoellipticity and number-theoretic aspects.

Contribution

It introduces new solution spaces for periodic differential equations on the torus and examines the hypoelliptic operators' properties with a focus on number theory.

Findings

Constructed solution spaces for periodic boundary value problems.

Identified conditions for hypoellipticity based on polynomial irreducibility.

Showed the significance of number-theoretic properties in differential operators on the torus.

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 smalls 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.