On differential operators and linear differential equations on torus
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929405474897920 |
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| author | Burskii, V. P. |
| author_facet | Burskii, V. P. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_01632 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On differential operators and linear differential equations on torus Burskii, V. P. Analysis of PDEs 35B10, 35D99, 58J15, 35H10, 26E35 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. |
| title | On differential operators and linear differential equations on torus |
| topic | Analysis of PDEs 35B10, 35D99, 58J15, 35H10, 26E35 |
| url | https://arxiv.org/abs/2407.01632 |