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Solvability of a Class of First Order Differential Operators on the Torus

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Abstract

This paper deals with Gevrey global solvability on the N-dimensional torus (\({\mathbb {T}}^{N}\simeq {\mathbb {R}}^{N}/2\pi {\mathbb {Z}}^{N}\)) to a class of nonlinear first order partial differential equations in the form \(Lu-au-b{\overline{u}}=f\), where a, b, and f are Gevrey functions on \({\mathbb {T}}^{N}\) and L is a complex vector field defined on \({\mathbb {T}}^{N}\). Diophantine properties of the coefficients of L appear in a natural way in our results. Also, we present results in \(C^\infty \) context.

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Acknowledgements

The authors are very grateful to the anonymous referee for interesting and valuable suggestions that improved the early version of this paper. The first author was supported in part by CNPq (grant 409306/2016-9), and the second author was supported in part by CNPq (grant 309496/2018-7) and FAPESP (grants 2018/15046-0 and 2018/14316-3).

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Sergipe, São Cristóvão, Sergipe, 49000-000, Brazil

    Marcelo F. de Almeida

  2. Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, São Carlos, São Paulo, 13560-970, Brazil

    Paulo L. Dattori da Silva

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  1. Marcelo F. de Almeida
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  2. Paulo L. Dattori da Silva
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Correspondence to Paulo L. Dattori da Silva.

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de Almeida, M.F., Dattori da Silva, P.L. Solvability of a Class of First Order Differential Operators on the Torus. Results Math 76, 104 (2021). https://doi.org/10.1007/s00025-021-01413-6

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