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Existence and Regularity of Periodic Solutions for a Class of Partial Differential Operators

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Abstract

We give a complete answer to the questions concerning existence and regularity of periodic solutions to a class of linear partial differential operators. The results depend on Diophantine conditions and also on a control on the sign of the imaginary part of the symbol, which is related to the Nirenberg–Treves condition (P). This control is based on the following aspects: the linear dependence of the imaginary part of the coefficients, the connectedness of certain sublevel sets, and the parity of the order of certain derivatives. For certain operators, the results are also influenced by the order of vanishing of the coefficients.

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Acknowledgements

The first author was supported in part by CNPq (grant 307541/2015-0) and FAPESP (grant 2012/03168-7). The second author was partially supported by CNPq (grant 300631/2003-0). The third author was supported by National Postdoctoral Program form CAPES-Brazil.

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Authors and Affiliations

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

    Adalberto P. Bergamasco

  2. Departamento de Matemática, Universidade Estadual de Maringá, Maringá, PR, 87020-900, Brazil

    Marcelo M. Cavalcanti & Rafael B. Gonzalez

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  1. Adalberto P. Bergamasco
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  2. Marcelo M. Cavalcanti
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  3. Rafael B. Gonzalez
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Correspondence to Rafael B. Gonzalez.

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Bergamasco, A.P., Cavalcanti, M.M. & Gonzalez, R.B. Existence and Regularity of Periodic Solutions for a Class of Partial Differential Operators. J Fourier Anal Appl 27, 52 (2021). https://doi.org/10.1007/s00041-021-09855-w

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