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June  2021, 20(6): 2323-2340. doi: 10.3934/cpaa.2021081

On local solvability for a class of generalized Mizohata equations

L. R. Nunes 1,, and  J. R. Dos Santos Filho 2,

1. 

Universidade Federal do Rio Grande, Instituto de Matemática, Estatística e Física, RS, Brazil
2. 

Universidade Federal de São Carlos, Departamento de Matemáica, SP, Brazil

* Corresponding author

Received  October 2020 Revised  March 2021 Published  June 2021 Early access  May 2021

Fund Project: The first author was partially supported by CAPES. The second author is a associate researcher of the Thematic project no. 2018/14316-3 (FAPESP)

The image in $ C^{\infty} $ for a class of complex vector fields, containing the Mizohata operator, was characterized.

Keywords: Partial differential equations, complex vector fields, local solvability, operators of Mizohata.
Mathematics Subject Classification: Primary: 35F05; Secondary: 35A01.
Citation: L. R. Nunes, J. R. Dos Santos Filho. On local solvability for a class of generalized Mizohata equations. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2323-2340. doi: 10.3934/cpaa.2021081
References:
[1]

M. S. Baouendi, C. H. Chang and F. Treves, Microlocal Hypo-Analytic and Extension of CR Functions, J. Differ. Geom., 18 (1983), 331-391.   Google Scholar

[2]

M. S. Baouendi and F. Treves, A local constancy principle for the solutions of certain overdetermined systems of first-order linear partial differential equations, in Advances in Mathematics Supplementary Studies, Academic Press, New York, (1981), 245-262.  Google Scholar

[3]

S. Berhanu, P. D. Cordaro and J. Hounie, An introduction to involutive structures, New mathematical monographs, University Press, (2008).  Google Scholar

[4]

P. D. Cordaro, Resolubilidade das Equações Diferenciais Parciais Lineares, Matemática Universitária, 14, (1992), 51-67. Google Scholar

[5]

P. D. Cordaro, Sistemas de Campos Vetoriais Complexos, Instituto de matemática pura e aplicada, (1986). Google Scholar

[6]

L. Garding and B. Malgrange, Opérateurs différentiels partiellement hypoelliptiques et partiellement elliptiques, Math. Scand., 9 (1961), 5-21.  doi: 10.7146/math.scand.a-10619.  Google Scholar

[7]

V. Grushin, A differential Equation without a solution, Mathematical Notes, 10 (1971), 499-501.   Google Scholar

[8]

N. Hanges, Almost mizohata operators, Trans. Amer. Math. Soc., 2 (1986), 663-675.  doi: 10.2307/2000030.  Google Scholar

[9]

G. Hoepfner and R. Medrado, Microlocal regularity for Mizohata type differential operators, J. Inst. Math. Jussieu, 19 (2020), 1185-1209.  doi: 10.1017/S1474748018000361.  Google Scholar

[10]

L. Hörmander, Differential equations without solutions, Math. Ann., 140 (1960), 169-173.  doi: 10.1007/BF01361142.  Google Scholar

[11]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Springer-Verlag, 1989. doi: 10.1007/978-3-642-61497-2.  Google Scholar

[12]

H. Lewy, An example of a smooth linear partial differential equation without solution, Ann. Math., 66 (1957), 155-158.  doi: 10.2307/1970121.  Google Scholar

[13]

S. Mizohata, Une remarque sur les opérateurs différentielshypoelliptiques et partiellement hypoelliptiques, J. Math. Kyoto Univ., 1-3 (1962), 411-423.  doi: 10.1215/kjm/1250525013.  Google Scholar

[14]

H. Ninomiya, A necessary and sufficient condition of local integrability, J. Math Kyoto Univ., 39-4 (1999), 685-696.  doi: 10.1215/kjm/1250517821.  Google Scholar

[15]

L. Nirenberg and F. Treves, On local solvability of linear partial differential equations, I. Necessary conditions, Commun. Pure Appl. Math, 23 (1970), 1-38.  doi: 10.1002/cpa.3160230102.  Google Scholar

[16]

L. Nirenberg and F. Treves, On local solvability of linear partial differential equations, II. Sufficient conditions, Commun. Pure Appl. Math, 23 (1970), 459-510.  doi: 10.1002/cpa.3160230314.  Google Scholar

[17]

L. Nirenberg and F. Treves, Solvability of a order linear partial differential equation, Commun. Pure Appl. Math, 16 (1963), 331-351.  doi: 10.1002/cpa.3160160308.  Google Scholar

[18]

L. Nunes, Resolubilidade Local Para Duas Classes de Campos de Vetores Suaves Complexos, Ph. D. thesis (in portuguese), UFSCar, 2016. Google Scholar

[19]

F. Treves, Demarks about certain first order linear PDE in two variables, Comm. Partial Differential Equations, 5 (1980), 381-425.  doi: 10.1080/0360530800882143.  Google Scholar

[20]

J. Sjöstrand, Note on a paper of F. Treves concerning Mizohata type operators, Duke Math. J., 47 (1980), 601-608.   Google Scholar

[21]

M. Yamamoto, On partially hypoelliptic operators, Osaka Math. J., 15 (1963), 233-247.   Google Scholar

show all references

References:
[1]

M. S. Baouendi, C. H. Chang and F. Treves, Microlocal Hypo-Analytic and Extension of CR Functions, J. Differ. Geom., 18 (1983), 331-391.   Google Scholar

[2]

M. S. Baouendi and F. Treves, A local constancy principle for the solutions of certain overdetermined systems of first-order linear partial differential equations, in Advances in Mathematics Supplementary Studies, Academic Press, New York, (1981), 245-262.  Google Scholar

[3]

S. Berhanu, P. D. Cordaro and J. Hounie, An introduction to involutive structures, New mathematical monographs, University Press, (2008).  Google Scholar

[4]

P. D. Cordaro, Resolubilidade das Equações Diferenciais Parciais Lineares, Matemática Universitária, 14, (1992), 51-67. Google Scholar

[5]

P. D. Cordaro, Sistemas de Campos Vetoriais Complexos, Instituto de matemática pura e aplicada, (1986). Google Scholar

[6]

L. Garding and B. Malgrange, Opérateurs différentiels partiellement hypoelliptiques et partiellement elliptiques, Math. Scand., 9 (1961), 5-21.  doi: 10.7146/math.scand.a-10619.  Google Scholar

[7]

V. Grushin, A differential Equation without a solution, Mathematical Notes, 10 (1971), 499-501.   Google Scholar

[8]

N. Hanges, Almost mizohata operators, Trans. Amer. Math. Soc., 2 (1986), 663-675.  doi: 10.2307/2000030.  Google Scholar

[9]

G. Hoepfner and R. Medrado, Microlocal regularity for Mizohata type differential operators, J. Inst. Math. Jussieu, 19 (2020), 1185-1209.  doi: 10.1017/S1474748018000361.  Google Scholar

[10]

L. Hörmander, Differential equations without solutions, Math. Ann., 140 (1960), 169-173.  doi: 10.1007/BF01361142.  Google Scholar

[11]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Springer-Verlag, 1989. doi: 10.1007/978-3-642-61497-2.  Google Scholar

[12]

H. Lewy, An example of a smooth linear partial differential equation without solution, Ann. Math., 66 (1957), 155-158.  doi: 10.2307/1970121.  Google Scholar

[13]

S. Mizohata, Une remarque sur les opérateurs différentielshypoelliptiques et partiellement hypoelliptiques, J. Math. Kyoto Univ., 1-3 (1962), 411-423.  doi: 10.1215/kjm/1250525013.  Google Scholar

[14]

H. Ninomiya, A necessary and sufficient condition of local integrability, J. Math Kyoto Univ., 39-4 (1999), 685-696.  doi: 10.1215/kjm/1250517821.  Google Scholar

[15]

L. Nirenberg and F. Treves, On local solvability of linear partial differential equations, I. Necessary conditions, Commun. Pure Appl. Math, 23 (1970), 1-38.  doi: 10.1002/cpa.3160230102.  Google Scholar

[16]

L. Nirenberg and F. Treves, On local solvability of linear partial differential equations, II. Sufficient conditions, Commun. Pure Appl. Math, 23 (1970), 459-510.  doi: 10.1002/cpa.3160230314.  Google Scholar

[17]

L. Nirenberg and F. Treves, Solvability of a order linear partial differential equation, Commun. Pure Appl. Math, 16 (1963), 331-351.  doi: 10.1002/cpa.3160160308.  Google Scholar

[18]

L. Nunes, Resolubilidade Local Para Duas Classes de Campos de Vetores Suaves Complexos, Ph. D. thesis (in portuguese), UFSCar, 2016. Google Scholar

[19]

F. Treves, Demarks about certain first order linear PDE in two variables, Comm. Partial Differential Equations, 5 (1980), 381-425.  doi: 10.1080/0360530800882143.  Google Scholar

[20]

J. Sjöstrand, Note on a paper of F. Treves concerning Mizohata type operators, Duke Math. J., 47 (1980), 601-608.   Google Scholar

[21]

M. Yamamoto, On partially hypoelliptic operators, Osaka Math. J., 15 (1963), 233-247.   Google Scholar

Figure 1.  Integration domain complexification
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