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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
| 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 |
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 and Applied Analysis,
2021, 20
(6)
: 2323-2340.
doi: 10.3934/cpaa.2021081
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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.
|
| [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. |
| [3] |
S. Berhanu, P. D. Cordaro and J. Hounie, An introduction to involutive structures, New mathematical monographs, University Press, (2008). |
| [4] |
P. D. Cordaro, Resolubilidade das Equações Diferenciais Parciais Lineares, Matemática Universitária, 14, (1992), 51-67. |
| [5] |
P. D. Cordaro, Sistemas de Campos Vetoriais Complexos, Instituto de matemática pura e aplicada, (1986). |
| [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. |
| [7] |
V. Grushin,
A differential Equation without a solution, Mathematical Notes, 10 (1971), 499-501.
|
| [8] |
N. Hanges,
Almost mizohata operators, Trans. Amer. Math. Soc., 2 (1986), 663-675.
doi: 10.2307/2000030. |
| [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. |
| [10] |
L. Hörmander,
Differential equations without solutions, Math. Ann., 140 (1960), 169-173.
doi: 10.1007/BF01361142. |
| [11] |
L. Hörmander, The Analysis of Linear Partial Differential Operators, Springer-Verlag, 1989.
doi: 10.1007/978-3-642-61497-2. |
| [12] |
H. Lewy,
An example of a smooth linear partial differential equation without solution, Ann. Math., 66 (1957), 155-158.
doi: 10.2307/1970121. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [18] |
L. Nunes, Resolubilidade Local Para Duas Classes de Campos de Vetores Suaves Complexos, Ph. D. thesis (in portuguese), UFSCar, 2016. |
| [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. |
| [20] |
J. Sjöstrand,
Note on a paper of F. Treves concerning Mizohata type operators, Duke Math. J., 47 (1980), 601-608.
|
| [21] |
M. Yamamoto,
On partially hypoelliptic operators, Osaka Math. J., 15 (1963), 233-247.
|
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.
|
| [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. |
| [3] |
S. Berhanu, P. D. Cordaro and J. Hounie, An introduction to involutive structures, New mathematical monographs, University Press, (2008). |
| [4] |
P. D. Cordaro, Resolubilidade das Equações Diferenciais Parciais Lineares, Matemática Universitária, 14, (1992), 51-67. |
| [5] |
P. D. Cordaro, Sistemas de Campos Vetoriais Complexos, Instituto de matemática pura e aplicada, (1986). |
| [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. |
| [7] |
V. Grushin,
A differential Equation without a solution, Mathematical Notes, 10 (1971), 499-501.
|
| [8] |
N. Hanges,
Almost mizohata operators, Trans. Amer. Math. Soc., 2 (1986), 663-675.
doi: 10.2307/2000030. |
| [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. |
| [10] |
L. Hörmander,
Differential equations without solutions, Math. Ann., 140 (1960), 169-173.
doi: 10.1007/BF01361142. |
| [11] |
L. Hörmander, The Analysis of Linear Partial Differential Operators, Springer-Verlag, 1989.
doi: 10.1007/978-3-642-61497-2. |
| [12] |
H. Lewy,
An example of a smooth linear partial differential equation without solution, Ann. Math., 66 (1957), 155-158.
doi: 10.2307/1970121. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [18] |
L. Nunes, Resolubilidade Local Para Duas Classes de Campos de Vetores Suaves Complexos, Ph. D. thesis (in portuguese), UFSCar, 2016. |
| [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. |
| [20] |
J. Sjöstrand,
Note on a paper of F. Treves concerning Mizohata type operators, Duke Math. J., 47 (1980), 601-608.
|
| [21] |
M. Yamamoto,
On partially hypoelliptic operators, Osaka Math. J., 15 (1963), 233-247.
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