Global solvability and global hypoellipticity for a class of complex vector fields on the 3-torus

被引：0

作者：

Adalberto P. Bergamasco

Paulo L. Dattori da Silva

Rafael B. Gonzalez

Alexandre Kirilov

机构：

[1] Universidade de São Paulo,Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação

[2] Universidade Federal do Paraná,Departamento de Matemática

来源：

Journal of Pseudo-Differential Operators and Applications | 2015年 / 6卷

关键词：

Global solvability; Global hypoellipticity; Complex vector field; Periodic solutions; Primary 35A01; Secondary 58Jxx;

D O I：

暂无

中图分类号：

学科分类号：

摘要：

This work deals with global solvability and global hypoellipticity of complex vector fields of the form L=∂/∂t+ib1(t)∂/∂x1+ib2(t)∂/∂x2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L=\partial /\partial t+ib_1(t)\partial /\partial x_1+ib_2(t)\partial /\partial x_2$$\end{document}, defined on T3≃R3/2πZ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {T}^3\simeq \mathbb {R}^{3}/2\pi \mathbb {Z}^{3}$$\end{document}, where both b1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_1$$\end{document} and b2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_2$$\end{document} belong to C∞(T1;R).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}^{\infty }(\mathbb {T}^1;\mathbb {R}).$$\end{document} The solvability and hypoellipticity depend on condition (P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal P$$\end{document}) and also on Diophantine properties of the coefficients.

引用

页码：341 / 360

页数：19

共 18 条

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[2] Bergamasco AP(1999)Remarks about global analytic hypoellipticity Trans. Am. Math. Soc. 351 4113-4126
[3] Bergamasco AP(2004)Global solvability for a class of complex vector fields on the two-torus Commun. PDE 29 785-819
[4] Cordaro PD(1977)Global solvability of an abstract complex Proc. Am. Math. Soc. 65 117-124
[5] Petronilho G(2002)Perturbations of vector fields on tori: resonant normal forms and Diophantine phenomena Proc. Edinb. Math. Soc. (2) 45 731-759
[6] Cardoso F(1972)Global hypoellipticity and Liouville numbers Proc. Am. Math. Soc. 31 112-114
[7] Hounie J(1979)Globally hypoelliptic and globally solvable first order evolutions equations Trans. Am. Math. Soc. 252 233-248
[8] Dickinson D(1982)Globally hypoelliptic vector fields on compact surfaces Commun. Partial Differ. Equ. 7 343-370
[9] Gramchev T(1963)Solvability of a first-order linear differential equation Commun. Pure Appl. Math. 16 331-351
[10] Yoshino M(1973)Orbits of families of vector fields and integrability of distributions Trans. Am. Math. Soc. 180 171-188

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