Solvability and hypoellipticity of first order partial differential operators and the Riemann-Hilbert problem

• Grant number: 15/20815-4
• Support Opportunities: Regular Research Grants
• Duration: December 01, 2015 - May 31, 2018
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Paulo Leandro Dattori da Silva
• Grantee: Paulo Leandro Dattori da Silva
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

Let X be smooth, connected, n-dimentional manifold and let L be a nonsingular smooth complex vector field defined on X.This project deals with the study of problems related with semiglobal/global solvability and global hypoellipticity of equations in the formLu=Au+B\overline{u}+fdefined in X, where A, B and f are smooth functions.Also, it deals with the study of generalized Riemann-Hilbert problemLu=Au+B\overline{u}+f, em U\subset R^2\Re(gu)=\chi, sobre \partialUwhere L is a smooth complex vector field defined on R^2, f\in C^\infty(R^2),g\in C^\alpha(\partialU, S^1) and \chi\in C^\alpha(\partialU, R).The problems mentioned above can be considered in others spaces of functions, for instance, L^p.This project also deals with the study of solvability and hipoellipticity of complex associated to a system of closed 1-formsdefined on compact manifolds and the calculus of relative cohomology. (AU)

Scientific publications

(4)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

CERNIAUSKAS, WANDERLEY A.; DATTORI DA SILVA, PAULO L.. Solvability near the characteristic set for a class of first-order linear partial differential operators. Mathematische Nachrichten, v. 291, n. 8-9, p. 1240-1268, JUN 2018. (12/03168-7, 15/20815-4)

BERGAMASCO, A. P.; DATTORI DA SILVA, P. L.; GONZALEZ, R. B.. Global solvability and global hypoellipticity in Gevrey classes for vector fields on the torus. Journal of Differential Equations, v. 264, n. 5, p. 3500-3526, MAR 1 2018. (12/03168-7, 15/20815-4)

BERGAMASCO, ADALBERTO P.; DATTORI DA SILVA, PAULO L.; GONZALEZ, RAFAEL B.. Existence of global solutions for a class of vector fields on the three-dimensional torus. BULLETIN DES SCIENCES MATHEMATIQUES, v. 148, p. 53-76, OCT 2018. (12/03168-7, 15/20815-4)

CAMPANA, C.; DATTORI DA SILVA, P. L.; MEZIANI, A.. A class of planar vector fields with homogeneous singular points: Solvability and boundary value problems. Journal of Differential Equations, v. 265, n. 10, p. 5297-5314, NOV 15 2018. (12/03168-7, 16/21969-8, 13/08452-8, 15/20815-4)