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

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)