Search

This paper deals with the solvability of planar complex vector fields with homogeneous degeneracies. Hölder continuous solutions are obtained via a Cauchy type integral operator associated to the vector field. An associated boundary value problem of Riemann–Hilbert type is also considered. [ABSTRACT FROM AUTHOR]

Abstract: Global solvability on the two-torus of a first order differential operator with complex coefficients is investigated. Diophantine properties of the coefficients are linked to the solvability. [Copyright &y& Elsevier]

Abstract: We study a class of complex vector fields defined on the two-torus of the form , , . We view as an operator acting on smooth functions and present conditions for to have either a closed range or a finite-codimensional range. Our results involve, besides condition () of Nirenberg and Treves, the behavior of near each one-dimensional Sussmann orbit homotopic to the unit circle. One of the main goals of our work is to provide some clarification about the role played by the coefficient a in the validity of the above properties of the range. [Copyright &y& Elsevier]