Abdelhamid Meziani

Global properties of a class of planar vector fields of infinite type

This paper studies some global and semi global properties of infinite type, planar, C-valued real analytic vector fields that are invariant under the rotation group. Results are proved on the integrability, kernel, range and classification of such operators.

Elliptic planar vector fields with degeneracies

This paper deals with the normalization of elliptic vector fields in the plane that degenerate along a simple and closed curve. The associated homogeneous equation Lu = 0 is studied and an application to a degenerate Beltrami equation is given.

Representation of solutions of a singular Cauchy–Riemann equation in the plane

This article deals with the Bers–Vekua type equations with a singular point. Series representations of solutions and Green functions are constructed. A generalized Cauchy formula and its consequences are obtained.

On First and Second Order Planar Elliptic Equations with Degeneracies

This paper deals with elliptic equations in the plane with degen- eracies. The equations are generated by a complex vector field that is elliptic everywhere except along a simple closed curve. Kernels for these equations are constructed. Properties of solutions, in a neighborhood of the degen- eracy curve, are obtained through integral and series representations. An application to a second order elliptic equation with a punctual singularity is given.

Generalized CR Equation with a Punctual Singularity: The Model Case

This paper deals with the generalized Cauchy–Riemann equation w[zbar] = Aw + B[wbar] + C in the presence of a punctual singularity. We use systems of differential equations with periodic coefficients and eigenfunctions expansion to construct and characterize the solution space.

HYPOELLIPTICITY OF NONSINGULAR CLOSED 1-FORMS ON COMPACT MANIFOLDS

ABSTRACT Hypoellipticity of nonsingular closed 1-forms on compact manifolds is characterized by a diophantine condition on the generators of the group of periods of the form.

The Mizohata complex

This paper deals with the local solvability of systems of first order linear partial differential equations defined by a germ ω at 0 ∈ Rn+1 of a Cvalued, formally integrable (ω∧dω = 0), 1-form with nondegenerate Levi form. More precisely, the size of the obstruction to the solvability, for (q − 1)-forms u, of the equation du ∧ ω = η ∧ ω, where η is a given q-form satisfying dη∧ω = 0 is estimated in terms of the De Rham cohomology relative to ω 0. Introduction This paper deals with the obstruction to the local solvability problem of a formally integrable form (or equivalently of a system of vector fields). More precisely, let ω be the germ at 0 ∈ R of a C-valued smooth 1-form satisfying ω(0) 6= 0 and dω ∧ ω = 0. (1) Two smooth germs η and η′ of q-forms are said to be ω-equivalent if η′ = η+α∧ω for some (q− 1)-form α. Denote by Ωω the quotient space, i.e. Ωω = Ωq/Ωq−1 ∧ω, where Ω is the space of germs of q-forms. It follows from (1) that the exterior derivative d : Ω −→ Ω induces a complex 0 −→ Ω −→ Ωω −→ · · · −→ Ωω −→ 0 whose cohomology we denote by H∗ ω. It is the De Rham cohomology relative to ω. Notice that H ω is the space of first integrals of ω, which is reduced to C when ω is without integrating factor. The only two situations in which this complex is fully understood are the real case (ω ∧ ω ≡ 0) and the elliptic case (ω ∧ ω 6= 0). In both cases the cohomology groups H ω are trivial when q ≥ 1. The planar case of a real analytic form ω has been studied in [Me1]. Otherwise the focus has been, so far, in finding conditions for the triviality of H ω, when ω = d(x + iφ(x, t)). (2) This was initiated by Nirenberg and Treves in [NT]. The tube case (φ independent on x) has been solved by Treves in [Tr1] who proved that the triviality of the (q − 1)-th cohomology of the level sets of φ is equivalent to the triviality of H ω. Among results in this direction (when φ depends effectively on x), we would like to mention [CH], [CT], [MT], [Tr2], [Tr3]. In particular, it is proved in [CT] that the nontriviality of the (q − 1)-th cohomology of the level set of x+ iφ implies the Received by the editors August 12, 1994 and, in revised form, June 27, 1995. 1991 Mathematics Subject Classification. Primary 35F05; Secondary 58A10. c ©1997 American Mathematical Society 1029 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Generalized CR Equation with a Singularity

This article deals with the solvability of the CR equation rw = (A0(θ) ☎ O(rα))w ☎ (B0(θ) ☎ O(rα)), where z = r eiθ and where A0, B0 are continuous and 2π-periodic. The solutions of this equation are shown to be similar to those of the model equation rw = A0(θ)w ☎ B0(θ). The solutions of the model equation are completely characterized by using dynamical systems and Fourier techniques.

The Riemann–Hilbert problem for elliptic vector fields with degeneracies

The boundary value problemis studied for a class of planar complex vector fields in a simply connected open set . The first integrals of are used to reduce the problem into a collection of classical Riemann–Hilbert problems with discontinuous data.

Riemann–Hilbert problem for a class of hypocomplex vector fields

This paper deals with a semilinear Riemann–Hilbert problem for a class of planar elliptic vector fields with degeneracies. Existence of Hölder continuous solutions is established when the associated index is nonnegative.