Class Of Complex Vector Fields Research Articles

Gevrey semiglobal solvability for a class of elliptic vector fields with degeneracies

AbstractWe deal with Gevrey solvability of a class of complex vector fields defined on , given by , , near the characteristic set . Diophantine conditions appear in a natural way in our results.

Gevrey semiglobal solvability for a class of complex vector fields

We deal with Gevrey solvability of a class of complex vector fields, defined on , where is the unit circle, given by , , near the characteristic set . We show that under certain condition involving the order of vanishing of the functions a and b at x = 0 the equation Lu = f does not have Gevrey solutions.

Class of hypocomplex structures on the two-dimensional torus

We study the Hölder solvability of a class of complex vector fields on the torus T 2 \mathbb {T}^2 . We make use of the Theta function to associate a Cauchy-Pompeiu type integral operator. A similarity principle for the solutions of the equation L u = a u + b u ¯ Lu=au+b\bar {u} is obtained.

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

This work deals with global solvability and global hypoellipticity of complex vector fields of the form \(L=\partial /\partial t+ib_1(t)\partial /\partial x_1+ib_2(t)\partial /\partial x_2\), defined on \(\mathbb {T}^3\simeq \mathbb {R}^{3}/2\pi \mathbb {Z}^{3}\), where both \(b_1\) and \(b_2\) belong to \(\mathcal {C}^{\infty }(\mathbb {T}^1;\mathbb {R}).\) The solvability and hypoellipticity depend on condition (\(\mathcal P\)) and also on Diophantine properties of the coefficients.

Solvability near the characteristic set for a special class of complex vector fields

This work deals with the solvability near the characteristic set Σ = {0} × S 1 of operators of the form $${L=\partial/\partial t + (x^na(x) + ix^mb(x))\partial/\partial x}$$ , $${b\not\equiv0}$$ and a(0) ≠ 0, defined on $${\Omega_\epsilon=(-\epsilon,\epsilon)\times S^1}$$ , $${\epsilon >0 }$$ , where a and b are real-valued smooth functions in $${(-\epsilon,\epsilon)}$$ and m ≥ 2n. It is shown that given f belonging to a subspace of finite codimension of $${C^\infty(\Omega_\epsilon)}$$ there is a solution $${u\in L^\infty}$$ of the equation Lu = f in a neighborhood of Σ; moreover, the L ∞ regularity is sharp.

Solvability in the large for a class of complex vector fields on the cylinder

This work deals with global solvability of a class of complex vector fields of the form L = ∂ / ∂ t + ( a ( x , t ) + i b ( x , t ) ) ∂ / ∂ x , where a and b are real-valued C ∞ functions, defined on the cylinder Ω = R × S 1 . Relatively compact (Sussmann) orbits are allowed. The connection with Malgrangeʼs notion of L -convexity for supports is investigated.

A Note about Analytic Solvability of Complex Planar Vector Fields with Degeneracies

AbstractThis paper deals with the analytic solvability of a special class of complex vector fields defined on the real plane, where they are tangent to a closed real curve, while off the real curve, they are elliptic.

Gevrey solvability near the characteristic set for a class of planar complex vector fields of infinite type

We study the Gevrey solvability of a class of complex vector fields, defined on Ω ϵ = ( − ϵ , ϵ ) × S 1 , given by L = ∂ / ∂ t + ( a ( x ) + i b ( x ) ) ∂ / ∂ x , b ≢ 0 , near the characteristic set Σ = { 0 } × S 1 . We show that the interplay between the order of vanishing of the functions a and b at x = 0 plays a role in the Gevrey solvability.

Nonexistence of global solutions for a class of complex vector fields on two-torus

The goal of this paper is study the global solvability of a class of complex vector fields of the special form L = ∂ / ∂ t + ( a + i b ) ( x ) ∂ / ∂ x , a , b ∈ C ∞ ( S 1 ; R ) , defined on two-torus T 2 ≅ R 2 / 2 π Z 2 . The kernel of transpose operator L t is described and the solvability near the characteristic set is also studied.

Solvability in the large for a class of vector fields on the torus

We study a class of complex vector fields defined on the two-torus of the form L = ∂ / ∂ t + ( a ( x , t ) + i b ( x , t ) ) ∂ / ∂ x , a , b ∈ C ∞ ( T 2 ; R ) , b ≢ 0 . We view L as an operator acting on smooth functions and present conditions for L to have either a closed range or a finite-codimensional range. Our results involve, besides condition ( P ) of Nirenberg and Treves, the behavior of a + i b 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.

Global Solvability for a Class of Complex Vector Fields on the Two-Torus

We consider complex vector fields L on the two-torus. We regard L as an operator acting on smooth functions and study conditions for L to have a closed range. We also give conditions for the range of L to have finite codimension. Our results involve condition (P) of Nirenberg and Treves. One-dimensional orbits diffeomorphic to the unit circle are allowed.

Normalization of complex-valued planar vector fields which degenerate along a real curve

Taking as a start point the recent article of Meziani [7], we present several results concerning the normalization of a class of complex vector fields in the plane which degenerate along a real curve. We mainly deal with operators with finite regularity and analyze both the local situation as well as the case of normalization near a circle. Some related questions (e.g., on semi-global solvability and on the normalization of a class of generalized Mizohata operators) are also discussed.