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 Omega(epsilon) = (-epsilon, epsilon) x S(1), given by L = partial derivative/partial derivative t + (a(x) + ib(x))partial derivative/partial derivative x, b not equivalent to 0, near the characteristic set Sigma = {0} x 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. (C) 2008 Elsevier Inc. All rights reserved.

C(k)-solvability near the characteristic set for a class of planar complex vector fields of infinite type

This paper deals with semi-global C(k)-solvability of complex vector fields of the form L = partial derivative/partial derivative t + x(r) (a(x) + ib(x))partial derivative/partial derivative x, r >= 1, defined on Omega(epsilon) = (-epsilon, epsilon) x S(1), epsilon > 0, where a and b are C(infinity) real-valued functions in (-epsilon, epsilon). It is shown that the interplay between the order of vanishing of the functions a and b at x = 0 influences the C(k)-solvability at Sigma = {0} x S(1). When r = 1, it is permitted that the functions a and b of L depend on the x and t variables, that is, L = partial derivative/partial derivative t + x(a(x, t) + ib(x, t))partial derivative/partial derivative x, where (x, t) is an element of Omega(epsilon).

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 = partial derivative/partial derivative t + (a + ib)(x)partial derivative/partial derivative x, a, b epsilon C(infinity) (S(1) ; R), defined on two-torus T(2) congruent to R(2)/2 pi Z(2). The kernel of transpose operator L is described and the solvability near the characteristic set is also studied. (c) 2008 Elsevier Inc. All rights reserved.

Maslov index in semi-Riemannian submersions

We study focal points and Maslov index of a horizontal geodesic gamma : I -> M in the total space of a semi-Riemannian submersion pi : M -> B by determining an explicit relation with the corresponding objects along the projected geodesic pi omicron gamma : I -> B in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary spacetime which is orthogonal to a timelike Killing vector field.