Solvability of the cohomological equation for regular vector fields on the plane

We consider planar vector field without zeroes X and study the image of the associated Lie derivative operator LX acting on the space of smooth functions. We show that the cokernel of LX is infinite-dimensional as soon as X is not topologically conjugate to a constant vector field and that, if the topology of the integral trajectories of X is ``simple enough'' (e.g. if X is polynomial) then X is transversal to a Hamiltonian foliation. We use this fact to find a large explicit subalgebra of the image of LX and to build an embedding of R^2 into R^4 which rectifies X. Finally we use this embedding to characterize the functions in the image of LX.Comment: 21 pages, 2 figure

Symmetry for solutions of two-phase semilinear elliptic equations on hyperbolic space

Assume that $f(s) = F'(s)$ where $F$ is a double-well potential. Under certain conditions on the Lipschitz constant of $f$ on $[-1,1]$, we prove that arbitrary bounded global solutions of the semilinear equation $\Delta u = f(u)$ on hyperbolic space \HH^n must reduce to functions of one variable provided they admit asymptotic boundary values on the infinite boundary of \HH^n which are invariant under a cohomogeneity one subgroup of the group of isometries of \HH^n. We also prove existence of these one-dimensional solutions.Comment: 24 page