Local Solvability in a Class of Overdetermined Systems of Linear PDE

Abstract

The simplest linear partial differential equations whose local solvability is not automatic are those defined by complex, smooth, nowhere vanishing vector fields in regions of the plane. Let $L = A\left( {{x_1},{x_2}} \right)\partial /\partial {x_1} + B\left( {{x_1},{x_2}} \right)\partial /{\partial _2}$ be such a vector field, defined in a domain Ω ⊂ R ². The (inhomogeneous) equation

$$Lu = f$$

(1)

is said to be locally solvable at a point O ∈ Ω if there is an open neighborhood U ⊂ Ω of O such that, given any f ∈ C ^∞(Ω), there is a distribution u ∈ D′(U) that satisfies (1) in the open set U. One can vary this definition by asking that u be a C ^∞ function, in which case one may talk of local solvability in C ^∞; or by asking that there be a distribution solution u in U, for each f ∈ D′(Ω). A moment of thought will also show that there is no loss of generality if we restrict our attention to right hand sides f whose support is compact and contained in U.