# Local Solvability in a Class of Overdetermined Systems of Linear PDE

Conference paper

## Abstract

The simplest linear partial differential equations whose is said to be

*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)

*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*.### Keywords

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© Springer Science+Business Media New York 1996