# 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

Vector Field Open Neighborhood OVERDETERMINED System Local Solvability Singular Fibre
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media New York 1996