Local Solvability in a Class of Overdetermined Systems of Linear PDE

  • François Treves
Conference paper
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 21)


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
$$Lu = f$$
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 fC (Ω), there is a distribution uD′(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 fD′(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.


Vector Field Open Neighborhood OVERDETERMINED System Local Solvability Singular Fibre 
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Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • François Treves
    • 1
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA

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