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
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.
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Treves, F. (1996). Local Solvability in a Class of Overdetermined Systems of Linear PDE. In: Hörmander, L., Melin, A. (eds) Partial Differential Equations and Mathematical Physics. Progress in Nonlinear Differential Equations and Their Applications, vol 21. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0775-7_22
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DOI: https://doi.org/10.1007/978-1-4612-0775-7_22
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