Global solvability for first order real linear partial differential operators |
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Authors: | José R dos Santos Filho Maurício Fronza da Silva |
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Institution: | a Departamento de Matemática, Universidade Federal de São Carlos, SP, Brazil b Departamento de Matemática, Universidade Federal de Santa Maria, RS, Brazil |
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Abstract: | F. Treves, in 17], using a notion of convexity of sets with respect to operators due to B. Malgrange and a theorem of C. Harvey, characterized globally solvable linear partial differential operators on C∞(X), for an open subset X of Rn.Let P=L+c be a linear partial differential operator with real coefficients on a C∞ manifold X, where L is a vector field and c is a function. If L has no critical points, J. Duistermaat and L. Hörmander, in 2], proved five equivalent conditions for global solvability of P on C∞(X).Based on Harvey-Treves's result we prove sufficient conditions for the global solvability of P on C∞(X), in the spirit of geometrical Duistermaat-Hörmander's characterizations, when L is zero at precisely one point. For this case, additional non-resonance type conditions on the value of c at the equilibrium point are necessary. |
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Keywords: | 35A05 35F05 |
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