Abstract
We prove the existence of a solution for a class of linear partial differential equations with smooth coefficients in a space of generalized functions larger than the space of Schwartz distributions. As an example, we show that H. Lewy’s equation has a solution in this space whenever its right hand side is a classical smooth function or a Schwartz distribution.
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References
J.F. Colombeau, “New Generalized Functions and Multiplication of Distributions”, North-Holland Math. Studies 84, 1984.
J.F. Colombeau, “New General Existence Result for Partial Differential Equations in the C ∞ Case”, Preprint of the Universite de Bordeaux, 1984.
J.F. Colombeau, A. Heibig, M. Oberguggenberger, “Generalized solutions to partial differential equations of evolution type”, Preprint of Ecole Normale Superieure de Lyon, France, 1991.
N. Dunford and J.T. Schwartz, “Linear Operators, Part I: General Theory”, Interscience Publishers, Inc. New York, 1958.
L. Ehrenpreis, “Solutions of some problems of division III”, Am. J. Math., 78(1956), p. 685.
L. Hörmander, “Linear partial differential operators”, Grund. Math. Wis., 116 (1963), Springer Verlag.
H. Lewy, “An example of a smooth linear partial differential equation without solution”, Annals of Mathematics, Vol. 66, No. 1, July, 1957.
T. Lindstrøm, “An Invitation to Nonstandard Analysis” in “Nonstandard Analysis and its Applications”, edited by Nigel Cutland, London Mathematical Society Student Texts 10, Cambridge University Press, 1988.
B. Malgrange, “Sur la propagation de la regularite des solutions des equations a coefficients constants”, Bull. Math. Soc. Sci. Math. Phys. R.P. Romuanie 3 (53) (1959), 433–440.
P. Schapira, Une equation aux derivees partielles sans solutions dans l’espace des hyperfonctions, C.R. Acad. Sci. Paris Ser. A-B 265 (1967), A665–A667. MR 36, #4112.
L. Schwartz, “Theorie des Distributions I, II”, Herman, Paris 1950, 1951.
T. Todorov, “Nonstandard Delta Function”, Proceedings of the American Mathematical Society, Volume 110, N o 4, December 1990, 1143–1144.
T. Todorov, “Pointwise Kernels of Schwartz Distributions”, Proceedings of the American Mathematical Society, Vol. 114, N o 3, March 1992, 817–819.
F. Treves, “On Local Solvability of Linear Partial Differential Equations”, Bulletin of the American Mathematical Society, Vol. 76, N o 3, May, 1970, 552–571.
F. Treves, “Ovcyannikov theorem and hyperdifferential operators”, Notas de Matematica, no. 46. Inst. Mat. Pura. Apl. con. Nac. Pesquisas, Rio de Janeiro, 1968.
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Todorov, T. (1995). An Existence Result for a Class of Partial Differential Equations with Smooth Coefficients. In: Albeverio, S.A., Luxemburg, W.A.J., Wolff, M.P.H. (eds) Advances in Analysis, Probability and Mathematical Physics. Mathematics and Its Applications, vol 314. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8451-7_9
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DOI: https://doi.org/10.1007/978-94-015-8451-7_9
Publisher Name: Springer, Dordrecht
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