An existence result for linear partial differential equations with 𝐶^{∞} coefficients in an algebra of generalized functions

Todorov, Todor

doi:10.1090/S0002-9947-96-01450-X

An existence result for linear partial differential equations with $C^\infty$ coefficients in an algebra of generalized functions
HTML articles powered by AMS MathViewer

by Todor Todorov

Trans. Amer. Math. Soc. 348 (1996), 673-689

DOI: https://doi.org/10.1090/S0002-9947-96-01450-X

PDF | Request permission

Abstract:

We prove the existence of solutions for essentially all linear partial differential equations with $C^\infty$-coefficients in an algebra of generalized functions, defined in the paper. In particular, we show that H. Lewy’s equation has solutions whenever its right-hand side is a classical $C^\infty$-function.

References

N. Aronszajn, Preliminary notes for the talk “Traces of analytic solutions of the heat equation”, Colloque International CNRS sur les Équations aux Dérivées Partielles Linéaires (Univ. Paris-Sud, Orsay, 1972) Astérisque, vol. 2, Soc. Math. France, Paris, 1973, pp. 5–34. MR 0603289
M. S. Baouendi, Solvability of partial differential equations in the traces of analytic solutions of the heat equation, Amer. J. Math. 97 (1975), no. 4, 983–1005. MR 390464, DOI 10.2307/2373684
Jean-François Colombeau, New generalized functions and multiplication of distributions, North-Holland Mathematics Studies, vol. 84, North-Holland Publishing Co., Amsterdam, 1984. Notas de Matemática [Mathematical Notes], 90. MR 738781
—, New general existence result for partial differential equations in the $C^\infty$ case, Universite de Bordeaux, preprint, 1984.
J. F. Colombeau, A. Heibig, and M. Oberguggenberger, Generalized solutions to partial differential equations of evolution type, Ecole Norm. Sup., 1991, preprint.
Jean-François Colombeau, Arnaud Heibig, and Michael Oberguggenberger, Le problème de Cauchy dans un espace de fonctions généralisées. I, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 9, 851–855 (French, with English and French summaries). MR 1246652
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
Yu. V. Egorov, On the theory of generalized functions, Uspekhi Mat. Nauk 45 (1990), no. 5(275), 3–40, 222 (Russian); English transl., Russian Math. Surveys 45 (1990), no. 5, 1–49. MR 1084986, DOI 10.1070/RM1990v045n05ABEH002683
Yu. V. Egorov, Generalized functions and linear differential equations, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2 (1990), 92–95 (Russian). MR 1064933
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
A. Kaneko, Introduction to hyperfunctions, Mathematics and its Applications (Japanese Series), vol. 3, Kluwer Academic Publishers Group, Dordrecht; SCIPRESS, Tokyo, 1988. Translated from the Japanese by Y. Yamamoto. MR 1026013
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
Nigel Cutland (ed.), Nonstandard analysis and its applications, London Mathematical Society Student Texts, vol. 10, Cambridge University Press, Cambridge, 1988. Papers from a conference held at the University of Hull, Hull, 1986. MR 971063, DOI 10.1017/CBO9781139172110
Bernard Malgrange, Sur la propagation de la régularité des solutions des équations à coefficients constants, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine (N.S.) 3(51) (1959), 433–440 (French). MR 161188
Edward Nelson, Internal set theory: a new approach to nonstandard analysis, Bull. Amer. Math. Soc. 83 (1977), no. 6, 1165–1198. MR 469763, DOI 10.1090/S0002-9904-1977-14398-X
Abraham Robinson, Non-standard analysis, North-Holland Publishing Co., Amsterdam, 1966. MR 0205854
Elemer E. Rosinger, Nonlinear partial differential equations, North-Holland Mathematics Studies, vol. 164, North-Holland Publishing Co., Amsterdam, 1990. An algebraic view of generalized solutions. MR 1091547
M. Oberguggenberger, Multiplication of distributions and applications to partial differential equations, Pitman Research Notes in Mathematics Series, vol. 259, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. MR 1187755
M. Oberguggenberger and E. E. Rosinger, Solution of continuous nonlinear PDEs through order completion, North-Holland Math. Studies, vol. 181, North-Holland, Amsterdam, 1994.
Pierre Schapira, Une équation aux dérivées partielles sans solutions dans l’espace des hyperfonctions, C. R. Acad. Sci. Paris Sér. A-B 265 (1967), A665–A667 (French). MR 221060
Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
Todor Todorov, Pointwise kernels of Schwartz distributions, Proc. Amer. Math. Soc. 114 (1992), no. 3, 817–819. MR 1081703, DOI 10.1090/S0002-9939-1992-1081703-0
—, An existence result for a class of partial differential equations with smooth coefficients, Advances in Analysis, Probability and Mathematical Physics; Contributions to Nonstandard Analysis, Vol. 314, (S. Albeverio, W. A. J. Luxemburg, M. P. H. Wolff, eds.), Kluwer Academic, Dordrecht, 1995, pp. 107–121.
François Trèves, On local solvability of linear partial differential equations, Bull. Amer. Math. Soc. 76 (1970), 552–571. MR 257550, DOI 10.1090/S0002-9904-1970-12443-0