## An existence result for linear partial differential equations with $C^\infty$ coefficients in an algebra of generalized functions

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- by Todor Todorov PDF
- Trans. Amer. Math. Soc.
**348**(1996), 673-689 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. Lewyâs equation has solutions whenever its right-hand side is a classical $C^\infty$-function.## References

- N. Aronszajn,
*Preliminary notes for the talk âTraces of analytic solutions of the heat equationâ*, Colloque International CNRS sur les Ăquations aux DĂ©rivĂ©es Partielles LinĂ©aires (Univ. Paris-Sud, Orsay, 1972) AstĂ©risque, vol. 2, Soc. Math. France, Paris, 1973, pp.Â 5â34. 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.Â 4, 983â1005. MR**390464**, DOI 10.2307/2373684 - Jean-FranĂ§ois Colombeau,
*New generalized functions and multiplication of distributions*, North-Holland Mathematics Studies, vol. 84, North-Holland Publishing Co., Amsterdam, 1984. Notas de MatemĂĄtica [Mathematical Notes], 90. MR**738781** - â,
*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-FranĂ§ois Colombeau, Arnaud Heibig, and Michael Oberguggenberger,
*Le problĂšme de Cauchy dans un espace de fonctions gĂ©nĂ©ralisĂ©es. I*, C. R. Acad. Sci. Paris SĂ©r. I Math.**317**(1993), no.Â 9, 851â855 (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.Â 5(275), 3â40, 222 (Russian); English transl., Russian Math. Surveys**45**(1990), no.Â 5, 1â49. MR**1084986**, DOI 10.1070/RM1990v045n05ABEH002683 - Yu. V. Egorov,
*Generalized functions and linear differential equations*, Vestnik Moskov. Univ. Ser. I Mat. Mekh.**2**(1990), 92â95 (Russian). MR**1064933** - Cahit Arf,
*Untersuchungen ĂŒber reinverzweigte Erweiterungen diskret bewerteter perfekter KĂ¶rper*, J. Reine Angew. Math.**181**(1939), 1â44 (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), 771â782. 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 rĂ©gularitĂ© des solutions des Ă©quations Ă coefficients constants*, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine (N.S.)**3(51)**(1959), 433â440 (French). MR**161188** - Edward Nelson,
*Internal set theory: a new approach to nonstandard analysis*, Bull. Amer. Math. Soc.**83**(1977), no.Â 6, 1165â1198. 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 Ă©quation aux dĂ©rivĂ©es partielles sans solutions dans lâespace des hyperfonctions*, C. R. Acad. Sci. Paris SĂ©r. A-B**265**(1967), A665âA667 (French). MR**221060** - Charles Hopkins,
*Rings with minimal condition for left ideals*, Ann. of Math. (2)**40**(1939), 712â730. MR**12**, DOI 10.2307/1968951 - Todor Todorov,
*Pointwise kernels of Schwartz distributions*, Proc. Amer. Math. Soc.**114**(1992), no.Â 3, 817â819. MR**1081703**, DOI 10.1090/S0002-9939-1992-1081703-0 - â,
*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. 107â121. - FranĂ§ois TrĂšves,
*On local solvability of linear partial differential equations*, Bull. Amer. Math. Soc.**76**(1970), 552â571. MR**257550**, DOI 10.1090/S0002-9904-1970-12443-0

## Additional Information

**Todor Todorov**- Affiliation: Department of Mathematics, California Polytechnic State University, San Luis Obispo, California 93407
- Email: ttodorov@oboe.calpoly.edu
- Received by editor(s): March 7, 1994
- Received by editor(s) in revised form: November 28, 1994
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**348**(1996), 673-689 - MSC (1991): Primary 35A05, 35D05, 35E20, 46S10, 46S20
- DOI: https://doi.org/10.1090/S0002-9947-96-01450-X
- MathSciNet review: 1316863