A general formula for fundamental solutions of linear partial differential equations with constant coefficients
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- by Gerhard May PDF
- Proc. Amer. Math. Soc. 122 (1994), 455-461 Request permission
Abstract:
In this note we present a formula which furnishes particular fundamental solutions of linear partial differential equations with constant coefficients. Our construction extends an explicit formula of König (to appear) after the procedure of Malgrange (1955-1956). The crucial point is that he works with equations rather than with estimations as in the classical proof of the Malgrange-Ehrenpreis theorem. Following his ideas, we obtain fundamental solutions which are regular in the sense of Hörmander (1983); they are of basic importance. Our formula is as explicit as the zeros of a polynomial in one variable are explicit as functions of the coefficients.References
- Lars Hörmander, Local and global properties of fundamental solutions, Math. Scand. 5 (1957), 27–39. MR 93636, DOI 10.7146/math.scand.a-10486
- Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
- Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
- Heinz König, An explicit formula for fundamental solutions of linear partial differential equations with constant coefficients, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1315–1318. MR 1197539, DOI 10.1090/S0002-9939-1994-1197539-8
- Bernard Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 271–355 (French). MR 86990
- Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 122 (1994), 455-461
- MSC: Primary 35E05
- DOI: https://doi.org/10.1090/S0002-9939-1994-1211585-7
- MathSciNet review: 1211585