A general formula for fundamental solutions of linear partial differential equations with constant coefficients. (English) Zbl 0808.35016
Summary: 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.
MSC:
| 35E05 | Fundamental solutions to PDEs and systems of PDEs with constant coefficients |
| 35D10 | Regularity of generalized solutions of PDE (MSC2000) |
Keywords:
Malgrange-Ehrenpreis theoremReferences:
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| [2] | 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. Lars Hörmander, The analysis of linear partial differential operators. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 257, Springer-Verlag, Berlin, 1983. Differential operators with constant coefficients. |
| [3] | 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. Lars Hörmander, The analysis of linear partial differential operators. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 257, Springer-Verlag, Berlin, 1983. Differential operators with constant coefficients. |
| [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. · Zbl 0792.35021 |
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