Abstract
Starting from first principles, all fundamental solutions (that are tempered distributions) for scalar elliptic operators are identified in this chapter. While the natural starting point is the Laplacian, this study encompasses a variety of related operators, such as the bi-Laplacian, the poly-harmonic operator, the Cauchy-Riemann operator, the Dirac operator, as well as general second order constant coefficient strongly elliptic operators. Having accomplished this task then makes it possible to prove the well-posedness of the Poisson problem (equipped with a boundary condition at infinity), and derive qualitative/quantitative properties for the solution. Along the way, Cauchy-like integral operators are also introduced and their connections with Hardy spaces is brought to light in the setting of both complex and Clifford analysis.
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Notes
- 1.
Named after the French mathematician and astronomer Pierre Simon de Laplace (1749–1827.)
- 2.
In the case n = 3, the expression for E was used in 1789 by Pierre Simon de Laplace to show that for f smooth, compactly supported, △(E ∗ f) = 0 outside the support of f (cf. [38]).
- 3.
In 1813, the French mathematician, geometer, and physicist Siméon Denis Poisson (1781–1840) proved that △(E ∗ f) = f in dimension n = 3 (cf. [55]), where \(E(x) = - \frac{1} {4\pi \vert x\vert }\), \(x \in {\mathbb{R}}^{3} \setminus \{ 0\}\), and \(f \in C_{0}^{\infty }({\mathbb{R}}^{3})\).
References
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K. Hoffman, Banach Spaces of Analytic Functions, Dover Publications, 2007.
P. S. Laplace, Mémoire sur la théorie de l’anneau de Saturne, Mém. Acad. Roy. Sci. Paris (1787/1789), 201–234.
M. Mitrea, Clifford Wavelets, Singular Integrals, and Hardy Spaces, Lecture Notes in Mathematics, 1575, Springer-Verlag, Berlin, 1994.
S. D. Poisson, Remarques sur une équation qui se présente dans la théorie de l’attraction des sphéroides, Bulletin de la Société Philomathique de Paris, 3 (1813), 388–392.
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Mitrea, D. (2013). The Laplacian and Related Operators. In: Distributions, Partial Differential Equations, and Harmonic Analysis. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8208-6_7
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DOI: https://doi.org/10.1007/978-1-4614-8208-6_7
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