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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A Laurent expansion for solutions to elliptic equations

Authors: Reese Harvey and John C. Polking
Journal: Trans. Amer. Math. Soc. 180 (1973), 407-413
MSC: Primary 35C10; Secondary 35J30
MathSciNet review: 0320494
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Abstract: Let $ P(\xi )$ be a homogeneous elliptic polynomial of degree $ m$. Let $ E$ be a fundamental solution for the partial differential operator $ P(D)$. Suppose $ \Omega $ is a neighborhood of 0 in $ {{\mathbf{R}}^n}$. Suppose $ f \in {C^\infty }(\Omega \sim \{ 0\} )$ satisfies $ P(D)f = 0$ in $ \Omega \sim \{ 0\} $. It is shown that there is a differential operator $ H(D)$ (perhaps of infinite order) and a function $ g \in {C^\infty }(\Omega )$ satisfying $ P(D)g = 0$ in $ \Omega $, such that $ f = H(D)E + g$ in $ \Omega \sim \{ 0\} $. This analog of the Laurent expansion for $ f$ is made unique by requiring that the Cauchy principal value of $ H(D)E$ be equal to $ H(D)E$.

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  • [1] Alberto-P. Calderón, Integrales singulares y sus aplicaciones a ecuaciones diferenciales hiperbolicas, Cursos y Seminarios de Matemática, Fasc. 3. Universidad de Buenos Aires, Buenos Aires, 1960 (Spanish). MR 0123834
  • [2] Fritz John, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955. MR 0075429
  • [3] Mikio Sato, Theory of hyperfunctions. II, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 387–437 (1960). MR 0132392
  • [4] Laurent Schwartz, Courant associé à une forme différentielle méromorphe sur une variété analytique complexe, Géométrie différentielle. Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1953, Centre National de la Recherche Scientifique, Paris, 1953, pp. 185–195 (French). MR 0066014
  • [5] -, Théorie des distributions, Publ. Inst. Math. Univ. Strasbourg, no. 9-10, Hermann, Paris, 1966. MR 35 #730.

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Keywords: Elliptic partial differential operator, fundamental solution, Laurent expansion, homogeneous polynomials, solid harmonics, hyperfunction, analytic functional, distribution, Cauchy principal value
Article copyright: © Copyright 1973 American Mathematical Society