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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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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