Regularization and derivatives of multipole potentials

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• 作者：Ricardo Estrada ^{restrada@math.lsu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Harmonic functions ; Harmonic polynomials ; Distributions ; Multipoles
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：770-785
• 全文大小：447 K

文摘

The only harmonic homogeneous functions defined in Rⁿ∖{0}Container hidden">${\mathbb{R}}^n\setminus \left\{\mathbf{0}\right\}$ are the harmonic polynomials and the so-called multipole potentials, namely functions of the type P(x)=p(x)/|x|^2k+n−2$P\left(\mathbf{x}\right)=p\left(\mathbf{x}\right)/{\left|\mathbf{x}\right|}^{2k+n-2}$ for some harmonic polynomial p of degree k  . The first aim of this article is to study the distributional regularization of multipole potentials. We show that even though the Hadamard regularization Pf(p(x)/|x|^2k+n−2)$\mathcal{P}f(p\left(\mathbf{x}\right)/{\left|\mathbf{x}\right|}^{2k+n-2})$ exists for any homogeneous polynomial of degree k  , the principal value p.v.(p(x)/|x|^2k+n−2)$\mathrm{p}.\mathrm{v}.(p\left(\mathbf{x}\right)/{\left|\mathbf{x}\right|}^{2k+n-2})$ exists if and only if p is harmonic; this means that if p is harmonic then for any test function ϕ the divergent   integral ${\int }_{{\mathbb{R}}^n}p\left(\mathbf{x}\right)\varphi \left(\mathbf{x}\right)/{\left|\mathbf{x}\right|}^{2k+n-2}\mathrm{d}\mathbf{x}$ can be computed by employing polar coordinates and performing the angular integral first. We also find the first and second order distributional derivatives of these regularizations and, more generally, of the regularizations of functions of the form P_l(x)=p(x)/|x|^k+l$P_l\left(\mathbf{x}\right)=p\left(\mathbf{x}\right)/{\left|\mathbf{x}\right|}^{k+l}$. We find many interesting formulas that hold precisely when p is a harmonic polynomial of degree k. In particular, we prove that generalizing the well known relation $\bar{\mathrm{\Delta }}\left(r^{2-n}\right)=(2-n)C\delta \left(\mathbf{x}\right)$, where C is the area of a sphere of radius 1. Actually formulas like this one hold for a homogeneous polynomial p of degree k if and only if p is harmonic.