کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4613843 1339273 2017 16 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Regularization and derivatives of multipole potentials
ترجمه فارسی عنوان
تنظیم و مشتقات پتانسیل های چندقطبی
کلمات کلیدی
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات آنالیز ریاضی
چکیده انگلیسی

The only harmonic homogeneous functions defined in Rn∖{0}Rn∖{0} are the harmonic polynomials and the so-called multipole potentials, namely functions of the type P(x)=p(x)/|x|2k+n−2P(x)=p(x)/|x|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)Pf(p(x)/|x|2k+n−2) exists for any homogeneous polynomial of degree k  , the principal value p.v.(p(x)/|x|2k+n−2)p.v.(p(x)/|x|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 ∫Rnp(x)ϕ(x)/|x|2k+n−2dx 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 Pl(x)=p(x)/|x|k+lPl(x)=p(x)/|x|k+l. We find many interesting formulas that hold precisely when p is a harmonic polynomial of degree k. In particular, we prove thatΔ‾p.v.(p(x)r2k+n−2)=(−1)k+1πn/22k−2Γ(n2+k−1)p(∇)δ(x), generalizing the well known relation Δ‾(r2−n)=(2−n)Cδ(x), 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.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Mathematical Analysis and Applications - Volume 446, Issue 1, 1 February 2017, Pages 770–785
نویسندگان
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