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Volume 43 Number 4
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Volume 43 Number 3 Volume 43 Number 4

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Research articles

ScienceAsia 43(2017): 267-274 |doi: 10.2306/scienceasia1513-1874.2017.43.267


Constant Riesz potentials on a circle in a plane with an application to polarization optimality problems


Nattapong Bosuwana,b, Pornrat Ruengrotc,*

 
ABSTRACT:     A characterization for a Riesz s-potential function of a multiset ωN of N points in ℝ2 is given when s=2−2N and the potential function is constant on a circle in ℝ2. The characterization allows us to partially prove a conjecture of Nikolov and Rafailov that if the potential function is constant on a circle Γ then the points in ωN should be equally spaced on a circle concentric to Γ. As an application of constant Riesz s-potential functions, we also find all maximal and minimal polarization constants and configurations of two concentric circles in ℝ2 for certain values of s.

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a Department of Mathematics, Faculty of Science, Mahidol University, Rama 6 Road, Ratchathewi District, Bangkok 10400 Thailand
b Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400 Thailand
c Mahidol University International College, 999 Phutthamonthon 4 Road, Salaya, Nakhonpathom 73170 Thailand

* Corresponding author, E-mail: pornrat.rue@mahidol.edu

Received 22 Jun 2017, Accepted 5 Sep 2017