Unconstrained Polarization (Chebyshev) Problems: Basic Properties and Riesz Kernel Asymptotics
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Unconstrained Polarization (Chebyshev) Problems: Basic Properties and Riesz Kernel Asymptotics Douglas P. Hardin1 · Mircea Petrache2 · Edward B. Saff1 Received: 3 April 2019 / Accepted: 26 August 2020 / © Springer Nature B.V. 2020
Abstract We introduce and study the unconstrained polarization (or Chebyshev) problem which requires to find an N -point configuration that maximizes the minimum value of its potential over a set A in p -dimensional Euclidean space. This problem is compared to the constrained problem in which the points are required to belong to the set A . We find that for Riesz kernels 1/|x − y|s with s > p − 2 the optimum unconstrained configurations concentrate close to the set A and based on this fundamental fact we recover the same asymptotic value of the polarization as for the more classical constrained problem on a class of d -rectifiable sets. We also investigate the new unconstrained problem in special cases such as for spheres and balls. In the last section we formulate some natural open problems and conjectures. Keywords Maximal Riesz polarization · Unconstrained polarization · Chebyshev constant · Riesz potential Mathematics Subject Classification (2010) Primary: 31C15, 31C20 ; Secondary: 30C80.
1 Introduction and Statement of Main Results Let A, B be two non-empty sets, and K : B × A → (−∞, +∞] be a kernel (or pairwise potential). For N ∈ N we consider the max-min optimization problem PK (A, ωN ) := inf
y∈A
N
K(xi , y),
PK (A, B, N ) := sup PK (A, ωN ),
i=1
Edward B. Saff
[email protected] Douglas P. Hardin [email protected] Mircea Petrache [email protected] 1
Vanderbilt University, Nashville, TN, USA
2
Pontificia Universidad Catolica de Chile, Santiago, Chile
ωN ⊂B
(1.1)
D.P. Hardin et al.
where the maximum is taken over N -point multisets ωN = {x1 , . . . , xN } ⊂ B . (Note that a multiset is a list where elements can be repeated.) The determination of Eq. 1.1 is called the two-plate polarization (or Chebyshev) problem (see Proposition 1.4 below for the link to the theory of Chebyshev polynomials, justifying this name). For background and motivation of the study of polarization problems, see [9, Chapter 14]. If A ⊂ A and B ⊂ B we note the basic monotonicity properties
PK (A , B, N ) PK (A, B, N ),
PK (A, B , N ) PK (A, B, N ).
(1.2)
The case A = B of Eq. 1.1, also known as the single-plate polarization (or Chebyshev) problem for A , has been the more studied so far (see [9, 16, 29]); and for it we introduce the notation PK (A, N ) := PK (A, A, N ). (1.3) A related quantity is the value of the minimum N -point K -energy,1 given by
EK (A, N ) := inf
ωN ⊂A
N N
K(xi , xj ).
(1.4)
i=1 j :j =i j =1
If N 2 , A ⊂ B are compact sets and K : B × B → (−∞, +∞] is a symmetric function, we have the following relation between the above quantities (see [9, Prop. 14.1.1], [16, Thm. 2.3])
EK (A, N + 1) EK (A, N ) . (1.5) N +1 N −1 The goal of this article is to study the case A ⊂ B = Rp of Eq. 1.1, in which the configurations ωN are
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