Green energy and extremal decompositions Текст научной статьи по специальности «Математика»

38

Probl. Anal. Issues Anal. Vol. 8(26), No3, 2019, pp. 38-44

DOI: 10.15393/j3.art.2019.6730

UDC 517.956.224

V. N. Dubinin

GREEN ENERGY AND EXTREMAL DECOMPOSITIONS

Abstract. We give two precise estimates for the Green energy of a discrete charge, concentrated in an even number of points on the circle, with respect to the concentric ring. The lower estimate for the Green energy is attained for the points with a nonstandard symmetry. The well-known Polya-Schur inequality for the logarithmic energy is a special case of this estimate. The proof is based on the application of dissymmetrization and an asymptotic formula for the conformal capacity of a generalized condenser in the case when some of its plates contract to given points. The upper bound is established for a charge that takes values of opposite signs. Its proof reduces to solving a problem on the so-called extremal decomposition of a circular ring with free poles on a circle. Key words: Green energy, discrete charge, dissymmetrization, extremal decompositions

2010 Mathematical Subject Classification: 31A15

1. Introduction and statement of results. There are many studies related to the extremal problems for different kinds of energy of a discrete charge (see, e.g., the papers [2-4], [9], [10], [12], and the references therein). In contrast to the previous research, we consider the Green energy (see also the recent articles [1] and [5]). In addition, an extremal problem with an alternating charge is studied.

Let 0k, k = 1,..., 2n, be real numbers, such that

0i <02 < ... <d2n <d1 + 2n (n ^ 2), and let Z = {zk}k=1, zk = exp(i0k), k = 1,..., 2n. Denote by

2n 2n

E (Z,B) = ^^ gB (zk ,zi)

k=i i=i

l=k

© Petrozavodsk State University, 2019

the Green energy of the collection Z with respect to the ring B := {z : Ri < |z| < R2}, 0 < Ri < 1 < R2 < ro. Here gB(z,zk) is the Green function of the domain B with poles at zk, k = 1,..., 2n. The following statement is valid.

Theorem 1. For any collection of points Z, and for any numbers Ri, R2, the inequality

E(Z,B) ^ E(Z*,B) (1)

holds, where Z* = {z*}2^, and the symmetrically located points z* are defined by the relations |z*| = 1, argz*j-i = —n/(2n) + 2nj/n,

n

arg z*j = n/(2n) + ^jV^ j = 1, . . . , n, V = X^J - d2j-i).

j=i

The proof of Theorem 1 is based on the theory of the condenser capacities and dissymmetrization. These ideas go back to the book [4]. By taking the limit n ^ 0 in (1), we obtain an inequality for the Green energy of a charge concentrated in an unrestricted number of points (not necessary even). Passing to the limit Ri ^ 0, R2 ^ ro, inequality (1) gives the classical Polya-Schur inequality

n n n n

nn |zk - zi| ^ nn |z* - z*l = nn,

k=i l=i k=il=i

l=k l=k

where zk, k = 1,..., n, are some points on the unit circle |z| = 1 and z* = exp(2nik/n), k = 1,..., n.

Theorem 2. Let Z and B be as above. Then, for the discrete Green energy

2 n 2 n

E(Z,B) := ^^(-1)k+lgB(zk,zi),

k=i l=i

l=k

we have

E(Z,B) ^ E(Z*,B), (2)

where Z* = {zk}k=i, z* = exp(nik/n), k = 1,..., 2n.

The proof of inequality (2) is carried out by reduction to the extremal decomposition problems, which have a long history and many applications [6], [11]. A significant contribution to the solution of such problems was

made by N. A. Lebedev [13]. By taking the limit R1 ^ 0, R2 ^ <x> in (2), we have

n n 2

n n izk - *i<-i)'+' > (n)n.

k=11=1

l=k

This inequality has been proved in [7] (see also [6, p. 127]).

2. Proof of Theorem 1. The notions and notation from the book [6] will be used in the proof below. For sufficiently small r > 0 the condenser

C*(r) := (B, {Ek}k=o, A*),

where E0 = dB, Ek = {z : |z - z*k | ^ r}, k = 1,..., 2n, and A* = = {0,1,..., 1} is well defined [6, p. 33]. Let $ be the group of symmetries of C, formed by the compositions of the reflections with respect to the rays {z : argz = 2nk/n}, k = 1,... ,n and the bisectors of the angles formed by these rays ("dihedral group"). The condenser C*(r) is symmetric with respect to the group $ (^-symmetric). By Lemma 4.3 from [6], there exist a dissymmetrization Dis, such that Dis E* = E. Here

n

h*

F* I I J II 1 n . 2nj' n 2nj\

E =U (z : |z| = 1, - 2n + ^ arg z ^ 2n +

j=1 V ^

n

E = U {z : |z| = 1, 02j-i ^ arg z ^ d2j} . j=i

According to Theorem 4.14 [6], we obtain the inequality for the capacities:

cap C* (r) ^ cap Dis C*(r). (3)

It is clear that

Dis C *(r) = (B,{Ek }kn=o,A*),

where E0 = dB and Ek = {z : |z — zk| ^ r}, k = 1,..., 2n.

In view of Theorem 2.1 from [6], the following asymptotic equalities

hold as r ^ 0:

n*( \ 4nn

cap C (r) = —

log r

2n 12 12 —g iog nB,zk)+e (z *-B^(iog^) +

capDis C * (r) = —

log r

12 12 Xlogr(B,zfc) + E(Z,B)}(—) + 0(' N

- log r/ VVlog r

where r(B,zk) is the inner radius of B with respect to the point zk, k = 1,..., n. Note that

r(B, z* ) = r(B,zk), k = 1,..., 2n.

Substituting these equalities in (3), we obtain the required inequality (1). 3. Proof of Theorem 2. Consider at the condenser

C(r) = (B, {Ek}k=o, A),

where B, Ek, k = 0,1,..., 2n, as in Section 2, A = {0,1, -1,1,..., -1}. Let u be the potential function of C(r) [6, p. 13]. In view of the asymptotic formula (2.10) [6], for r ^ 0

[ |Vu|2 = cap C(r) = -14^-

log r

B\ufc=iEfc

2n 12 12 -glogr<B-zk) + E<Z-B>}(logr) + «(Gog;))• (4)

Consider the family of functions {uk}|=i that are defined in the sectors Dk, Dk = {z G B : 0k < arg z < 0k+i}, k = 1,..., 2n, respectively ($2n+i = + 2n). For every k, the function uk is continuous in Dk, harmonic in Dk\(Ek U Ek+i),

(-1)k+l, z G Ek n Dk;

uk(z)= <{ (-1)k+2, z G Ek+i n Dk;

.0, z G (dDk) n (dB),

and satisfies conditions: duk/dn = 0 on (dDk) n B\(Ek U Ek+i), k = 1,..., 2n. By Dirichlet's principle,

J |Vu|2 > J |Vuk |2, k =1,..., 2n.

Dk\(EfcUEfc+i) Dk\(Ek UEfc+i)

The symmetry principle for the harmonic functions gives

J |Vufc |2 = J |2,

Dk\(Ek UEk+i) Dk\Hk

where Hk = {z : |z — (k\ ^ r}, (k = exp(i(0k+1 — 6k)/2), and the function uk is continuous in Dk, harmonic in Dk\Hk, is equal to 1 on Hk and to zero on dDk, k = 1,..., 2n. Once again, using formula (2.10) [6], we conclude that

J ^nf ^ j |V^k|2 =

Dk\(Ek UEk+i) Dk\Hk

= — ionr— 2n|logr(Dk)](¡sgr)2+o((iogr)2)'r ^0' (5)

k = 1,..., 2n. Note that a suitable branch of the logarithm w = log(z/Zk) maps the sector Dk conformally and univalently onto the rectangle Gk = {w : log R1 < Re w < log R2, |Im w| < (dk+1 — 9k )/2}, k = 1,..., 2n. The result of the Marcus radial averaging transformation [14] of the family {Gk}k=1 (with weights ak = 1/(2n), k = 1,..., 2n) belongs to the rectangle G = {w : logR1 < Re w < log R2, |Imw| < n/(2n) (also, see [6, p. 83] and [8]). By the Marcus theorem,

2n 2n

n r(Dk, Zk) = 11 r(Gk, 0) ^ r2n(G, 0).

k=1 k=1 Taking into account (4) and (5), we find

2n

J]logr(B,zk) + E(Z,B) ^ 2nlogr(G, 0). k=1

It is straightforward to see that in the case zk = exp(nik/n), k = 1,..., 2n, we have the equality sign in the last relation. This yields the required inequality.

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Received July 18, 2019. In revised form, October 14, 2019. Accepted October 30, 2019. Published online November 7, 2019.

Far Eastern Federal University

8 Sukhanov str., Vladivostok, 690090, Russia

Institute of Applied Mathematics, Far Eastern Branch of RAS

7 Radio str., Vladivostok, 690041, Russia

E-mail: dubinin@iam.dvo.ru