International Journal of Mathematical Analysis
Vol. 9, 2015, no. 56, 2763 - 2773
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ijma.2015.510259
Extremal Decomposition Problems in
the Euclidean Space
K.A. Gulyaeva
Far Eastern Federal University, Vladivostok, Russia
S.I. Kalmykov
Far Eastern Federal University, Vladivostok, Russia
&
Institute of Applied Mathematics
Far Eastern Branch of the Russian Academy of Sciences, Vladivostok, Russia
E.G. Prilepkina
Far Eastern Federal University, Vladivostok, Russia
&
Institute of Applied Mathematics
Far Eastern Branch of the Russian Academy of Sciences, Vladivostok, Russia
c 2015 K.A. Gulyaeva, S.I. Kalmykov and E.G. Prilepkina. This article is
Copyright distributed under the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Abstract
Composition principles for reduced moduli are extended to the case
of domains in the n-dimensional Euclidean space, n > 2. As a consequence analogues of extremal decomposition theorems of Kufarev, Dubinin and Kirillova in the planer case are obtained.
Mathematics Subject Classification: 31B99
Keywords: reduced modulus, Robin function, Neumann function,
nonoverlapping domain, extremal decomposition problem.
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K.A. Gulyaeva, S.I. Kalmykov and E.G. Prilepkina
Introduction and notations
Extremal decomposition problems have a rich history and go back to
M.A. Lavrentiev’s inequality for the product of conformal radii of nonoverlapping domains. There exist two methods of their study: the extremalmetric method and the capacitive method. The first one has been systematically developed in papers by G.V. Kuz’mina, E.G. Emel’yanov, A.Yu. Solynin,
A. Vasil’ev, and Ch. Pommerenke [9, 14, 6, 11]. The second approach is developed mainly in works of V.N. Dubinin and his students [4, 5, 2, 3]. In
particular, a series of well-known results about extremal decomposition follows one way from composition principles for generalized reduced moduli (see
[1, p. 56] and [12]). In the present paper we extend the mentioned composition
principles to the case of spatial domains. As a consequence we get theorem
about extremal decomposition for the harmonic radius [7] obtained earlier in
[5].
Throughout the paper, Rn denotes the n-dimensionalpEuclidean space consisting of points x = (x1 , . . . , xn ), n ≥ 3, and |x| = x21 + · · · + x2n is the
length of a vector x ∈ Rn . We introduce the following notations:
B(a, r) = {x ∈ Rn : |a − x| < r},
S(a, r) = {x ∈ Rn : |a − x| = r}, a ∈ Rn ;
ωn−1 = 2π n/2 /Γ(n/2) is the area of the unit sphere S(0, 1);
λn = ((n − 2)ωn−1 )−1 .
D is a bounded domain in Rn , Γ is a closed subset of ∂D. The pair (D, Γ) is
admissible if there exists the Robin function, gΓ (z, z0 , D) harmonic in D \ {z0 },
continuous in D \ {z0 } and
∂gΓ
= 0 on (∂D) \ Γ,
∂n
gΓ = 0 on Γ,
(1)
(2)
and in a neighborhood of z0 there is an expansion
gΓ (z, z0 , D) = λn |z − z0 |2−n − r(D, z0 , Γ)2−n + o(1) , z → z0 ,
(3)
where ∂/∂n means the inward normal derivative on the boundary. In what
follows all such pairs are assumed to be admissible.
In the case Γ = ∅ we change the condition (1) by the condition
∂gΓ
1
=
on ∂D,
∂n
µn−1 (∂D)
where µn−1 (∂D) is the area of boundary.
By analogy with the definition of the Robin radius for plain domains from
the paper [3] we will call the constant r(D, z0 , Γ) the Robin radius of the
Extremal decomposition problems in the Euclidean space
2765
domain D and the set Γ. Note that in the case of Γ = ∂D we get the harmonic
radius [7, 10, 5].
m
Let ∆ = {δk }m
k be a collection of real numbers and Z = {zk }k=1 be points
of the domain D. For Γ = ∅ we additionally require
m
X
δk = 0.
k=1
Define the potential function for the domain D, the set Γ, the collection of
points Z, and numbers ∆:
u(z) = u(z; Z, D, Γ, ∆) =
m
X
δk gΓ (z, zk , D).
k=1
Note that for Γ = ∅ the function gΓ (z, zk , D) is defined up to an additive
constant. Nevertheless, the function u(z) is defined uniquely and characterized
by the condition
∂u
= 0 on ∂D.
∂n
It is clear from the definition of the potential function that in a neighborhood of zk we have
u(z) = δk λn |z − zk |2−n + ak + o(1), k = 1, . . . , m,
where
ak = −δk λn r(D, zk , Γ)
2−n
+
m
X
δl gΓ (zl , zk , D).
l=1
l6=k
Now if we introduce the following notation
gΓ (zk , zk , D) = −λn r(D, zk , Γ)2−n ,
then the constant in the expansion of the potential function in a neighbourhood
of zk is
m
X
ak =
δl gΓ (zl , zk , D).
(4)
l=1
A function v(z) is admissible for D, Z, ∆, and Γ if v(z) ∈ Lip in a neighbourhood of S
each point of D except maybe finitely many such points, continuous in D \ m
k=1 {zk }, v(z) = 0 on Γ, and in neighbourhood of zk there is an
expansion
v(z) = δk λn |z − zk |2−n + bk + o(1), z → zk.
(5)
The Dirichlet integral is the following
Z
I(f, D) =
|∇f |2 dµ,
D
where dµ = dx1 . . . dxn .
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K.A. Gulyaeva, S.I. Kalmykov and E.G. Prilepkina
Main results
Lemma 2.1 The asymptotic formula
!
m
m
X
X
I(u, Dr ) =
δk2 λn r2−n +
δk ak + o(1), r → 0,
k=1
k=1
is true, where u is the
S potential function and ak , k = 1, . . . , m are defined
in (4) and Dr = D \ m
k=1 B(zk , r).
Proof. The Green’s identity
Z
Z
2
|∇u| dµ = −
V
u
∂V
∂u
ds
∂n
gives
m Z
X
∂u
∂u
u ds = −
I (u, Dr ) = −
u ds.
∂n
∂Dr ∂n
k=1 S(zk ,r)
Z
The second equality in (6) holds because u
(6)
∂u
= 0 on ∂D. Note that
∂n
u = δk λn r2−n + ak + o(1), z → zk
in a neighbourhood of zk .
R
∂u
ds. Let u(z) = h(z) + g(z), where
We calculate the integral S(zk ,r) u ∂n
2−n
h(z) = λn δk |z − zk |
and g(z) is a harmonic function. Note that g(zk ) = ak .
For |z − zk | = r we have the following correlations
∂h
∂g
∂h
∂g
n−1 ∂u
n−1
=r
h
+h
+g
+g
r u
∂n
∂n
∂n
∂n
∂n
= (2 − n)λ2n δk2 r2−n + rλn δk
=−
Therefore
Z
∂g
∂g
+ (2 − n)gλn δk + g rn−1
∂n
∂n
λn δk2 2−n g(zk )δk
r
−
+ o(1), r → 0.
ωn−1
ωn−1
∂u
u ds =
S(zk ,r) ∂n
Z
u
S(0,1)
∂u n−1
r ds
∂n
= −λn δk2 r2−n − δk ak + o(1), r → 0.
Substituting it in (6) we get the lemma. Extremal decomposition problems in the Euclidean space
2767
Lemma 2.2 For an admissible function v and the potential function u we
have
m
X
I(v − u, Dr ) = I(v, Dr ) − I(u, Dr ) − 2
δk (bk − ak ) + o(1), r → 0.
k=1
Proof. One may observe that
Z
I(v − u, Dr ) =
|∇v|2 + |∇u|2 − 2∇u∇v dµ
Z
ZDr
∂u
2
2
(v − u) ds
|∇v| − |∇u| dµ + 2
=
∂n
∂Dr
Dr
Z
m
X
∂u
= I(v, Dr ) − I(u, Dr ) + 2
(v − u) ds
∂n
k=1 S(zk ,r)
= I(v, Dr ) − I(u, Dr ) − 2
m
X
δk (bk − ak ) + o(1), r → 0.
k=1
R
∂u
Here we calculated the integral S(zk ,r) (v − u) ∂n
a similar way as in the proof
of lemma 2.1 and used the Green’s identity
Z
Z
∂u
(∇u · ∇v)dµ = −
v ds,
Dr
∂Dr ∂n
where n is the inner normal vector. The quantity
m
m X
m
X
X
δk ak =
δk δl gΓ (zl , zk , D)
k=1
k=1 l=1
we call the reduced modulus and denote it by M (D, Γ, Z, ∆). According to
lemma 2.1
!
!
n
X
M (D, Γ, Z, ∆) = lim I(u, Dr ) −
δk2 λn r2−n .
r→0
k=1
m
Theorem 2.3 Let sets D, Γ, collections Z = {zk }m
k=1 , ∆ = {δk }k=1 , be
as in the definition of the reduced modulus M = M (D, Γ, Z, ∆), u(z) be the
potential function for D, Γ, Z, ∆, and let Di ⊂ D be pairwise non-overlapping
i
i
subdomains of D, Γi , Zi = {zij }nj=1
, ∆i = {δij }nj=1
, be from the definition of
the reduced moduli Mi = M (Bi , Γi , Zi , ∆i ), ui (z) be the potential function for
Di , Γi , Zi , ∆i , i = 1, ..., p. Assume that the following conditions are fulfilled:
1)(D ∩ ∂D
i , i = 1, ..., p;
Spi ) ⊂ ΓS
Sp
D
;
2) Γ ⊂ S
( i=1 Γi )
Rn \
i
i=1
p
3)Z = i=1 Zi , that is each point zk ∈ Z coincides with some point
zij ∈ Zi for k = k(i, j) and vice versa;
4) δk = δij for k = k(i, j).
2768
K.A. Gulyaeva, S.I. Kalmykov and E.G. Prilepkina
Then the inequality
M≥
p
X
Mi +
i=1
p
X
I(u − ui , Di ) ≥
i=1
p
X
Mi
i=1
holds.
Proof. Consider the function
ui (z),
v(z) =
0,
z ∈ Di , S
z ∈ D \ ( pi=1 Di ) .
TheS condition 1) guarantees that the function v(z) is continuous in
D\ m
k=1 {zk }. From the conditions 2) and 3) it follows that v(z) = 0 for
z ∈ Γ and in a neighbourhood of zk , k = 1, ..., m, there is the expansion (5).
Applying lemma 2.2, we get
I(v − u, D) = I(v, Dr ) − I(u, Dr ) − 2
p
X
δk (bk − ak ) + o(1), r → 0,
(7)
k=1
here ak and bk from (4) and (5) respectively. By lemma 2.1
I(v, Dr ) = λn r
2−n
p
ni
X
X
δij2
+
i=1 j=1
p
X
Mi + o(1) = λn r
2−n
i=1
I(u, Dr ) = λn r2−n
m
X
m
X
δk2
+
k=1
δk2 + M + o(1),
p
X
Mi + o(1),
i=1
r → 0,
k=1
taking into account 3), we have
m
X
δk (bk − ak ) =
k=1
p
X
Mi − M.
i=1
Substituting the obtained correlations in (7), we see that the inequality
p
X
I(u − ui , Di ) ≤ I(v − u, D) = M −
i=1
p
X
Mi + o(1),
r → 0,
i=1
is true. Theorem is proved. m
Theorem 2.4 Let sets D, Γ, collections Z = {zk }m
k=1 , ∆ = {δk }k=1 , be
as in the definition of the reduced modulus M := M (D, Γ, Z, ∆), u(z) be the
potential function for D, Γ, Z, ∆, and let Di ⊂ D, i = 1, ..., p, be pairwise noni
i
overlapping domains, Γi , Zi = {zij }nj=1
, ∆i = {δij }nj=1
, be from the definition
2769
Extremal decomposition problems in the Euclidean space
of the reduced moduli Mi = M (Di , Γi , Zi , ∆i ), ui (z) be the potential function
S
for Di , Γi , Zi , ∆i , i = 1, ..., p. Assume that Γi ⊂ Γ, i = 1, ..., p, Z = m
i=1 Zi ,
(that is each point zk ∈ Z coincides with some point zij ∈ Zi for k = k(i, j)
and vice versa), δk = δij . Then the inequality
p
X
Mi ≥ M +
p
X
I(u − ui , Di ) ≥ M
i=1
i=1
holds.
Proof. The function u is admissible for Di , i = 1, . . . , p. Let bk be constants from the expansion of the function u in a neighbourhood of zk , bij = bk
if k = k(i, j). Applying lemmata 2.1 and 2.2 with the potential functions uk
for Dk we get
p
p
p
ni
ni
X
X
X
X
X
2 2−n
δij aij + o(1) =
I(ui , (Di )r ) =
(δij ) r λn +
i=1 j=1
i=1 j=1
p
=
X
I(u, (Di )r ) − 2
i=1
ni
X
i=1
!
δij (bij − aij ) − I(u − ui , (Di )r )
j=1
≤ I(u, Dr ) −
p
X
I(u − ui , (Di )r ) − 2
p
ni
X
X
i=1
p
=
+ o(1)
ni
XX
i=1 j=1
p
p
(δij )2 r2−n λn +
i=1 j=1
ni
XX
δij (bij − aij ) + o(1)
δij bij − 2
ni
XX
i=1 j=1
δij (bij − aij )
i=1 j=1
p
−
X
I(u − ui , (Di )r ) + o(1), r → 0.
i=1
It implies that
p
ni
X
X
i=1 j=1
δij bij ≤
p
ni
X
X
δij aij −
i=1 j=1
p
X
I(u − ui , Di )
i=1
or equivalently
p
X
i=1
I(u − ui , Di ) + M (D, Γ, Z, ∆) ≤
p
X
M (Di , Γi , Zi , ∆i ).
i=1
Here we used the fact that the function u − ui has no singularity in Di . Denote by r(Dl , xl ) = r(Dl , xl , ∂D) the harmonic radius. Directly from
theorem 2.3 we get theorem 2 of the paper [5]
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K.A. Gulyaeva, S.I. Kalmykov and E.G. Prilepkina
Corollary 2.5 For any non-overlapping domains Dl ⊂ Rn , n ≥ 3, points
xl ∈ Dl and real numbers δl , l = 1, . . . , m the inequality
−
m
X
δl2 r (Dl , xl )2−n
≤
m X
m
X
δl δp |xl − xp |2−n
l=1 p=1
p6=l
l=1
holds true.
Proof. The Green’s function of the ball B(0, ρ) is
λn
2−n !
|x
|x
ρx
0
0
−
.
|x − x0 |2−n − ρ
|x0 | Denote by Dl (ρ) the intersection Dl ∩ B(0, ρ). By theorem 2.3
M (ρ) ≥
m
X
Ml (ρ),
l=1
where M (ρ) is the modulus of the ball B(0, ρ), the collections {xl }m
l=1 , ∆ =
m
{δl }l=1 , and Γ = ∂B,
Ml (ρ) = −δl2 r (Dl (ρ), xl )2−n λn .
It is sufficient to take a limit as ρ → ∞. Theorems 2.3 and 2.4 imply for p = 1 monotonicity of the quadratic form
m
m X
X
δl δp gΓ (zl , zp , D)
l=1 p=1
under extension of a domain. Following [2] we will say that a domain D̃ is
obtained by extending a domain D across a part of its boundary γ ⊂ ∂D if
D ⊂ D̃ and (∂D) ∩ D̃ lies in γ.
Corollary
2.6 If D̃ is obtained by extending D across Γ, Γ̃
S n
Γ (R \ D) , then for any real numbers δl and points zl ∈ D
m X
m
X
δl δp gΓ̃ zl , zp , D̃ ≥
l=1 p=1
m X
m
X
δl δp gΓ (zl , zp , D) + I(u − ũ, D)
l=1 p=1
≥
m X
m
X
l=1 p=1
δl δp gΓ (zl , zp , D) .
⊂
Extremal decomposition problems in the Euclidean space
2771
If D̃ is obtained by extending D across the part of (∂D) \ Γ, Γ̃ = Γ, then
m X
m
m X
m
X
X
δl δp gΓ̃ zl , zp , D̃ ≤
δl δp gΓ (zl , zp , D) − I(u − ũ, D)
l=1 p=1
l=1 p=1
≤
m X
m
X
δl δp gΓ (zl , zp , D) ,
l=1 p=1
m
here u and ũ are the potential functions for D, Γ, Z = {zl }m
l=1 , ∆ = {δl }l=1
m
m
and D̃, Γ̃, Z = {zl }l=1 , ∆ = {δl }l=1 , respectively.
In [3] the notion of the Robin radius
r (D, z0 , Γ) = exp lim (gD (z, z0 , Γ) + log |z − z0 |)
z→z0
was introduced. This quantity generalized the notion of the conformal radius.
An analogue of Kufarev’s theorem (see [8]) for non-overlapping domains D1 ,
D2 lying in the unit disk U under the condition ((∂Dk ) ∩ U ) ⊂ Γk ⊂ ∂Dk ,
ak ∈ Dk , k = 1, 2 is the inequality
"
2 #−1
a
−
a
2
1
r (D1 , a1 , Γ1 ) r (D2 , a2 , Γ2 ) ≤ |a2 − a1 |2 1 − .
1 − a1 a2 By setting in theorem 2.3 p = 2, Γ = ∅, we obtain in Rn the following inequality.
Corollary 2.7 Let D1 and D2 be non-overlapping and lie in the ball U =
B(0, 1), ak ∈ Dk , (∂Dk ) ∩ U ⊂ Γk ⊂ ∂Dk , k = 1, 2. Then
−λn r (D1 , a1 , Γ1 )2−n − λn r (D2 , a2 , Γ2 )2−n ≤ M (U, ∅, {a1 , a2 } , {1, −1}) . (8)
To calculate M (U, ∅, {a1 , a2 } , {1, −1}) we need to know the Neumann function of the unit ball. Note that it is a quite comlicated problem in Rn . In
particular, for n = 3 (see [13])
1
|y|
|x|y|2 − y| 1
+
− log 1 − (x, y) +
g∅ (x, y, U ) =
.
4π |x − y| |x|y|2 − y|
|y|
In [13] there is an analytic view of g∅ (D, x, y) for n = 4, 5. So, for n = 3 the
inequality (8) has the following form
2
2|a2 |
−
− r (D1 , a1 , Γ1 )−1 − r (D2 , a2 , Γ2 )−1 ≤ −
|a1 − a2 | |a1 |a2 |2 − a2 |
|a1 |a2 |2 − a2 | 1
1
+ 2 log 1 − (a1 , a2 ) +
+
+
2
|a2 |
1 − |a1 |
1 − |a2 |2
− log(4(1 − |a1 |2 )(1 − |a2 |2 )).
Acknowledgements. This work was supported by the Russian Science
Foundation under grant 14-11-00022.
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K.A. Gulyaeva, S.I. Kalmykov and E.G. Prilepkina
References
[1] V.N. Dubinin, Condenser Capacities and Symmetrization in Geometric
Function Theory, Birkhäuser-Springer, Basel, 2014.
http://dx.doi.org/10.1007/978-3-0348-0843-9
[2] V.N. Dubinin and N.V. Eyrikh, Some applications of generalized condensers in the theory of analytic functions, Journal of Mathematical Sciences, 133 (2006), no. 6, 1634-1647.
[3] V.N. Dubinin and D.A. Kirillova, On extremal decomposition problems, Journal of Mathematical Sciences, 157 (2009), no. 4, 573-583.
http://dx.doi.org/10.1007/s10958-009-9342-1
[4] V.N. Dubinin and L.V. Kovalev, The reduced module of the complex
sphere, Journal of Mathematical Sciences, 105 (2001), no. 4, 2165-2179.
http://dx.doi.org/10.1023/a:1011377024516
[5] V.N. Dubinin and E.G. Prilepkina, Extremal decomposition of spatial
domains, Journal of Mathematical Sciences, 105 (2001), no. 4, 2180-2189.
http://dx.doi.org/10.1023/a:1011329108587
[6] E.G. Emel’yanov, On problems on the extremal decomposition, Zapiski
Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta Imeni V. A. Steklova, 154 (1986), 76-89.
[7] J. Hersch, Transplantation harmonique, transplantation par modules,
et théoremes isoperimetriques, Commentarii Mathematici Helvetici, 44
(1969), 354-366.
[8] P.P. Kufarev, On the question of conformal mappings of complementary
domains, Dokl. Akad. Nauk SSSR, 73 (1950), 881-884.
[9] G.V. Kuz’mina, Methods of geometric function theory. I, II, St. Petersburg
Mathematical Journal, 9 (1998), no. 3, 455-507; 9 (1998), no. 5, 889-930.
[10] B.E. Levitskii, The reduced p-module and the inner p-harmonic radius,
Dokl. Akad. Nauk SSSR, 316 (1991), 812-815.
[11] Ch. Pommerenke and A. Vasil’ev, Angular derivatives of bounded univalent functions and extremal partions of the unit disk, Pacific J. Math.,
206 (2002), no. 2, 425-450. http://dx.doi.org/10.2140/pjm.2002.206.425
[12] E.G. Prilepkina, On composition principles for reduced moduli, Siberian
Mathematical Journal, 52 (2011), no. 6, 1079-1091.
http://dx.doi.org/10.1134/s0037446611060139
Extremal decomposition problems in the Euclidean space
2773
[13] M.A. Sadybekov and B.T. Torebek and B.Kh. Turmetov, Representation of Green’s function of the Neumann problem for a multi-dimensional
ball, Complex Variables and Elliptic Equations, 61 (2015), 104-123.
http://dx.doi.org/10.1080/17476933.2015.1064402
[14] A.Yu. Solynin, Modules and extremal metric problems, St. Petersburg
Mathematical Journal, 11 (2000), no. 1, 1-65.
Received: November 2, 2015; Published: December 12, 2015