CC BY
482
Probl. Anal. Issues Anal. Vol. 7 (25), No. 2, 2018, pp. 82-97
DOI: 10.15393/j3.art.2018.5190
UDC 517.54
B. E. Levitskii
REDUCED p-MODULUS, p-HARMONIC RADIUS AND p-HARMONIC GREEN'S MAPPINGS
Abstract. We consider the definitions and properties of the metric characteristics of the spatial domains previously introduced by the author, and their connection with the class of mappings, the particular case of which are the harmonic Green's mappings introduced by A. I. Janushauskas. In determining these mappings, the role of the harmonic Green's function is played by the p-harmonic Green's function of the n-dimensional region (1 < p < to), the existence and properties of which are established by S. Kichenassamy and L. Veron. The properties of p-harmonic Green mappings established in the general case are analogous to the properties of harmonic Green's mappings (p = 2, n = 3). In particular, it is proved that the p-harmonic radius of the spatial domain has a geometric meaning analogous to the conformal radius of a plane domain.
Key words: reduced p-modulus, p-harmonic inner radius, p-harmonic Green function, p-harmonic Green's mapping 2010 Mathematical Subject Classification: 31B15, 30C65, 58E20
1. Introduction. By definition, the conformal radius of a plane simply connected domain with respect to a fixed interior point is the radius of the disk onto which this domain can be conformally mapped, so that the indicated point passes to the origin, and its derivative at this point is equal to one. It is known that the conformal radius coincides with the inner radius of the region with respect to this point, determined by the (harmonic) Green's function [6]. The notion of p-harmonic inner radius of a spatial domain, introduced by the author (see [10]), is a natural generalization of the inner (conformal) radius of a plane domain. This concept has been applied in a number of works on the potential theory (see [4,8,18]). We show that the p-harmonic inner radius of a spatial
© Petrozavodsk State University, 2018
domain homeomorphic to a ball has a geometric meaning analogous to the conformal radius. The role of the conformal mapping is played by the p-harmonic Green's mapping. This class of maps is defined by analogy with harmonic Green's mappings [19] and has a number of similar properties. The connection between the reduced p-module of the domain with respect to the interior point and the inner p-harmonic radius ([10]) makes it possible to extend geometric estimates and properties established by using the moduli method for other classes of maps to the case of p-harmonic Green's mappings (see [10,12]).
2. The reduced p-module. Let En be the u-dimensional Euclidean space and En = En U ro be its one-point compactification. Denote by x = (x1,x2,...,xn) a vector in En, |x| = \Jx\ + ... + xn is the length
of x. Bn(x0,{) is an open ball centered at a point x0 E En with radius t; Bn(œ,t) = {x E En : |x| > t}; Sn-1(x,t) = dBn(x,t),x E En; Un is the
volume of an u-dimensional ball of unit radius, uun is the area of its surface.
The concept of p-capacity and its generalizations in different versions is encountered in the works of many authors (see, for example, [2,5,7,13]). We consider a condenser, which is a ring domain D E En, the complement of which consists of two connected components C0 and C1 (condenser plates).
For 1 < p < ro, we define the p-capacity of the condenser D by the formula
where inf is taken over the class of continuous functions in D that are continuously differentiable in D and take values 0 on Co and 1 on Ci. It is known that if C0 and C1 are nondegenerate, then there is a unique potential function u0(x), which is the extremal for p-capacity of the condenser D and is p-harmonic, that is, it satisfies the p-Laplace equation (in the generalized sense).
Lemma 1. For almost all t E (0,1) we have:
(1)
D
St(u o )
where St(u0) = {x E D : u0(x) = t} is a level surface of the potential function and dS is the surface area element.
Proof. Note that if p(x) is a twice continuously differentiable function defined in a domain G with a piecewise smooth boundary, and ^(x) is a continuously differentiable function in G, then, integrating by parts, we obtain:
J^div (|Vp|p-2 Vp) du =
G
= /^p-2dnds-/|v,r2£gg* (3)
dG G k=1
where ~rt is the vector of the external normal to dG. Let Da b be a domain bounded by level surfaces Sa(u0) and Sb(u0) (0 < a < b < 1). Due to the fact that uo(x) is a monotone function (see [15]), Da,b is a ring domain. Applying formula (3) in this domain in the case =1 and p = u0, we obtain
/ iVuol'-2 dS = J |Vuo|p-2 dndS + J |Vuo|p-2 dS = 0.
9Da,b Sa(uo) Sb(uo)
It follows that
J Vuof-2 dS = -J |Vuo |p-2 dS = J Vuof-1 dS = I.
Sb(u-o) Sa'(u o) Sa(u o)
The value of I, therefore, does not depend on the choice of the level surface of the potential function, except for the case when the gradient uo(x) vanishes on this surface. Applying formula (3) again in the domain Da,b when ^ = p = uo, we find
J V^^ du = a J Vuor2 dS + bj |Vuo|p-2 dS = (b - a)I.
Da,b Sa(u o) Sb(uo)
Passing to the limit as a t 0 and b i 1, we obtain the relation (2). □ A convenient metric characteristic of a condenser is the quantity
m»dp D = (5-1 ■ (4)
which is called the p-module of a condenser D.
Let Mp(x,x0) = — x0|) = ^p(t) be a fundamental solution of the p-Laplace equation:
Apu = — div(|Vu|p-2 Vu) = nwra#(x — x0), (5)
where $(x — x0) is the Dirac measure or Dirac ^-function at x0 G En. With x0 = we have
m J — ln ^ P = n; ra\
^(t) = { 1 t-Y, p = n, (6)
where y = ^-p. If x0 = to, then the role of mp(x, to) is played by the function
MP°°(M) = mT (t) = —^p(1/t). (7)
Note that the p-module of the spherical ring KR, bounded by the concentric spheres of radii r and R > r is
modp KR = Mp(r) — Mp(R) = r
1 R
ln r, p = n, — 1 (R-7 — r-Y), p = n.
We will need the following well-known property from the potential theory, formulated here for the case of p-modules of ring domains.
Lemma 2. If ring domains D^ D2,..., Dm are pairwise disjoint and each of them separates the boundary components of a ring domain D, then
m
modp D ^ ^^ modp Dk. (9)
fc=i
Proof. Let uk be an admissible function for the ring domain Dk, ak ^ 0 and m=1 ak = 1. Then u = m=1 uk is an admissible function for the ring domain D and
m
J |Vu|p dw = ak J |Vuk|p dw.
D k=1 Dk
Hence,
m
Capp D ^ ^ ap Capp Dk. (10)
fc=i
Assuming
-1
(Capp D J pak = ~m-zr
p-i
ki
E(CaPp D
k=1
from (10), we obtain (9). □
Let G C be a domain homeomorphic to a ball, x0 G G, Gt = G\Bra(x0,t). If x0 = to, by Lemma 2, for sufficiently small 0 < t1 < t2 we have
modp Gtl ^ modp Gt2 + modp K^,
where Kt2 is the ring bounded by concentric spheres of radii t1 and t2 with center at x0. Therefore,
modp Gtl — Mp(t1) ^ modp Gi2 — Mp(t2). Consequently, the following limit exists:
lim [mod Gt — Mp(t)] = mp(x0,G). (11)
Similarly, when x0 = to,
t-i
modp Gt-i ^ modp Gt-i + modp Kt-1,
where K\ is the ring bounded by concentric spheres of radii t2 1 and t-1
2
(t-1 < t-1) with the center at the origin. Hence, taking (8) into account, we find
modp G1 — M°(t1) ^ modp G1 — ^^2).
tl t2
Consequently, the following limit exists:
lim
t^0
mod G1 — MT(t) = mp(TO,G). (12)
In the general case, the quantity mp(x0, G) = hp(x0, G) will be called the reduced p-module of the domain G with respect to the point x0. If p > n and x0 = to, we have
mp(x0, G) = limmodp Gt.
If 1 < p < n and x0 = oo, then
_ 1
nun
p-1
mJoo,G) = limmodP G1 = (-==n—) ,
p( , ) t—o p 1 V Cp(En\G)y ,
where Cp(A) is the p-capacity of the compact A C En, see [13], defined by
Cp(A) = in^y |Vu|p du.
En
Here the infimum is taken over the class of continuously differentiable functions, greater than or equal to 1 on A, with compact support in En.
The notion of reduced modulus of a plane domain (p = n = 2) appeared for the first time in Teichmiiller's article [17]. Various generalizations of the concept of the reduced module and their applications were considered in [1,10-12,14]. The definition of the reduced p-module of a domain with respect to a point, given in this article above, can be extended to the case of domains of arbitrary connectivity. To do this, we use the definition of the p-module of the domain, connected either to the p-capacity of the corresponding condenser, or to the corresponding modules of families of curves or surfaces (see [7,16]).
3. The inner p-harmonic radius. Let G be a domain with the regular boundary in En, x0 G G. From the results of S. Kichenassamy and L. Veron [9] it follows that in the domain G there exists a unique (generalized) solution u = uc(x,x0) G C 1,a(G\x0), a > 0 of the Dirichlet problem for equation (5), which equals zero on the boundary of the domain G, and such that the function
hp(x, x0) = uG(x, x0) — ^p(x, x0) G L™(G).
In addition, there is the limit
lim hp(x, x0) = hp(x0, G) (13)
x—xo
and
n-1
lim |x — x0| p-1 (VuG(x, x0) — V^p(x, x0)) = 0. (14)
x- xo
The function uG(x,x0) will be called the p-harmonic Green's function of the domain G with a pole at the point x0, and the function hp(x0, G) will
be called the Roben p-function of the domain G. Note that when p > n we have hp(xo,G) = uG(xo,xo). By definition [10], the inner p-harmonic radius of the domain G at the point xo is the value of Rp(xo, G) for which
f -^p(Rp(xo, G)) , xo = to, hp(xo, G) = < (15)
[-^~(rp(to, G)), xo = to.
Thus, Rn(xo, G) = exp{hn(xo, G)} and for p = n
f—Yhp (xo,G)) 1/7, xo = to,
r>(xo,g) = < ; Yp( , // , = , (16)
(7hp(TO,G)) , xo = to.
The inner p-harmonic radius of an arbitrary domain G C En at the point xo is the number Rp(xo, G) = sup Rp(xo, G'), where the supremum is taken over all domains G' C G with the smooth boundary.
Theorem 1. [10] For any domain G C En with regular boundary and any xo G G we have mp(xo, G) = hp(xo, G).
Proof. Let G C En be a domain with regular boundary, xo = to, and ^«(uc) = {x G G : uG(x,xo) ^ a}. Let (uG) be a closed bounded set. We show that Qa(uG) is star-shaped with respect to the point xo for sufficiently large a. It follows from (14), that for any direction l
Um |x — xo|n-1 (^ — ^^pi^l) = o. (17)
In particular, passing to spherical coordinates and calculating the derivative along the radius — — xo for p = |x — xo| we obtain:
n-i duG
lim pn- dU^ = — 1. (18)
dp
It follows that for small p the function uG(x,xo) decreases monotonically with respect to p and the level surface S«(«G) = d^a(uG) is star-shaped with respect to the point xo. We consider the condenser G(a) = G\^a(uG). The extremal function for the p-capacity of the condenser G(a) has the form va(x) = «uG(x,xo). By Lemma 1,
Capp G(a) = ——[ J |vug|p_1 dS. (19)
Sa(uG)
Applying formula (3) to the domain bounded by the surface Sa(uG) and sphere S(x0,t), where t > 0 is sufficiently small, and setting ^ = 1, and p = uG(x, x0), we find
Capp G(a) = a-i f IVug|p_1 dS = -J |VuG|p_2 dS.
Sa(uG) S(xo,t)
(20)
du
It follows from (14) that = -t^ (1 + o(t)), t ^ 0. Thus,
Capp G(a)I t1-n(1 + a(t))p 2 (1 + o(t)) dS = aP J
S(xo,t)
^ (1 + a(t))p-2(l + o(t))
ap-
and passing to the limit as t ^ 0, we find that Capp G(a) = nun/ap~1 or modp G(a) = a. We consider now the condenser Gt = G\Bn(x0,t), where t > 0 is sufficiently small, and the values a1 < a2 are such that the level surface Sai(uG) contains the sphere Sn~l(x0,t) and touches it, and the level surface Sa2 (uG) lies inside this sphere and touches it from within. Such values a1 and a2 exist except for the trivial case when the domain G is a ball with center at the point x0.
Since the p-module of the condenser does not decrease as it expands, then modp G(a2) ^ modp Gt ^ modp G(a1) or
a1 ^ modp Gt ^ a2. (21)
It follows from (13) that for any e > 0 there exists 5 > 0, such that
^p(|x - X0I) + hp(x0, G) - e < uG(x,X0) < ^p(|x - X01) + hp(x0, G) + e
for any x, such that |x - x01 <5. Choosing t sufficiently small and using (21), we have
hp(x0, G) - e < modp Gt - ^p(t) < hp(x0, G) + e,
so mp(x0,G) = hp(x0,G). The proof of the theorem in the case of an arbitrary domain with a regular boundary, as well as the consideration of the case x0 = 00 is obtained by modifying the above arguments. □
Note that from the definition of the inner p-harmonic radius of an arbitrary domain G C En at the point x0 and the relation (15), and also the well-known property of continuity of the p-capacity (p-module) with respect to the monotonic convergence of sets (see, for example, [7]), it follows that Rn(x0, G) = exp{mn(x0, G)} and for p = n
4. p-Harmonic Green's mappings. Let G and G be homeomorphic to a ball domains regular boundaries in En. Let ug(x,x0) and ug(y,y0) be p-harmonic Green's functions for these domains with poles at points x0 G G (x0 = to) and y0 G G (y0 = to), respectively, 1 < p ^ n. Consider the mapping f : G ^ G such that:
• f (x0) = y0;
• the level set St(ug) is mapped onto the level set St(ug);
• the trajectory of the gradient field Vug(x,x0) that enters the pole x0 corresponds to the trajectory of the gradient field Vug(y,y0) that enters the pole y0.
Such mappings are constructed by analogy with the Green's mappings (p = n = 3) considered in the monograph by A. I. Januszauskas [19], as a special case of harmonic mappings with respect to M. A. Lavrentyev. It follows from relation (17) that p-harmonic Green's functions of G and G have the property that for any ray l from the point x0 G G (respectively, y0 G G) there is the unique trajectory of the field Vug(x, x0) (respectively, the unique trajectory of the field Vug(y,y0)), entering x0 (respectively, y0) with the tangent l. Let a : S ^ S be the rotation (linear mapping) of the unit sphere S = S(0,1) under which a point X G S mapped to the point a(X). If l is the ray from the center of S passing through the point X, then a(l) denotes the ray from the center of S passing through the point a(X).
p-Harmonic Green's mapping f : G ^ G is defined in a sufficiently small neighborhood U(x0) of the pole X0 , as follows. If l is the tangent at the point x0 of the trajectory of gradient field V«G(x,x0) that enters the pole x0 and passes through the point x G St(«G), then y = f (x) G St(ug) belongs to the trajectory of the gradient field V«g(y,y0), that enters the
x0 = to,
x0 = to.
(22)
pole y0 with the tangent a (I). The constructed mapping is a homeomor-phism of a sufficiently small neighborhood U(x0) onto a sufficiently small neighborhood of U(y0). This homeomorphism can be extended outside these neighborhoods by means of the following construction, similar to that described in [19]. If the function uG(x,x0) has no critical values a : a ^ a < 00 in the domain G(a) = G\Qa(uG), or VuG(x,x0) = 0 in some points on the level surface Sa(uG), then the whole domain G(a) is homeomorphic to the ball. The same is true for the domain G(a) = G\na(ug). Let a0 > a1 > ... > ak > 0 be the critical values of the function uG in the domain G. There are a finite number of such values, provided that VuG(x,x0) = 0 on dG. Analogously, let [0 > [1 > ... > [m > 0 be the critical values of the function ug in the domain G. Let 7 = max(a0,[0). Consider the field VuG(x,x0) that enters the pole x0 with the tangent I and has the level surface Sa(uG), and the field Vug (y,y0) that enters the pole y0 with the tangent a (I) and has the level surface Sa(uG). Consider a point x G G(y) at the intersection of the trajectory of the field VuG(x,x0) and associate it with the point y G G(y) at the intersection of the trajectory of the field Vug(y,y0). Thus, the extension of the mapping f from the neighborhood U(x0) to the homeomorphism of the domain G(y) to the domain G(y) is defined. Further extended beyond G(y) along such trajectories, this mapping may have singularities, because different trajectories of the gradient field intersect at critical points. Such construction is possible only if both functions uG(x,x0) and ug(y,y0) have singularities at the points x0 and y0, or ug (y0,y0) = ug(x0,x0).
Let ft be the trace of the mapping f on the level surface St(uG), Jft (x) be the Jacobian of the trace ft and Jf (x) be the Jacobian of f. The following theorem extends the properties established in [19] for p =2 and n = 3 to the case of p-harmonic Green's mappings.
Theorem 2. The following relations hold:
1) |f(x0)| = lim JfM-M =K(G,G), p = n; (23)
x ^xo |x - x01 11, p < n.
2) lim
Vug |
= 1. (24)
UG=UQ =t
|VuG| lim Jf (x)
x ^xo 11, p < n.
3) lim Jf (x)HAn(G,G), p = n; (25)
A\ 7 f \ T f \ „ |Vug(x,x0)| ^ Q f ,
4) Jf (x) = f (x) x ——T| , x G St(«g).
|Vug(f (x),y0)|
(26)
5) Jft(x) = <
|Vug(x,x0)|
n— 1
|V«g (f (x),y0)| |Vug(x,x0)| xp-1
x An(G, G), p = n,
x G St(ug). (27)
p < n,
V|Vug(f (x),y0)|
Here An(G, G) = exp hn(y0, G) - hn(x0, G)
Proof. The following representations hold for the p-harmonic Green's functions of the domains G and G in the neighborhood of the poles x0 and f (x0) = y0 due to (13):
ug(x,x0) = ^p(x,x0) + hp(x0, G) + O(|x - x01)
and
ug(f (x),y0) = Mf (x),y0) + hp(y0,GG) + O(|f (x) - y0|).
On the corresponding level surfaces ug(f (x),y0) = ug(x,x0); thus
|f (x) - y0|
|x — x0| exp hp(y0,G) — hp(x0,G) + O(|x — x0|)
p = n,
|x — x0N 1 — |x — x0|Y hp(y0,G) — hp(x0,G) + o(|x — x0|Y) >, p < n.
This implies the first relation.
Due to (14), we have for all 1 < p ^ n:
|Vug(x,x0)| = |x — x01p—1 (1 + O(|x — x01)), and, respectively,
|Vug(f(x),y0)| = |f(x) — y0|^ (1 + O(|f(x) — y0|)) .
Hence, taking (28) into account, we obtain (24).
Let us prove the equality (25). Let the ball Bn(x0,r) C G and B be its image under the mapping f. Let
Rti = max |y — y01
yes
|yti — y0|
where ytl G Stl (ug), and
Rt2 =min |y - yo1 = |yii - yo1 ,
y£B
where yt2 G St2(uG), 0 ^ ti < t2 < ro. We set xu = f-1(ytv)Lv = 1, 2. For the n-dimensional Lebesgue measure mn(B) of the domain B we have the inequality
R2 mn(D) Rn , ,
_Ji ^ ^ _Ji. (29)
rn unr'
From relation (27) we easily find
n
n
lim Rl = lim Rl J exp n |^hn(yo, G) - hn (xo, G)] , p
r—0 rn r—0 rn I 1, p < n.
Note that Jf(x0) = limmn(D)/unrn; then (29) implies (25). By con-
r—0
struction of the map f, Jf (x) = Jft (x)Kt, where Kt is the coefficient of extension along the orthogonal trajectories of the mapping f on the level surface St(uG). The increase rate of a function along orthogonal trajectories to level surfaces is proportional to the length of its gradient, then Kt = |VuG(x,x0)| x [|Vug(f(x),y0)|] for x G St(uG), that is, equality (26) is satisfied. Moreover, by virtue of (24), limKt = 1. We consider
t—y^o
two level surfaces St(uG) and Stl(uG), where 0 ^ t < t1 < ro. Let point X G St(uG) and 9(X) G Stl (uG) be its image lying on the trajectory of the field VuG(x,x0), passing through X. If there are no critical points in the layer bounded by these surfaces, then the mapping 9 : St(uG) ^ Stl (uG) is a homeomorphism. Let U(X) C St(uG) be an open connected neighborhood of X, and V(X) C Stl (uG) be its image under the mapping 9. Denote by Q(X) the domain that represents the part of the flow tube of the vector field VuG(x,x0), enclosed between U(X) and V(X). Applying formula (3) in the domain Q(X) in the case when ^ = 1 and p = uG(x, x0), we obtain
|Vug|p-2 dSv = |Vug|p-2 dp dSu =
dn J dn
V(X) U(X)
I |Vug|p-2dn [Jo(x)]-1 dSv,
V(X)
where J? (x) is the Jacobian of the mapping 0. Applying the mean-value theorem and contracting the neighborhood U(X) to the point X, we find
| G| dn
| G| dn
[J? (X )]-1
UG=t
Hence,
J?(X) = |VuG|p-1|uG=t x [|V«g|p-1|«g =tl]-1. Analogously, for the mapping Q : Stl («g) ^ St2 («g) we obtain
J?r(X) = |V«g|p-1|uG=t x [|V«g|p-1|uG=ti]-1. As /t = Q-1 o /ti o 0,
J/t (x) = / (x) x
|V«g(x,xq)|
p-1 |v«g(/(x), yo)|p-1
|V«g (/(x),yo )|p-1
x
|V«g(x,xq)|p-1
UG=UG =t
. (30)
UG=UG =tl
Passing to the limit in (30) for t1 ^ to and taking (24) and (25) into account, we obtain (27). □
From relation (23) and our reasoning, we obtain
Corollary 1.
(|//(x0)| Rn(x0,G) P = ^ RP(y0,G) = \ -1 (31)
t[Rp-Y(xq,G) + Ap(xq)] y , p< where Ap(xq) = lim [|/(x) - yo|-Y - |x - xq|-7] .
n,
Theorem 3. Assume that G is a ball of radius R centered at y0; for p = n we have |//(xq)| = 1 if and only if Rn(x0, G) = R, and for p < n we
n-1
have J/(x) = 1 + O(|x — xq| p-1) if and only if Rp(x0, G) = R.
Proof. The first part of the statement follows immediately from (31). Further, since
~ Jin R - ln |y - yo| , p = n,
uG (yo,G)=i Y I,, (32)
I-1R-Y + 1 |y - yo| , p < n,
for y = /(x), we have
|/(x) _ | i Rn(R0,G) Ix - x0| (1 + 0(|x - XQ|)), p = n 1
\ |x - Xq| [1 + C |x - Xo|Y + O(|x - xq|7+1)]- 1 , p < n.
(33)
Here C = R-Y - Rp-Y(xq, G). It follows from (32) and (14) that
n — 1
VuG(yo,G) = |y - yo| p—1
and, respectively,
n-1 p— 1
|v«g(x, xq)| = |x - xq| (1 + 0(|X - xq |)). From this, using (33), we find:
|v«g(x,xq)| J Rra(R°,G)(1 + O(|x-^^ p = n,
|VuG (/(x),yo)| |1 - n-C |x - Xq |Y + O(|x - Xq|7+1), p<n.
n - p
Since, by virtue of (26) and (27),
(34)
J/ (x)
|v«g(x,xq)| \n f R \n
x -fn—, p = n;
|VuG (/(x),yo)|/ \Rn (xq,g) |V«G(x, xq)| xp
\|VuG(/(x),yo)| from (34) we deduce
p < n,
1 + O(|x - xq| , p = n, J/ (x) =4 (35)
1 - pn^pC |x - xq|y + O(|x - xq|7+1 ), p < n, from which the assertion to be proved follows. □
Remark. For p = n the construction of p-harmonic Green's mappings described above and the assertion of Theorem 2 can be extended to the case where one or both poles xq or yQ are equal to to.
References
[1] Aseev V. V., Lazareva O. A. On the continuity of the reduced module and the transfinite diameter. Izv. Vyssh. Uchebn. Zaved. Mat., 2006, vol.10, pp. 10-18. (in Russian); translation in: Russian Math. (Iz. VUZ), 2006, 50:10, pp. 8-16.
[2] Dubinin V. N. Capacities and Geometric Transformations of Subsets in n-Space. Geom. Funct. Anal., 1993, vol. 33, no. 4, pp. 342-369. DOI: https: //doi.org/10.1007/BF01896260.
[3] Dubinin V. N. Condenser capacities and symmetrization in geometric function theory. Basel: Birkhauser/Springer, 2014, 344 p.
[4] Flucher M. Variational Prolems with Concentration. Progr. Nonlinear Differential Equations Appl., vol. 36, Birkhauser, Basel, 1999, 163 p.
DOI: https://doi.org/10.1007/978-3-0348-8687-1.
[5] Gol'dstein V. M., Reshetnyak Yu. G. Introduction to the Theory of Functions with Generalized Derivatieves and Quasiconformal Mappings. Nauka, Moskow, 1983, 284 p. (in Russian).
[6] Goluzin G. M. Geometric Theory of Functions of a Complex Variable. (Translations of Mathematical Monographs). Amer. Math. Soc., 1969, 676 p.
[7] Hesse J. A p-extremal length and p-capacity equality. Ark. Mat., 1975, vol. 13, no. 1, pp. 131-144.
[8] Kalmykov S.I., Prilepkina E.G. Extremal decomposition problems for p-harmonic radius. Analysis Mathematica, 2017, vol. 43(1), pp. 49-65. DOI: https://doi .org/10.1007/s10476-017-0103-y.
[9] Kichenassamy S., Veron L. Singular Solutions of the p-Laplace Equation. Math. Ann., 1986, vol. 275, pp. 599-615; Erratum: Math. Ann., 1987, vol. 277 (2), p. 352. DOI: https://doi.org/10.1007/BF01459140.
[10] Levitskii B. E. Reduced p-modulus and the Interior p-harmonic radius. Dokl. Akad. Nauk SSSR, 1991, vol. 316(4). pp. 812-815. (in Russian); translation in: Soviet Math. Dokl., 1991, vol. 43(1). pp. 189-192.
[11] Levitskii B. E., Miklyukov V. M. The reduced module on the surface. Vestn. Tomsk. Gos. Univ., 2007, vol. 301, pp. 87-91 (in Russian).
[12] Levitskii B. E., Mityuk I. P. "Narrow" theorems on spatial modules. Dokl. Akad. Nauk SSSR, 1979, vol. 248(4), pp. 780-783 (in Russian).
[13] Maz'ja V. G., Sobolev Spaces. Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985, 486 p.
[14] Mityuk I. P. Generalized reduced module and some of its applications. Izv. Vyssh. Uchebn. Zaved. Mat., 1964, no. 2, pp. 110-119 (in Russian).
[15] Mostow G. D. Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms. Publ. Math. Inst. Hautes Etudes Sci., 1968, vol. 34, pp. 53-104.
[16] Shlyk V. A. The equality between p-capacity and p-modulus, Sib. Math. J., 1993, vol. 34, no. 6, pp. 1196-1200. DOI: https://doi.org/10.1007/ BF00973485.
[17] Teichmuller O. Untersuchungen über konforme und quasikonforme Abbildungen. Deutsche Math., 1938, no. 3, pp. 621-678.
[18] Wang W. N-Capacity, N-harmonic radius and N-harmonic transplantation. J. Math. Anal. Appl., 2007, vol. 327(1), pp. 155-174. DOI: https://doi. org/10.1016/j.jmaa.2006.04.017.
[19] Yanushauskas A. I. Three-dimensional Analogs of Conformal mappings. Novosibirsk, Nauka, 1982, 173 p. (in Russian).
Received August 19, 2018. In revised form, November 08, 2018. Accepted November 12, 2018. Published online November 23, 2018.
B. E. Levitskii Kuban State University
149 Stavropolskaya str., Krasnodar 350040, Russia E-mail: [email protected]
