CC BY
12388 Probl. Anal. Issues Anal. Vol. 7 (25), Special Issue, 2018, pp. 88-100
DOI: 10.15393/j3.art.2018.5511
The paper is presented at the conference "Complex analysis and its applications" (COMAN 2018), Gelendzhik - Krasnodar, Russia, June 2-9, 2018.
UDC 517.518
N. A. KUDRYAVTSEVA, S. K. VODOPYANOV
ON THE CONVERGENCE OF MAPPINGS WITH k-FINITE DISTORTION.
Abstract. We prove that a locally uniform limit of a sequence of homeomorphisms with finite k-distortion is also a mapping with finite fc-distortion. We obtain also an estimation for the distortion coefficient of the limit mapping.
Key words: mapping with k-finite distortion, distortion coefficient, passing to the limit, differential form.
2010 Mathematical Subject Classification: 30C65
1. Introduction. It is well known that the limit of a uniformly converging sequence of analytic functions is an analytic function. Reshetnyak generalized this result to mappings with bounded distortion: the limit of a locally uniformly converging sequence of mappings with bounded distortion is a mapping with bounded distortion.
Definition 1. [8] A mapping f : Q ^ Rn is called mapping with bounded distortion if f is continuous, f G Wn ioc(Q), the Jacobian J(x, f) does not change the sign in the domain Q and
|Df (x)|n < K| J(x,f)| for almost allx G Q. (1)
The smallest constant in this inequality is called the distortion coefficient of the mapping f and is denoted by the symbol K(f). It is clear that
K(f )=sup{ JXf : x G Q, J(x,f )=0}.
Reshetnyak used the weak convergence of Jacobians to prove the following theorem on the limit of a sequence of mappings with bounded distortion.
© Petrozavodsk State University, 2018
Theorem 1. [8] Let fm : Q ^ Rn, m = 1, 2,..., be an arbitrary sequence of mappings with bounded distortion, locally converging in Ln(Q) to a mapping f0 : Q ^ Rn. Assume that the sequence of distortion coefficients K(fm), m = 1, 2,..., is bounded. Then the limit mapping f0 is a mapping with bounded distortion and the following inequality holds:
K(fo) < lim K(fm). (2)
k^w
We briefly outline the proof in the case of non-negative Jacobians. For a test function < e CW(Q) we have
i \Dfo(x)\n<(x) dx < lim / |Dfm(x)\n<(x) dx <
J m^w J
n n
^ K lim J(x,fm)<(x) dx = K J(x,fo)<(x) dx.
m^w J J
nn
To justify the limit in the last equality, we apply the weak convergence of Jacobians. Consequently, for the limit mapping, the point-wise inequality \Df0(x)\n < KJ(x,f0) holds a.e. in Q.
More recently, research has begun on mappings with finite distortion. They are a natural generalization of mappings with bounded distortion.
Definition 2. [7] Let a mapping f : Q ^ Rn belong to the Sobolev class W1nioc(Q) and J(x,f) > 0. We define the pointwise distortion coefficient K(x, f) of the mapping f as a value
f Jf f J (xf) > 0, K(x,f) = i lJ(x,f >1 ( f ) ,
I 1 otherwise.
The mapping f : Q ^ Rn is called mapping with finite distortion (f e e FD(Q)) if
\Df (x)\n < K(x,f )J(x,f) where 1 < K(x,f) < to for almost all x e Q.
Remark. In other words, the condition of finite distortion is that the partial derivatives of the mapping f e W1 loc(Q) vanish a.e. on the set of zeros of the Jacobian J(x,f).
For the first time, the essential properties of mappings with finite distortion were investigated in the paper [15] in the study of homeomor-phisms inducing a bounded composition operator. The name "mapping with finite distortion" was proposed much later in the paper [7].
In [5] F. Gehring and T. Iwaniec showed that the limit of a weakly converging sequence of mappings with finite distortion is also a mapping with finite distortion, and they obtained also an estimation for the distortion coefficient of the limit mapping.
Theorem 2. [5] Let fm : l ^ Rn, m = 1,2,..., be an arbitrary sequence of mappings with finite distortion converging weakly in W1 ioc(l) to a mapping f0 : l ^ Rn. Assume that
where l 3 x ^ M(x) G is a measurable function. Then the limit
mapping fo is a mapping with finite distortion and the inequality
More precisely, in this paper existence of a subsequence fmk such that
was shown. Here the limit is understood in the sense of so-called biting convergence.
Definition 3. [2] Let h and hk, k G N, be Lebesgue measurable functions defined on a set E c Rn. The sequence hk is said to converge in the biting sense on E to h if there exists an increasing sequence Ev of measurable subsets of E,
K (x,fm) < M (x) < to, m = 1, 2,...
K(x,fo) < M(x)
holds.
K(x,fo) < b * lim K(x, fmk)
Ue = e,
such that
for any function < G L^(EV).
In the paper [4] the limit of a sequence of homeomorphisms with finite distortion converging weakly in W/ was shown to have, also, a finite distortion under the condition that the limit mapping is a homeomorphism.
Theorem 3. [4] Let Q, Q' be bounded domains in Rn, fj : Q ^ Q', j e N, f : Q ^ Q', be homeomorpisms belonging to W11(Q), and fj ^ f weakly in W^Q). Assume that
\Dfj(x)\n < K(x,fj)J(x,fj) for almost all x e Q,
where K(x,fj) : Q ^ [1,to) are Borel functions for all j, and the sequence K(x,fj) converges in the biting sense to K(x) as j ^ to. Then the limit mapping f is a mapping with finite distortion and the inequality K(x,f) ^ K(x) holds for almost all x e Q.
In the paper [1] mappings with bounded (q, p)-distortion (n — 1 < < q < p < to) were defined and investigated; these mappings coincide with the class of mappings with bounded distortion if q = p = n.
In the paper [17] a locally uniform limit of a sequence of mappings with bounded (9,1)-weighted (q,p)-distortion was shown to be, also, a mapping with a bounded (9,1)-weighted (q, p)-distortion and an estimation similar to (2) was established. The proofs of the theorems like those in the article [17] and in this work are based on the method developed in [11] for extending Reshetnyak's result to Carnot groups.
In this paper we extend the above-mentioned assertions to the class of mappings with k-finite distortion, which arise naturally in the problem of operating with differential forms of degree k (see [12]).
2. Preliminaries. Let U be a domain in Rn. We consider the Banach space Lp(U,Ak) of differential forms w of degree k, k = 1,... ,n, with measurable coefficients, which have the following finite norm: ||w||p =
= (fu \w\p dxfp.
A mapping f : U ^ Rn is said to be approximate differentiate at a point x e U [3], if there exists a linear mapping L : Rn ^ Rn such that
Um \{y e B(x,r) : \f(y) — f(x) — L(y — x)\ >£}\ = 0 r^0 rn
for any e > 0. Here the symbol \ • \ denotes the Lebesgue measure. It is well known that the approximate differential is unique [3] if x is a density point. In what follows, it is denoted by the symbol ap Df (x).
In our paper we consider the mappings belonging to some Sobolev class. Such mappings are unconditionally approximately differentiable.
Let w = Y1 wp dy3 be any k-form, k = 1,... ,n, in W with continuous coefficients wp : W ^ R, where the summation is over all k-dimensional ordered multi-indices ft = ..., ftk), 1 ^ < ... < ^ n, and dy3 = dy(il A dyp2 A ... A dypk. Let a mapping f = (fi,..., fn) : U ^ W of Euclidean domains U, W c Rn be approximately differentiable almost everywhere in U. We write the pull-back of the k-form w in the following way:
f *w(x) = Y wp (f (x)) dfPl Adfp2 A...AdfPk = YY wp (f (x))M? (x) dxa.
3 a 3
In other words, it is a k-form with measurable coefficients, which are
defined for almost all x G U (here dfpk = ^ dxi and the partial
i=i *
derivatives are understood in the approximate sense, M3(x) are (k x k)-minors of the matrix apDf (x) = ijxr), ',3 = 1,... ,n, with ordered lines and columns).
We recall that the approximate differential ap Df (x) : TxU ^ Tf (x)W is defined a.e. in U. It generates canonically the linear mapping Ak f (x) : AkTxU ^ AkTf (x)W of the spaces of k-vectors, and the pullback operation f * of k-forms. We denote the norm of the last linear mapping by the symbol |Akf (x) |.
The minimal analytic and geometric properties of the mapping f were obtained in [12] for generating a bounded pullback operator
f * : Lp(W,Ak) ^ Lq(U,Ak), 1 < q < p < to, (4)
of differential forms of degree k = 1,... ,n.
We say that an approximately differentiable mapping f : U ^ W has k-finite distortion, 1 < k < n, (shortly f G CDk (U; W)) if rankap Df (x) < < k almost everywhere on a set Z. (Hereinafter Z = {x G U :detap Df (x) = = 0}.) For k = 1 (k = n - 1) and f G Wii,1oc(U) (f G W^^U)), this notion is well-known in literature: it is just the class of Sobolev mappings with finite distortion (codistortion), which is characterized by the property: ap Df (x) = 0 (adj Df (x) = 0) almost everywhere on Z (see [10], [14] for the second notion).
Besides of the property of k-finite distortion we consider mappings with a certain behavior of some characteristics of the distortion containing in
itself the ratio ^Aj(xf) , where J(x,f) = detapDf (x) [12]: operator (4)
is bounded if and only if the mapping f G CDk(U; W) and the distortion function W 3 y ^ Hk,q(y) =
( £ Jf) ^ if f-1 (y) \ (e u z) =
) otherwise,
belongs to LK(W) where K = 1 — P if q < p, and k = to (k = q) if q = p
K q P
(p = to). Moreover, the norm of the operator f* is comparable with the value \\Hk,q(•) | Lk(W)||:
a
q P
\Hk,q (•) I Lk (fi)|| < \\f *\\ < \\Hkq (•) | Lk (fi)|
where aq p is some constant.
Hereinafter E C U is a set of measure zero outside of which the mapping f has the Luzin property N.
For homeomorphic mappings one can use a simpler characteristic.
Corollary 1. [12] Let f : U ^ W be an approximate differentiable homeomorphism. The operator f *: LP(W, Ak) ^ Lq(U^k), 1 ^ q ^p ^ to, k = 1,... ,n, is bounded if and only if the following conditions are satisfied:
1) f : U ^ W has the k-finite distortion;
( |Afc/(x)| ¡f J(x f)=0
2) the function Kk,p(x,f) = \ |J(*'f)|1/p ( ,f ) = , belongs to
otherwise,
LK(U), where — = 1 — p if q < p, and k = to (k = q) if q = p
K q p
(p = to).
In this case, the norm of the operator f * is comparable with \\Kk,p(;f) I Lk (U )\\ : aqp\\Kk,p(;f) I Lk (n)\\ < \\f *\\ < \\Kk,p(;f) I Lk (ty^ where aq,P is some constant. 3. Main results.
Definition 4. [12] An approximately differentiable homeomorphism f : U ^ W belongs to the class CV'k P(U; W) if the following conditions hold
1) f G CDk(U; W);
2) Kk,p(-,f ) G Lk(U) where K = 1 - p, 1 < q < P< œ.
Theorem 4. Let fm G CDkp(U; W), m G N, be a sequence of homeo-morphisms of the Sobolev class Wl1loc(U) with k < l, q < l/k, 1 <q <p < < œ. Suppose that the sequence fm is locally bounded in WI1(U), and locally uniformly converges to a homeomorphism f : U ^ W as m ^ œ. Assume also that there exists a sequence of functions U 3 x ^ Mm(x),
belonging to LK(U), that is bounded in LK(U), 1 = 1 — p, for which the
K q p
inequality
Kk,p(x,fm) < Mm(x) for almost all x G U (5)
is true.
Then there exists a function U 3 x ^ M(x) of LK(U) such that some subsequence
(i) in the case 1 < q < p < œ: of functions {Mm(x)K}m£N converges in the biting sense to M(x)K;
(ii) in the case 1 < q = p < œ: of numbers {||Mm | Lœ(U)||}meN converges to M = lim ||Mm | Lœ(U)||;
the limit mapping f belongs to CVkqpp(U; W) and Kk,p(-,f ) G LK(U),
where 1 = 1 — 1.
K q p
Moreover, the inequalities
{Kk,p(x,f ) < M(x) in the case q < p, Kk,p(x,f ) < M in the case q = p,
hold for almost all x G U.
In the proof we use some arguments from the paper [17], where Theorem 4 is proved for k = 1.
Proof. It follows from the conditions of the theorem that f G WI1loc(U). First, we show that the limit mapping f belongs to CDk(U; W). For doing this, we show that the mapping f induces a bounded operator f * : Lp(W, Ak) ^ Lq(U, Ak), 1 < q < p < œ. Since every mapping fm G CVkp(U; W), it follows from Corollary 1 that the homeomorphism fm : U ^ W induces the bounded operator fm : Lp(W, Ak) ^ Lq(U, Ak), 1 < q < p < œ, m G N. Moreover, the norms of the operators fm are totally bounded
||fm|| < HKkA-Jm) | Lk(U)|| < 11Mm(■ ) | Lk(U)|| < M < œ.
Take a k-form w G Lp(W,Ak) nC(W,Ak) and set am = f*n(w). Since \\fm|| < M, the sequence of forms am is bounded in Lq(U, Ak). Therefore, we can extract a weakly converging subsequence. We assume that the sequence am converges weakly in Lq(U, Ak) to a form <r0. The weak convergence of forms means that coefficients of the forms am converge weakly in Lq(U) to the corresponding coefficients of the form a0. Since the sequence am converges weakly in Lq(U, Ak) to a0 as m — to, we have
IK | Lq (U )\\ < Hm \am | Lq (U )\\ = Hm \f*w | Lq (U )\\ <
m—^^o m—^^o
< Hm \\fm\H\w | Lp(U)\\ < M .\\w | Lp(U)\\. (6)
k—w
The following lemma is proved in the book [8, Chapter 2, §4].
Lemma 1. [8] Suppose U is an open subset in Rn, and suppose that <m = (<m1, <m2,..., <mk), 1 ^ k ^ n, m = 1, 2,..., is a sequence of vector-functions of Wl1loc(U), k < I, locally bounded in WI1(U). Assume that, as m — to, the functions <m converge in L1,ioc to a vector function <0 = (<£>01, <02,..., <0k), and set Wm = d<m1 A d^m2 A ... A d^mk. Then the sequence of forms wm weakly converges in Ll/kloc(U) to a form w01.
Since the homeomorphisms fm locally uniformly converge to f and the form w has continuous coefficients, the functions wp(fm(x)) converge locally uniformly to wp(f (x)) as m — to. Lemma 1 implies that the minors of the matrices Dfm converge weakly in Ll/kloc(U) to minors of the matrix Df. Therefore, the forms am converge weakly in Ll/k loc(U) to f*(w).
It is not hard to see that both limits a0 and f *(w) coincide: a0 = = f *(w). In view of (6) the mapping f induces a bounded operator f * : £P(W, Ak) —y Lq(U, Ak), 1 <q < p< to. By Corollary 1, f eCVk(U; W).
First, we consider the case q < p. The following lemma is valid.
Lemma 2. [2] Every sequence of mappings hm, m = 1, 2,..., that is bounded in L1(U), contains a subsequence, converging in the biting sense to some function h G L1(U).
This lemma implies existence of a function U B x — M(x) of LK(U) such that some subsequence of the function hm = Mm(x)K converges
xIt means that the sequence of forms wm converges weakly in Lj/j.(D) to a form wo on every subdomain D g U.
in the biting sense to h = M(x)K. We assume that the given sequence Mm(x)K converges in the biting sense to the function M(x)K (the set of the Definition 3 is denoted by Ev).
Now we estimate the distortion coefficient of the limit mapping f. For this, we consider estimates on a set Ev. Let Zm be the set of zeros of the Jacobian of the mapping fm. Since the rank of the matrix Dfm on the set Zis less then k, it follows that all k-th-order minors are equal to zero on the set Zm.
Applying the Holder inequality, and taking into account that K + p = 1 on each intersection Ev n B(x0,r), where x0 G Ev, B(x0,r) c U, in view of (5) we have
f \Akfm(x)\q dx = f fm(x)9q \J(x,fm)\p dx <
J J \J(x,fm)\ p
Ev nB(xo,r) (Ev nB(x0,r))\Zm
< ( / fK dx) K ( / \J <x^)\pp dx) P =
(Ev nB(xo,r))\Zm Ev nB(xo,r)
q q
J (Kk,P(fm))K (x) dx^j y \ J (x,fm) \ dx^j ' <
EvnB(xo,r) EvnB(xo,r)
q q
J MKK (x) dx^j J \ J (x,fm) \ dx^j P . (7)
EvnB(xo,r) B(xo ,r)
Elements of the matrix Ak (fm)(x) are the k-th-order minors of ap Dmf (x). In view of Lemma 1, they converge weakly in L[/k,loc (U) to elements of the matrix Ak(f )(x). Since q < l/k, Ak(fm)(x) converges weakly in L9,joc(U) to Ak(f )(x). Since the norm is semicontinuous in the Banach space L9, the left-hand side of the inequality can be estimated as
i \ Akf (x)\ 9 dx < lim i \ Ak fm(x)\ 9 dx.
J m^tt J
Ev nB(xo,r) Ev nB(xo,r)
We have also / \ J(x, fm)\ dx < \ fm(B(xo,r)\ .
B(xo,r)
Since \ f (B(x0,r))\ < to and the mapping f is a homeomorphism, the images f (S(x0,r)) of the spheres S(x0,r) do not intersect under different
r. It follows that the n-measure of the image of any sphere is zero for almost all r: |f (S(x0,r))| = 0. We fix r so that
If (S(xo,r))l =0
and surround the image of the sphere f (S(x0,r)) by an e-neighborhood U£. Since the mappings fm converge locally uniformly to the mapping f, it follows that, starting from a number m0, the images of the spheres fm(S(x0,r)), m > m0, are contained in this e-neighborhood. It is clear that |Ue| ^ 0 as e ^ 0, and hence Ifm(B(x0,r)I ^ |f (B(x0,r)l as m ^ to.
Taking into account that Mm(x)K converge in the biting sense to M(x)K, we pass to the lower limit in (7) as m ^ to. We get
q
J A f^XEv (x) dx ^ J MK (x)xEv (x) dx^j ^ f(B(x0,r)|p.
B(xo,r) B(xo,r)
Dividing both sides of this inequality by the measure of the ball B (x0, r), we obtain the following inequality
1 / |Ak f (x)|qXEv (x) dx <
|B(x0,r)|
B(xo,r)
^ ( Ttt7~—tt i Mk (x)xe (x) d^ K(|f (B(x0,r^) P ■ (8) ^ {B^r^ J ( )XEv ( ) J V |B(x0,r)U
B(x0,r)
Since the homeomorphism f is Sobolev differentiable, then by [13, Section 2.3, formula (2.5)] we have
|f^r^0^ ^ |J(x0,f)| as r ^ 0 for almost all x0 e Ev. |B(x0,r)|
Hence, by the Lebesgue differentiability theorem, letting r go to 0 we obtain that
|Akf (x)|q < Mq(x)| J(x,f)|p for almost all x e Ev. (9)
As U = U EV, the point-wise inequality (9) holds in U almost everywhere.
V
In the case 1 < q = p < to we can assume that a sequence of numbers j||Mm | L^(U)||}meN converges to M = Hm ||Mm | L^(U)|| e R. In
m—^oo
this case, instead of (7), for any e > 0 there exists m1 such that for all m ^ m1 we have
J IAk fm(x)lq dx C (M + e)^ J IJ (x, fm) I dx
Corollary. [16] Let fm G CV'k,p(U; W), m G N, be a sequence of home-omorphisms of the Sobolev class Wl1loc(U) with k < l, q C l/k, 1 < q C
kf
B(xo,r) B(xo,r)
Further, proceeding as in the case q < p, we obtain the estimation
Kk,p(x,f) C (M + e) for almost all x G U.
Since e > 0 is an arbitrary number, we get the desired estimation.
By the Corollary 1, we have proved that the limit mapping f belongs to CVkqp(U; W). □
As a straightforward consequence of Theorem 4 we get the following
k
p(U
oc(U) with k < ^ q c i/k C p < to. Suppose that the sequence fm is locally bounded in Wl1(U), and converge locally uniformly to a homeomorphism f : U — W as m — to.
Assume also that there exists a function M(x) G LK(U), — = 1 — p, such
K q p
that
Kk,p(',fm)(x) c M(x) for all m G N
in U almost everywhere. Then the limit mapping f belongs to CV'k p(U; W) and Kk,p(-,f) G Lk(U), where K = 1 — p.
Moreover, the inequality Kk,p(x,f) C M(x) holds almost everywhere.
Remark. There exists a misprint in the paper [16]: in the statement of the main result the condition q C l/k is missing.
Acknowledgment. This work was supported by Russian Foundation for Basic Research, agreement 17-01-00875 for the first author, and by the program of fundamental scientific researches of the SB RAS I.1.2., project 0314-2016-0006 for the second author.
References
[1] Baykin A. N., Vodop'yanov S. K., Capacity estimates, Liouville's theorem, and singularity removal for mappings with bounded (p,q)-distortion. Sib. Math. J., 2015, vol. 56, no. 2, pp. 237-261. DOI: http://doi.org/10.1134/S0037446615020056.
[2] Brooks J. K., Chacon R. V., Continuity and compactness of measures. Adv. Math., 1980, vol. 37, no.1, pp. 16-26.
DOI: http://doi.org/10.1016/0001-8708(80)90023-7.
[3] Federer H. Geometric Measure Theory. Springer-Verlag, 1969.
[4] Fusco N., Moscariello G., Sbordone C., The limit of W1,1 homeomorphisms with finite distortion. Calc. Var., 2008, vol. 33, no. 3, pp. 377-390.
DOI: http://doi.org/10.1007/s00526-008-0169-2.
[5] Gehring F. W., Iwaniec T., The limit of mappings with finite distortion. Ann. Acad. Sci. Fenn. Math., 1999, vol. 24, pp. 253-264.
[6] Hajlash P., Change of variables formula under minimal assumptions. Col-loq. Math., 1993, vol. 64, no. 1, pp. 93-101.
DOI: http://doi.org/10.4064/cm-64-1-93- 101.
[7] Iwaniec T., Sverak V., On mappings with integrable dilatation. Proc. Amer. Math. Soc., 1993, vol. 118, no. 1, pp. 181-188.
DOI: http://doi.org/10.1090/S0002-9939- 1993- 1160301-5.
[8] Reshetnyak Yu. G. Space Mappings with Bounded Distortion. Amer. Math. Soc., 1989.
[9] Stein E. M. Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, 1970.
[10] Troyanov M., Vodop'yanov S. K., Liouville type theorems for mappings with bounded (co)-distortion. Ann. Inst. Fourier, Grenoble. 2001. vol. 52, no. 6, pp. 1753-1784.
[11] Vodop'yanov S. K., Closure of classes of mappings with bounded distortion on Carnot groups. Sib. Adv. Math., 2004, vol. 14, no. 1, pp. 84-125.
[12] Vodop'yanov S. K., Spaces of differential forms and maps with controlled distortion. Izvestiya: Mathematics, 2010, vol. 74, no. 4, pp. 663-689. DOI: http://doi.org/10.1070/IM2010v074n04ABEH002502.
[13] Vodop'yanov S. K., Regularity of mappings inverse to Sobolev mappings. Sbornik: Mathematics, 2012, vol. 203, no. 10, pp. 1383-1410.
DOI: http://doi.org/10.1070/SM2012v203n10ABEH004269.
[14] Vodop'yanov S. K., Quasiconformal analysis of two-indexed scale of spatial mappings and its applications. Abstracts International Conference on Complex Analysis and its Applications (Krasnodar, June, 02-09), 2018, pp. 25-27.
[15] Vodop'yanov S. K., Gol'dstein V. M., Quasiconformal mappings and spaces of functions with generalized first derivatives. Sib. Math. J., 1976, vol. 17, no. 3, pp. 399-411. DOI: http://doi.org/10.1007/BF00967859.
[16] Vodop'yanov S. K., Kudryavtseva N. A., On the convergence of mappings with k-finite distortion. Math. Notes, 2017, vol. 102, no. 5-6, pp. 878-883. DOI: http://doi.org/10.1134/S0001434617110281.
[17] Vodopyanov S. K., Molchanova A. O., Lower semicontinuity of mappings with bounded (6,1)-weighted (p,q)-distortion. Sib. Math. J., 2016, vol. 57, no. 5, pp. 778-787. DOI: http://doi.org/10.1134/S0037446616050062.
Received May 31, 2018. In revised form, June 18, 2018. Accepted September 18, 2018. Published online September 27, 2018.
N. A. Kudryavtseva Novosibirsk State University 1 Pirogova str., Novosibirsk 630090, Russia E-mail: [email protected]
S. K. Vodopyanov
Sobolev Institute of Mathematics
4 Akademika Koptyuga pr., Novosibirsk 630090, Russia E-mail: [email protected]
