On mappings satisfying the inverse Poletsky inequality
E. A. Sevost'yanov S. O. Skvortsov O. P. Dovhopiatyi




lemma



corollary



remark



theorem


example





proposition












517.5

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  ,       , . )

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   )

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 ­,       , . ')

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   ­)

E.A. Sevost'yanov (Zhytomyr Ivan Franko State University,
Institute of Applied Ma­the­ma­tics and Mechanics of NAS of
Ukraine, Slov'yans'k)

S.O. Skvortsov, O.P. Dovhopiatyi (Zhytomyr Ivan Franko State
University)

,    

,     

On mappings satisfying the inverse Poletsky inequality

,
      .
 ,  ,   
        .

,
       .
,  ,   '          .

We have studied the local and boundary behavior of mappings satisfying one estimate of the distortion of the modulus of families of paths.
In particular, we have obtained conditions under which the families of the indicated mappings are equicontinuous inside and on the boundary of a domain.


1. .  ,      ( )
      


    M(Γ)⩽ N(f, A)· K· M(f(Γ)) ,


 M –    Γ   D, 
    N(y, f, A) =  card {x∈ A:
    f(x)=y} ,      N(f, A) = sup_y∈
    R^n N(y, f, A) ,


A –     D,  K⩾ 1 –
 ,      K= ess
sup  K_O(x, f), K_O(x, f)=‖ f^ '(x)‖^n/J(x, f)
 J(x, f) 0; K_O(x, f)=1  f^ '(x)=0,  K_O(x,
f)=∞  f^ '(x) 0,  J(x, f)=0 (., .,
<cit.>  <cit.>).
        
.  ,     
     


    M(Γ)≤∫_f(E)K_I(y, f^ -1,
    E)·ρ_*^n(y) dm(y) ,


 E –     D, Γ
–     E  ρ_*(y)∈
adm f(Γ) (., ., <cit.>).  
     ,
      
 (<ref>).   .  y_0∈ R^n,
0<r_1<r_2<∞ 


    A(y_0, r_1,r_2)={ y ∈  R^n:
    r_1<|y-y_0|<r_2} .


   E, F⊂ R^n  
D⊂ R^n   Γ(E,F,D)  
 γ:[a,b]→ R^n , 
γ(a)∈ E,γ(b)∈ F  γ(t)∈ D  t ∈ (a,
b).  f:D→ R^n –  ,
y_0∈ f(D)  0<r_1<r_2<∞,   Γ_f(y_0, r_1,
r_2)      γ   D
,  f(γ)∈Γ(S(y_0, r_1), S(y_0, r_2),
A(y_0,r_1,r_2)).  Q: R^n→ [0, ∞] –
   .   ,  f
      y_0∈
f(D),  


    M(Γ_f(y_0, r_1, r_2))⩽∫_f(D) Q(y)·η^n (|y-y_0|)  dm(y)


       η:
(r_1,r_2)→ [0,∞ ] , 


    ∫_r_1^r_2η(r)  dr⩾ 1 .


   (<ref>)   
   <cit.>. ,  
     
 (<ref>)      Q, .,
., <cit.>.    ,   
,     .  , 
     (<ref>)
        
    <cit.>.  
     .

D⊂ R^n, n⩾ 2,   
  Q: R^n→ [0, ∞]  
F_Q(D)     
f:D→ R^n ,   (<ref>)
    y_0∈ f(D).  ,
  , .


   Q∈ L^1( R^n).   
C_n>0,      n,  
  x_0∈ D   0<2r_0< dist (x_0,
∂ D)   


    |f(x)-f(x_0)|⩽C_n· (‖
    Q‖_1)^1/n/log^1/n(1+r_0/|x-x_0|)  ∀  x∈
    B(x_0, r_0) ,  ∀   f∈F_Q(D) ,


 ‖ Q‖_1 –   Q  L^1( R^n). 
,  F_Q(D)   
D.


,   D⊂ R^n  ­ x_0∈∂ D,    
U  x_0   V⊂ U  x_0 ,
 V∩ D .  D    ∂ D,
 D      x_0∈∂ D. 
 D     x_0∈∂ D,    P>0     U
 x_0   V⊂ U    ,
 M(Γ(E, F, D))>P    E, F⊂
D,  ∂ U  ∂ V.   D
  ,     
   D.

δ>0,  D, D^ '⊂ R^n,
n⩾ 2,  A⊂ D^ '  
    Q:D^ '→ [0, ∞]
  S_δ, A, Q (D, D^ ')
       f
 D  D^ ',   (<ref>)
  y_0∈ f(D),  ,  h(f^ -1(A),
∂ D)=inf_x∈ f^ -1(A), y∈∂ Dh(x, y)⩾ δ. 
 .


,   D    . 
Q∈ L^1(D^ '),   D^ '  
  ,   f∈ S_δ, A, Q (D,
D^ ')     
f:D→D^ ',
f|_D=f,  ,
f(D)=D^ '  
S_δ, A, Q (D, D^ '),
    
f:D→D^ ',
   D. 



  <ref>    
  ,  ,  
ε>0  δ=δ(ε, x_0)>0 ,
  |x-x_0|<δ  
|f(x)-f(x_0)|<ε   f∈F_Q(D). 
 <ref>    
  , ..,   ε>0
 δ=δ(ε, x_0)>0 ,  
h(x, x_0)<δ   h(f(x),
f(x_0))<ε   f∈
S_δ, A, Q (D, D^ ').


2.      
.

 <ref>.  x_0∈ D,
0<r_0< dist (x_0,
∂ D)  f∈F_Q(D).  x∈ B(x_0, r)  


    |f(x)-f(x_0)|:=ε_0 .



   f(x)  f(x_0) 
r=r(t)=f(x_0)+(f(x)-f(x_0))t, -∞<t<∞
(.  <ref>).



 γ^1:[1, c)→ D, 1<c⩽∞ –
 f-  r=r(t), t⩾ 1,   
 x,   <cit.>.   



    h(γ^1(t), ∂ D)→ 0


 t→ c-0,  h(A, B)=sup_x∈ A,y∈ Bh(x,
y)  h(x, y) –     x, y∈ R^n (. <cit.>).

,  γ^2:(d, 0]→ D, -∞⩽
d<0 –  f-  r=r(t), t⩽ 0, 
   x_0, 
 <cit.>. ,    (<ref>) 
, 


    h(γ^2(t), ∂ D)→ 0


 t→ d-0.   (<ref>)  (<ref>), 
 <cit.>,   
1⩽ t^1<c  1⩽ t^2<d  
x^1:=γ^1(t^1)  x^2:=γ^2(t^2)∈ S(x_0, 2r_0). 
 ,  ,  γ^1(t)∈ B(x_0,
2r_0)   t∈ [1, t^1]  γ^2(t)∈ B(x_0, 2r_0) 
 t∈ [t^2, 0].




    β^1:=γ^1|_[1, t^1],    β^2:=γ^2|_[1,
    t^2],    Γ:=(|β^1|, |β^2|, B(x_0, 2r_0)) .



     <cit.>


    M(Γ)⩾ (1/2)· M(Γ(|β^1|, |β^2|, 
    R^n)) ,


   ,  <cit.>


    M(Γ(|β^1|, |β^2|,  R^n))⩾
    c_n·log(1+1/m) ,


 c_n>0 –  ,    n,


    m=d(|β^1|, |β^2|)/min{d(|β^1|),
    d(|β^2|)},   d(|β^1|, |β^2|)=inf_x∈
    |β^1|, y∈|β^2||x-y| .


,  diam (|β^i|)=sup_x,
y∈|β^i||x-y|⩾ r_0, i=1,2. 
 (<ref>)  (<ref>)  ,  d(|β^1|,
|β^2|)⩽ |x-x_0|,   , 


    M(Γ)⩾c_n·log(1+r_0/d(|β^1|, |β^2|))⩾c_n·log(1+r_0/|x-x_0|) ,


 c_n>0 –  ,   
n.

M(Γ).  P – 
 r(t),    f(x) 
z^1:=f(x^1)=f(γ^1(t^1)),  Q –   r(t),
   f(x_0) 
z^2:=f(x^2)=f(γ^2(t^2)). 



    A:=A(z^1, ε^1, ε^2)={x∈ R^n: ε^1<|x-z^1|<ε^2} ,


 ε^1:=|f(x)- z^1|, ε^2:=|f(x_0)-z^1|.

, 


    f(Γ)>Γ(S(z^1, ε^1), S(z^1, ε^2), A) .


  ,  γ∈Γ.  f(γ)∈
f(Γ), f(γ)=f(γ(s)):[0, 1]→ R^n,
f(γ(0))∈ P, f(γ(1))∈ Q  f(γ(s))∈ f(D)
 0<s<1.  q>1 – ,  


    z^1=f(x_0)+(f(x)-f(x_0))q .
 
f(γ(0))∈ P,  1⩽ t⩽ q , 
f(γ(0))=f(x_0)+(f(x)-f(x_0))t. ,


    |f(γ(0))-z^1|=|(f(x)-f(x_0))(q-t)|⩽



    ⩽
    |(f(x)-f(x_0))(q-1)|=|(f(x)-f(x_0))q+f(x_0)-f(x))|=



    =|f(x)-z^1|=ε^1 .


  ,  f(γ(1))∈ Q, 
p⩽ 0 , 

    f(γ(1))=f(x_0)+(f(x)-f(x_0))p .
   ,
 , 


    |f(γ(1))-z^1|=|(f(x)-f(x_0))(q-p)|⩾



    ⩾ |(f(x)-f(x_0))q|=|(f(x)-f(x_0))q +f(x_0)-f(x_0)|=



    =|f(x_0)-z^1|=ε^2 .


, 


    |f(x_0)- f(x)|+ε^1=|f(x_0)- f(x)|+|f(x)-z^1|=
    |z^1-f(x_0)|=ε^2 ,


, , ε^1<ε^2.   (<ref>)
, 


    |f(γ(1))-z^1|>ε^1 .


 f(γ(0))∉S(z^1, ε^1), 
 (<ref>)  (<ref>) ,  |f(γ)|∩ B(z^1,
ε^1)∅ (f(D)∖ B(z^1,
ε^1))∩|f(γ)|.   , 
 <cit.>  t_1∈ (0, 1) , 
f(γ(t_1))∈ S(z^1, ε^1).   ,
  ,  f(γ(t))∉B(z^1, ε^1)
 t∈ (t_1, 1).  α^1:=f(γ)|_[t_1, 1].

,  ε^1<ε^2 
f(γ)_1(t_1)∈ S(z^1, ε^1),  , 
|α^1|∩ B(z^1, ε^2)∅. 
 (<ref>)  ,  (f(D)∖ B(z^1,
ε^2))∩ |α^1|∅. ,
 <cit.>  t_2∈ [t_1, 1) ,
 α^1(t_2)∈ S(z^1, ε^2).

  ,   ,  f(γ)_1(t)∈
B(z^1, ε^2)  t∈ (t_1, t_2). 
α^2:=α^1|_[t_1, t_2].  f(γ)>α^2 
α^2∈Γ(S(z^1, ε^1), S(z^1, ε^2),
A).  ,  (<ref>) .

 (<ref>) ,  Γ>Γ_f(z^1, ε^1,
ε^2).  


    η(t)= {[         1/ε_0, t∈ [ε^1, ε^2],;             0, t∉[ε^1, ε^2] . ].


,  η   (<ref>) 
r_1=ε^1  r_2=ε^2.   ,
 (<ref>)  (<ref>) , 


    r_1-r_2=ε^2-ε^1=|f(x_0)-z^1|-|f(x)-
    z^1|=

    =|f(x)-f(x_0)|=ε_0 .
 
∫_ε^1^ε^2η(t) dt=(1/ε_0)·
(ε^2-ε^1)⩾ 1. 
 (<ref>),    z^1 
     F_Q(D) 
, 


    M(Γ)⩽ M(Γ_f(z^1, ε^1, ε^2)) ⩽



    ⩽1/ε_0^n∫_ R^n
    Q(y) dm(y)=‖ Q‖_1/|f(x)-f(x_0)|^n .


 (<ref>)  (<ref>) , 


    c_n·log(1+r_0/|x-x_0|)⩽‖ Q‖_1/|f(x)-f(x_0)|^n .


    
 (<ref>), 
C_n:=c_n^-1/n. 

3.      
.      
   (<ref>)   , .,
., <cit.>, <cit.>  <cit.>.  
 –     , 
.    .


D⊂ R^n, n⩾ 2, – ,
   ,   D^ '⊂
R^n     . , f – 
     D  D^ ',
  (<ref>)    y_0∈
f(D),  Q∈ L^1(D^ ').   f
    
f:D→D^ ', 
, f(D)=D^ '.



    x_0∈∂ D.
     
f   x_0.    
 φ:∞↦ 0   
 M     (<ref>)
(. <cit.>),   ,  x_0∞.

,      
f   x_0   .     
 x_i, y_i∈ D, i=1,2,… , , 
x_i, y_i→ x_0  i→∞,  ,
h(f(x_i), f(y_i))⩾ a>0   a>0   i∈ N,  h –  ,
. <cit.>.    
R^n   ,  
f(x_i)  f(y_i)   i→∞  z_1 
z_2, ,  z_1∞.   
f ,     ,
. <cit.>,  z_1, z_2∈∂
D^ '.   D^ '   
 ,    U_1  U_2
 z_1  z_2 ,  W_1=D^ '∩ U_1 
W_2=D^ '∩ U_2  .  , 
W_1  W_2  ,    
U_1  U_2     (.,
., <cit.>). .  <ref> 
 .



 , 


    U_1⊂ B(z_*, R_0),    B(z_*, 2R_0)∩U_2=∅ ,     R_0>0 ,


 z_*∈ D^ ' –  ,   
z_1.    ,  f(x_i)∈ W_1  f(y_i)∈ W_2
  i=1,2,… .   f(x_i)  f(x_1) 
α_i:[0, 1]→ D^ ',   f(y_i) 
f(y_1)  β_i:[0, 1]→ D^ ' , 
|α_i|⊂ W_1  |β_i|⊂ W_2  i=1,2,…
.  α_i:[0, 1]→ D^ ' 
β_i:[0, 1]→ D^ ' – 
  α_i  β_i     x_i 
y_i,  (  
 <cit.>). ,    f(x_1) 
f(y_1)   D      
   f, . <cit.>. 
 r_0>0 ,  α_i(1),
β_i(1)∈ D∖ B(x_0, r_0)  
i=1,2,… .    D  
,   P>0  i=i_P⩾ 1 , 


    M(Γ(|α_i|, |β_i|,
    D))>P    ∀  i⩾ i_P .


,   (<ref>)  
 f  (<ref>).   ,  
 (<ref>)   <cit.>


    f(Γ(|α_i|, |β_i|,
    D))>Γ(S(z_*, R_0), S(z_*, 2R_0), A(z_*, R_0, 2R_0)) .


 (<ref>) , 


    Γ(|α_i|, |β_i|, D)
    >Γ_f(z_0, R_0, 2R_0) .


 η(t)= {[          1/R_0, t∈ [R_0, 2R_0],;              0,   t∉[R_0, 2R_0] ]. .

,  η   (<ref>) 
r_1=R_0  r_2=2R_0.   (<ref>)  (<ref>) 
, 


    M(Γ(|α_i|, |β_i|, D))
    ⩽ M(Γ_f(z_0, R_0, 2R_0)) ⩽



    ⩽1/R_0^n∫_D^ '
    Q(y) dm(y):=c<∞ ,


 Q∈ L^1(D). ,  (<ref>)
  (<ref>).  
       f 
 x_0.

  
f(D)=D^ '. ,
f(D)⊂D^ '. ,
 D^ '⊂f(D). 
 ,  y_0∈D^ ',  
y_0∈ D^ ',  y_0∈∂ D^ '. 
y_0∈ D^ ',  y_0=f(x_0)  y_0∈f(D),    f – 
 D  D^ '. ,  y_0∈∂
D^ ',    y_k∈
D^ ' ,  y_k=f(x_k)→ y_0 
k→∞  x_k∈ D.    
R^n   ,  x_k→ x_0,
 x_0∈D. ,  x_0∈∂ D,  
 f  .  f(x_0)=y_0∈f(∂ D)⊂f(D). 
 . 


4.      
 .

 <ref>.  f∈
S_δ, A, Q (D, D^ ').   <ref>
 f    
f:D→D^ ', 
, f(D)=D^ '.
   S_δ, A, Q
(D, D^ ')  D 
  <ref>.   
    ∂ D^ '.

   . ,  
x_0∈∂ D,  ε_0>0,  x_m∈D,
   x_0   
f_m∈ S_δ, A, Q (D,
D) , 


    h(f_m(x_m),f_m(x_0))⩾ε_0,  
    m=1,2,… .


 f_m:=f_m|_D.  f_m 
   ∂ D,  ,  x_m∈
D. , f_m(x_m)=f_m(x_m).  ,
  x^ '_m∈ D , 
x^ '_m→ x_0  m→∞ 
h(f_m(x^ '_m),f_m(x_0))→ 0 
m→∞.   R^n
,   ,  f_m(x_m)  f_m(x_0)
  m→∞.  f_m(x_m)→x_1  f_m(x_0)→x_2
 m→∞.     (<ref>),
x_1x_2.   , 
,  x_1∞.   f_m
,     (. <cit.>),
 x_2∈∂ D.  x_1 
x_2    A,    
   x_1.  <cit.> 
  x_1, x_1  x_2,
x_2    γ_1:[0, 1]→D  γ_2:[0, 1]→D ,
 |γ_1|∩ |γ_2|=∅, γ_1(t),
γ_2(t)∈ D  t∈ (0, 1), γ_1(0)=x_1,
γ_1(1)=x_1, γ_2(0)=x_2 
γ_2(1)=x_2.  D^ ' 
   ∂ D^ ',   U_1 
U_2  x_1  x_2,   
,  ,  W_i:=D^ '∩ U_i
 .     , 
U_1⊂ B(x_1, δ_0) 


    B(x_1,
    δ_0)∩|γ_2|=∅=U_2∩|γ_1| ,
      B(x_1, δ_0)∩U_2=∅ .


  ,  f_m(x_m)∈ W_1  f_m(x^ '_m)∈
W_2   m∈ N.  a_1  a_2 –  
,  |γ_1|∩ W_1  |γ_2|∩ W_2.
 0<t_1, t_2<1 ,  γ_1(t_1)=a_1 
γ_2(t_2)=a_2.   a_1  f_m(x_m) 
α_m:[t_1, 1]→ W_1 ,  α_m(t_1)=a_1 
α_m(1)=f_m(x_m). ,  a_2 
f_m(x^ '_m)  β_m:[t_2, 1]→ W_2,
β_m(t_2)=a_2  β_m(1)=f_m(x^ '_m) (.
 <ref>).







    C^1_m(t)=  {[      γ_1(t), t∈ [0, t_1],;      α_m(t),  t∈ [t_1, 1] ]. ,    
    C^2_m(t)=  {[      γ_2(t), t∈ [0, t_2],;      β_m(t),  t∈ [t_2, 1] ]. .


 D^1_m  D^2_m –    |C^1_m| 
|C^2_m|     x_m  x^ '_m,
.     <cit.>, 
,    h(f_m^ -1(A),
∂ D)=inf_x∈ f_m^ -1(A), y∈∂ Dh(x,
y)⩾ δ,     S_δ, A, Q (D, D^ '),   D^1_m 
D^2_m,    b_m^1  b_m^2,  
 D  ,   δ.

 ,   |C^1_m|  |C^2_m|  
C^1_m  C^2_m, . 


    l_0=min{ dist (|γ_1|,
    |γ_2|),  dist (|γ_1|, U_2)} ,


  A_0:=⋃_x∈ |γ_1|B(x,
l_0/4)  |γ_1|  . 
|γ_1| –  ,    
 1⩽ N_0<∞   
z_1,…, z_N_0∈ |γ_1| ,  |γ_1|⊂
B_0:=⋃_i=1^N_0B(z_i, l_0/4).

  ,


    |C^1_m|⊂ U_1∪ |γ_1|⊂B(x_1, δ_0)∪⋃_i=1^N_0B(z_i, l_0/4) .


 Γ_m –   ,  |C^1_m| 
|C^2_m|  D^ '.   , 


    Γ_m=⋃_i=0^N_0Γ_mi ,


 Γ_mi –    γ:[0, 1]→
D^ ' ,  γ(0)∈ B(z_i, l_0/4)∩ |C^1_m| 
γ(1)∈ |C_2^m|  1⩽ i⩽ N_0. ,
 Γ_m0 –   γ:[0, 1]→
D^ ' ,  γ(0)∈ B(x_1,
δ_0)∩ |C^1_m|  γ(1)∈ |C_2^m|. 
 (<ref>)  σ_0>δ_0>0 , 


    B(x_1,
    σ_0)∩|γ_2|=∅=U_2∩|γ_1| ,
      B(x_1, σ_0)∩U_2=∅ .


 ,     <ref> 
 <cit.>,   , 


    Γ_m0>Γ(S(x_1, δ_0), S(x_1, σ_0), A(x_1, δ_0,
    σ_0)) ,




    Γ_mi>Γ(S(z_i, l_0/4), S(z_i, l_0/2), A(z_i, l_0/4,
    l_0/2)) .


  x_*∈ D^ ', δ_*>0  σ_*>0
,  A(x_*, δ_*, σ_*)⊂ A(x_1,
δ_0, σ_0), 


    Γ(S(x_1, δ_0), S(x_1, σ_0),
    A(x_1, δ_0, σ_0))>Γ(S(x_*, δ_*),
    S(x_*, σ_*), A(x_*, δ_*, σ_*)) .





    η(t)= {[             4/l_0, t∈ [l_0/4, l_0/2],;                 0, t∉[l_0/4, l_0/2] , ].     η_0(t)= {[   1/(σ_*-δ_*), t∈ [δ_*, σ_*],;             0, t∉[δ_*, σ_*] . ].



 Γ^ *_m:=Γ(|D_m^1|, |D_m^2|, D). ,
 f(Γ^ *_m)⊂Γ_m.    (<ref>),
(<ref>)  (<ref>)


    Γ^ *_m>(⋃_i=1^N_0Γ_f_m(z_i,
    l_0/4, l_0/2))∪Γ_f_m(x_*, δ_*, σ_*) .


  f_m   (<ref>)
 f_m(D),  (<ref>) , 



    M(Γ^ *_m)⩽
    (4N_0/l_0^n+(1/(σ_*-δ_*))^n)‖ Q‖_1:=c<∞ .



,   (<ref>)   
  .   ,  


    h(|D^1_m|)⩾ h(x_m, b_m^1) ⩾
    (1/2)· h(f^ -1_m(A), ∂ D)>δ/2 ,



    h(|D^2_m|)⩾ h(x^ '_m, b_m^2) ⩾ (1/2)·
    h(f^ -1_m(A), ∂ D)>δ/2


  m⩾ M_0    M_0∈ N.

 U:=B_h(x_0, r_0)={y∈ R^n: h(y,
x_0)<r_0},  0<r_0<δ/4   δ  
 (<ref>). ,  |D^1_m|∩
U∅ |D^1_m|∩ (D∖ U)  
m∈ N,  h(|D^1_m|)⩾δ/2  x_m∈
|D^1_m|, x_m→ x_0  m→∞. ,
|D^2_m|∩ U∅ |D^2_m|∩ (D∖ U).
 |D^1_m|  |D^2_m| – ,


    |D^1_m|∩∂ U∅,    |D^2_m|∩∂ U∅ ,


., ., <cit.>.  ∂ D
 ,    P>0   V⊂
U  x_0 , 


    M(Γ(E, F, D))>P


   E, F⊂ D ,  E∩∂ U∅ E∩∂ V  F∩∂
U∅ F∩∂ V. ,   
 m∈ N


    |D^1_m|∩∂ V∅,    |D^2_m|∩∂ V∅ .


  , x_m∈ |D^1_m|  x^ '_m∈ |D^2_m|, 
x_m, x^ '_m→ x_0∈ V  m→∞.
  , |D^1_m|∩ V∅ |D^2_m|∩ V 
  m∈ N. ,  h(V)⩽
h(U)⩽ 2r_0<δ/2.   (<ref>)
h(|D^1_m|)>δ/2. , |D^1_m|∩ (D∖
V)∅ , , |D^1_m|∩∂ V∅
(., ., <cit.>). ,
h(V)⩽ h(U)⩽ 2r_0<δ/2.  (<ref>)
,  h(|D^2_m|)>δ, , |D^2_m|∩
(D∖ V)∅.  <cit.> 
,  |D^2_m|∩∂ V∅.  ,
 (<ref>) . 
 (<ref>), (<ref>)  (<ref>),  ,
 M(Γ^ *_m)=M(Γ(|D^1_m|, |D^2_m|, D))>P. 
  (<ref>),    . 

5. .


    B^n 
f_m(z)=mz, m=1,2,… . ,    
f_m,     f^ -1_m 
 (<ref>)     Q≡ 1
(., ., <cit.>). , 
 f_m    , 
   Q≡ 1  R^n. 
  ,   f_m^ -1:
R^n→ R^n    R^n,
     <ref>.   ,
   y_0∈ R^n  
 g_m:=f^ -1_m|_B(y_0, r_0), r_0>0.  
   m_0∈ N : g_m: B(y_0,
r_0)→ B^n  m⩾ m_0.  
 Q(x)≡ 1  x∈ B^n  Q(x)≡ 0 
x∉ B^n,  ,  g_m
 (<ref>)  Q∈ L^1( R^n). 
,    <ref> .




  <cit.>  <cit.>  
 ,   (<ref>).
        
.  p⩾ 1  ,   2/p
 1,  ,  , α∈ (0, 2/p) – 
.    f_m: B(0,
2)→ B^2  : f_m(z)=(f∘f_m)(z), f(z)=z^2,


    f_m(z) = {[ (|z|-1)^1/α/|z|· z ,    1+1/(m^α)⩽|z|⩽ 2,;   1/m/1+(1/m)^α· z ,   0<|z|<1+1/(m^α)  . ].


 ,  
 <cit.>,  , 
  K_O(z, f_m)  f_m   z∈
B^2   K_O(z,
f_m)=1/α·|z|/|z|-1  z∈ B(0,
2)∖ B^2, K_O(z, f_m)=1  z∈
B^2. ,    z∈ B^2∖ B(0,
1/m)       f_m,  ,
w=re^iφ↦±((√(r))^ α+1)e^iφ/2.
 w∈ B(0, 1/m)      
 f_m, , z^1_m  z^2_m.  
      w=0.

 ,  w∈ B^2    α


    K_I(w, f_m^ -1,  B^2):=∑_z∈ f_m^ -1(w)∩ B^2K_O(z, f_m)
    ⩽2/α·(√(r))^α+1/(√(r))^α⩽4/α
    r^α∈ L^p( B^2) .


 <cit.>  f_m 
 (<ref>)  
Q(y)=4/α|y|^α, y∈ B^2; Q(y)≡
0  y∈ R^2∖ B^2. 
,     {f_m}_m=1^∞
    <ref>.    
    <ref>,  f_m 
     B^2  
m⩾ 2.



    , 
 <ref>,    
 n⩾ 2.  ,   p⩾ 1
,  n/p(n-1),    α∈
(0, n/p(n-1)).  f(x)=(rcos 2φ, rsin 2φ,
x_3,…, x_n), x=(x_1,x_2,…, x_n),
r=√(x_1^2+x_2^2).   
f_m: B(0, 2)→ B^n  :
f_m(x)=(f∘f_m)(x), 


    f_m(x) = {[ (|x|-1)^1/α/|x|· x ,    1+1/(m^α)⩽|x|⩽ 2,;   1/m/1+(1/m)^α· x ,   0<|x|<1+1/(m^α)  . ].


 ,  
 <cit.>,  , 
K_O(x,
f_m)⩽2^n-1/α^n-1·|x|^n-1/(|x|-1)^n-1
 x∈ B(0, 2)∖ B^n, K_O(x,
f)=2^n-1  x∈ B^n. ,    x∈ B^n∖ B(0, 1/m)     
 f_m,  ,


    w=(rcosφ, rsinφ,x_3,…,x_n)↦1+|x|^α/|x|· (± rcosφ/2, ±
    rsinφ/2, x_3,…, x_n) .


  ,


    K_I(w, f_m^ -1,  B^n):=∑_x∈ f_m^ -1(w)∩ B^nK_O(x, f_m)
    ⩽2^n-1/α^n-1·(r^α+1)^n-1/(r^α(n-1)⩽2^n-14/α^n-1 r^α(n-1)∈ L^p(
    B^n) .


 <cit.>  f_m 
 (<ref>)  
Q(y)=2^n-14/α^n-1|y|^(n-1)α, y∈
B^n; Q(y)≡ 0  y∈ R^n∖
B^n.  ,    
{f_m}_m=1^∞     <ref>.
        <ref>,
 f_m      
B^n   m⩾ 2.


6.   .    
     <cit.>, .  <cit.>. 
       ,
    .


  n⩾ 2, D  D^ ' –   R^n,  , D    ,   
     ,   D^ '
    .   A – 
  D^ '  δ>0. , f_m –
 ,    
 D  D^ '   :  
m=1,2,…   A_m⊂ D, m=1,2,… ,
,  f_m(A_m)=A  h(A_m)⩾δ>0.  
 f_m   (<ref>)  
 y_0∈ D^ ' ,  , Q∈ L^1(D^ '),
  δ_1>0 , 


    h(A_m,
    ∂ D)>δ_1>0  ∀   m∈
    N ,


 h(A_m,
∂ D)=inf_x∈ A_m, y∈∂ Dh(x, y).



    R^n 
 D     ,   
h(A_m,
∂ D)  .

. ,  
   .    k∈ N 
 m=m_k ,  h(A_m_k,
∂ D)<1/k.  ,   m_k   k.
 A_m_k – ,   x_k∈ A_m_k 
y_k∈∂ D ,  h(A_m_k,
∂ D)=h(x_k, y_k)<1/k (.  <ref>).



 ∂ D –  ,   ,
 y_k→ y_0∈∂ D  k→∞;  


x_k→ y_0∈∂ D  k→∞.


 K_0 –   ∂ D,  
y_0. , K_0 –   R^n.
 ∂ D  ,  D^ ' 
   ,    <ref> 
f_m_k   
f_m_k:D→D^ '.  , 
f_m_k    
D    k,  
f_m_k    D. 
  ε>0 
δ_k=δ_k(ε)<1/k , 



    |f_m_k(x)-f_m_k(x_0)|<ε  ∀   x,x_0∈D,   h(x, x_0)<δ_k ,   δ_k<1/k .


 ε>0 , 


    ε<(1/2)·  dist (∂ D, A) .


 B_h(x_0, r)={x∈ R^n: h(x, x_0)<r}.
  k∈ N, 


    B_k:=⋃_x_0∈ K_0B_h(x_0, δ_k) ,   k∈ N .


 B_k –   K_0,
 <cit.>   U_k  K_0,
,  U_k⊂ B_k  U_k∩ D .  , 
U_k – ,   U_k∩ D   
(. <cit.>).  h(K_0):=sup_x,
y∈ K_0h(x, y)=m_0.   z_0, w_0∈ K_0 , 
h(K_0)=h(z_0, w_0)=m_0. ,  
y_k∈ U_k∩ D, z_k∈ U_k∩ D  w_k∈ U_k∩
D ,  z_k→ z_0, y_k→ y_0
 w_k→ w_0  k→∞.  , 


    h(z_k, w_k)>m_0/2  ∀   k∈ N .


  U_k∩ D  ,   
 z_k, y_k  w_k,   
γ_k∈ U_k∩ D.  ,   
|γ_k|  ()  γ_k   D. 
f_m_k(|γ_k|) –    D. 
x∈|γ_k|,   x_0∈ K_0 ,  x∈ B(x_0,
δ_k).  ω∈ A⊂ D. 
x∈|γ_k| ,  , x –   D, 
   f_m_k(x) 
f_m_k(x).   (<ref>)  (<ref>),
    ,  , 



    |f_m_k(x)-ω|⩾
    |ω-f_m_k(x_0)|-|f_m_k(x_0)-f_m_k(x)|⩾



    ⩾  dist (∂ D^ ', A)-(1/2)·
    dist (∂ D^ ', A)=(1/2)·  dist (∂
    D^ ', A)>ε


   k∈ N,  dist (∂
D, A):=inf_x∈∂ D, y∈ A|x-y|.  
inf  (<ref>)   x∈ |γ_k|  ω∈ A, 
, 


    dist (f_m_k(|γ_k|), A):=inf_x∈
    f_m_k(|γ_k|), y∈ A|x-y|>ε,   ∀  
    k=1,2,… .



  A  B(x, ε/4), x∈ A.
 A ,   ,  A⊂⋃_i=1^M_0B(x_i, ε/4), x_i∈ A,
i=1,2,…, M_0, 1⩽ M_0<∞.  , M_0
   A,  , M_0    k. 


    Γ_k:=Γ(A_m_k, |γ_k|, D) .


 Γ_ki:=Γ_f_k(x_i, ε/4,
ε/2),  , Γ_ki   
 γ:[0, 1]→ D,  
Γ_k,   γ(0)∈ S(x_i, ε/4),
γ(1)∈ S(x_i, ε/2)  γ(t)∈ A(x_i,
ε/4, ε/2)  0<t<1. , 


    Γ_k>⋃_i=1^M_0Γ_ki .



  ,  γ∈Γ_k. 
γ:=f_m_k(γ)∈Γ(A, f_m_k(|γ_k|),
D^ ').   B(x_i, ε/4), 1⩽
i⩽ M_0,    A,  i∈ N ,  γ:[0, 1]→ D, γ(0)∈
B(x_i, ε/4)  γ(1)∈ f_m_k(|γ_k|). 
 (<ref>), |γ|∩ B(x_i,
ε/4)∅ |γ|∩ (D^ '∖
B(x_i, ε/4)). ,
 <cit.>  0<t_1<1 , 
γ(t_1)∈ S(x_i, ε/4).  , 
γ(t)∉B(x_i, ε/4)  t>t_1. 
γ_1:=γ|_[t_1, 1].  (<ref>) , 
|γ_1|∩ B(x_i, ε/2)∅
|γ_1|∩ (D∖ B(x_i, ε/2)). ,
 <cit.>  t_1<t_2<1 , 
γ(t_2)∈ S(x_i, ε/2).  , 
γ(t)∈ B(x_i, ε/2)   t<t_2. 
γ_2:=γ|_[t_1, t_2], ,   γ_2
  γ,   Γ(S(x_i,
ε/4), S(x_i, ε/2), A(x_i, ε/4,
ε/2)).  ,  (<ref>)
. 


    η(t)= {[           4/ε, t∈ [ε/4, ε/2],;             0, t∉[ε/4, ε/2] . ].


,  η   (<ref>) 
r_1=ε/4  r_2=ε/2.   f
  (<ref>),    y_0=x_i,
:


    M(Γ_f_k(x_i, ε/4, ε/2))⩽
    (4/ε)^n·‖ Q‖_1<c<∞ ,


where c –     ‖ Q‖_1
–  L_1-  Q  D^ '.  (<ref>) 
(<ref>),     ,
:


    M(Γ_k)⩽4^nM_0/ε^n∫_D^ 'Q(y) dm(y)⩽
    c· M_0<∞ .


  ,  ,   
 (<ref>)    (<ref>), 
,  M(Γ_k)→∞ 
k→∞,   (<ref>). 
  . 


7.     . ,  
 <ref>      
  ,     .  
   ,     
     .    
 .   D, D^ '⊂
R^n,  a∈ D, b∈ D^ '   
   Q:D^ '→ [0, ∞] 
 S_a, b, Q(D, D^ ')  
,     f  D 
D^ ',   (<ref>)  
y_0∈ f(D),   f(a)=b.  
.


,   D    , 
     .  Q∈
L^1(D^ ')   D^ '    
,  f∈ S_a, b, Q(D, D^ ') 
   
f:D→D^ ',
f|_D=f,  ,
f(D)=D^ '  
S_a, b, Q(D, D^ '),
    
f:D→D^ ',
   D. 


    S_a, b, Q(D,
D^ '),     
 f∈ S_a, b, Q(D, D^ ')  
f(D)=D^ '  
 <ref>  <ref>.   
   
f:D→D^ ' 
   D.

. ,  
S_a, b, Q(D, D^ ') 
      x_0∈∂
D.    x_m∈ D   f_m∈
S_a, b, Q(D, D^ '), m=1,2,…
,   x_m→ x_0  m→∞ , 
,   ε_0>0


    h(f_m(x_m), f_m(x_0))⩾ε_0 ,   m=1,2,… .


    y_0∈ D^ ', y_0
b,      b    D^ ',
   α.  A:=|α|.  A_m –
   α   f_m   
 a (   <cit.>). ,
 h(A_m, ∂ D)>0    f_m. 
  :  h(A_m):=sup_x, y∈
A_mh(x, y)→ 0  m→∞, 
h(A_m_k)⩾δ_0>0  k→∞ 
    m_k  
δ_0>0.

    , , h(A_m, ∂ D)⩾δ>0   δ>0.   
{f_m}_m=1^∞     x_0 
 <ref>,    (<ref>).

  ,  h(A_m_k)⩾δ_0>0 
k→∞,   ,  h(A_m_k, ∂
D)⩾δ_1>0   δ_1>0  
 <ref>.  ,   <ref>  , 
 {f_m_k}_k=1^∞    
 x_0,     (<ref>).

,          
 (<ref>),       
    S_a, b,
Q(D, D^ ')  D.  . 


.    <ref> 
     
 D,      . ?




99



MRV_1 Martio O., Rickman S., and Väisälä J. Definitions for
quasiregular mappings // Ann. Acad. Sci. Fenn. Ser. A1. – 1969. – V. 448. – P. 1–40.

Ri Rickman S. Quasiregular mappings. – Berlin etc.:
Springer-Verlag, 1993.

MRSY Martio O., Ryazanov V., Srebro U. and Yakubov
E. Moduli in Modern Mapping Theory. – New York: Springer Science +
Business Media, LLC, 2009.

Pol  ..     
 // . . – 1970. – 83,  2. –
. 261–272.

Sev_3 Sevost'yanov E.A. On open and discrete mappings with a
modulus condition // Ann. Acad. Sci. Fenn. – 2016. –
41. – P. 41–50.

MRV_3 Martio O., Rickman S., and Väisälä J.
Topological and metric properties of quasiregular mappings // Ann.
Acad. Sci. Fenn. Ser. A1. – 1971. – 488. – P. 1–31.

Va Väisälä J. Lectures on n-Dimensional Quasiconformal
Mappings. – Lecture Notes in Math., 229, Berlin etc.:
Springer–Verlag, 1971.

Ku  . , . 2. – .: , 1969.

MRSY_1 Martio O., Ryazanov V., Srebro U.
and Yakubov E. On Q-ho­me­o­mor­phisms
// Ann. Acad. Sci. Fenn. Math. – 2005. – 30, no. 1. – P. 49–69.

RS  ..,  ..       
 // . . . – 2007. – 4,  2. –
. 199–234.

Sm  ..   
Q-­­­­­    // .
. . – 2010. – 62,  5. – P. 682–689.


Vu_1 Vuorinen M. Exceptional sets and boundary behavior of quasiregular
mappings in n-space // Ann. Acad. Sci. Fenn. Ser. A 1. Math.
Dissertationes. – 1976. –  11. – P. 1–44.


SevSkv_1  ..,  ..    
       
// . . . – 2018.  – . 70,   7. –
. 952–967; translation ”On the Convergence of Mappings in Metric
Spaces with Direct and Inverse Modulus Conditions” in Ukr. Math. J.
– 2018. – V. 70, no. 7. – P. 1097–1114.

SevSkv_2  ..,  ..     
  // . .  (  ).

HK Herron J. and Koskela P. Quasiextremal distance domains
and conformal mappings onto circle domains // Compl. Var. Theor.
Appl. – 1990. – 15. –  P. 167–179.

SevSkv_3 Sevost'yanov E.A., Skvortsov S.A. On
mappings whose inverse satisfy the Poletsky inequality // Ann. Acad.
Sci. Fenn. (accepted to print).


Vu_2 Vuorinen M. On the existence of angular limits of
n-dimensional quasiconformal mappings // Ark. Math. – 1980. –
18. – P. 157–180.

Vu_3 Vuorinen M. Conformal Geometry and Quasiregular Mappings. – Lecture Notes in
Math., 1319,  Berlin etc.: Springer–Verlag, 1988.




 

 

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