BOUNDARY VALUE PROBLEMS FOR ANALYTIC AND
HARMONIC FUNCTIONS IN THE FRAME OF VARIABLE
EXPONENT ANALYSIS
Vakhtang Kokilashvili and Vakhtang Paatashvili
A. Razmadze Mathematical Institute, Tbilisi, Georgia
Abstract. The Dirichlet and Neumann problems in the domains with piecewisesmooth boundaries are solved, when the boundary functions are taken from variable
Lebesgue spaces. We establish the solvability conditions. Along the uniquely
solvability we study cases when the problem is solvable non-uniquely or is
unsolvable. In all solvability cases we construct the solutions in explicit form in terms
of Cauchy type integrals. The main feature of our research is to reveal the influence of
all parameters of given problem on the solvability conditions. We emphasize that the
picture of solvability largelly depends on the values of angles at angular points of
boundary, values of exponent at angular points and values of exponents from the
weight function.
Our talk is related to the solution of the Riemann-Hilbert problem in the
following setting: Find a holomorphic function   K p D,   whose boundary
values   t  satisfy the boundary condition


Re at   ibt   t   ct  a.a. t   .
()
Here D is a simply connected domain not containing z   and bounded by a simple
piecewise-smooth closed curve  and K p  D ,  denotes the set of all functions
z  representable in the form
z  
1 1
 z  2i
 t 
tz
dt ,
  L p  

when
 z  
i

z  t k  k ,
tk   ,
t k  t j when k  j ,  k  R .
k 1
In boundary condition () a(t) and b(t) are real piecewise-continuous functions
defined on  .
We obtained the complete picture of the solvability – the conditions for the
problem to be solvable are derived and solutions are constructed. These conditions
and solutions essentialy depend both on values p(t) at the angular points of  and on
the angle sizes at these points, an important contribution to the picture of solvability is
made by the exponent  k of the weight function and the jump value at the
discontinuity points of the coefficients a(t) and b(t).
Reflecting our results to save the time we will specify the case when a(t)1,
b(t)0 i. e. when we have the Dirichlet problem. The Neumann problem is also
reduced to the above-mentioned case.
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THE DIRICHLET AND NEUMANN PROBLEMS FOR HARMONIC
FUNCTIONS IN THE SMIRNOV CLASS WITH VARIABLE EXPONENT
1
DEFINITIONS
1.1 Boundaries
Here and subsequently   t : t  t s ,0  s  l denotes the simple pieciwisesmooth curve bounding finite domain D with angular points A1, A2 ,, An at which
angle values with respect to the domain D are equal to k , 0  k  2, k  1, n . The
set of such curves is denoted by C1  A1 , A2 ,, An ;1 ,, n  . For n=1 we assume that
A1  A , 1   and in this case we have the class of curves - C 1  A;  .
The set of piecewise-Lyapunov curves (i.e. t s   Lip   Ak , Ak 1  , k  1, n
An 1  A1 contained is that class is denoted by C 1, L  A1 , A2 ,, An ;1 ,, n  .
1.2 The classes Lp ;  
Let p=p(t) be positive measurable function on  and
j
 t  
 t  t 
k
k
,
tk   ,
k  R
(1)
k 1
We set
L
p 
 l



p t s 
;    f : f t s  t s 
ds   .


 0

~
~
Class P   . We say that function p belongs to the class P   if
1) p satisfies the Log-Hölder condition, i. e. exist positive constants A and  such

that
pt1   pt 2  
t1,t2  
2)   min pt   1 .
A
ln t1  t 2
1 
(2)
t 
Class P  . The set of functions p for which   1 and (2) is satisfied for
  0 , we denote by P  .
1.3 Classes of holomorphic and harmonic functions
Denote by E D ,   0 , the Smirnov class of holomorphic functions in D.
This is the set of all holomorphic functions  for which

sup
0  r 1


z  dz   ,
r
7-98
where r is the image of circumfence w  r by the conformal mapping of disk
U  w : w  1 on D. Every function  from this class possesses angular boundary
value  t  for almost all t   forming a function from L   .
We set
E , p D;    :   E  D;    Lp ;  ;
e , p D;   u : u  Re ;   E , p D;  ;





e , p  D   e , p  D;1 .
2

THE DIRICHLET PROBLEM
2.1 Formulation of the problem and axuallary results
Define the function u satisfying the conditions
~

u  e1, p  D,  ,
p  P  ,
u  0,
 
p 

t  .
u t   f t , f  L ;  ,
(3)
where u t  denotes angular boundary value of the function u t  at the point t and
the equality in (3) is understood almost everywhere on  .
We establish the solvability conditions. Along the uniquely solvability we study
cases when the problem is solvable non-uniquely or is unsolvable. In all solvability
cases we construct the solutions in explicit form in terms of Cauchy type integrals and
conformal mapping U onto D.
p  
Moreover, when the problem (3) is unsolvable for arbitrary f  L ;   we
establish the necessary and sufficient condition for the right side to be problem
solvable. When   1 we indicate sufficiently large subset of Lp  for which the
latter condition is satisfied.
The main feature of our research is to reveal the influence of all parameters of
given problem on the solvability conditions. We emphasize that the picture of
solvability largelly depend on the values of angles at angular point of boundary,
values of exponent p(t) at angular points and values of exponents  k from the weight
function  .

2.2 The case   C1 A;  ,   1
Our goal is to study all possible cases:
(i)
 0
0    p A ;
(ii)
  p A;
(iii)
  p A;
(iv)
(i) If 0    p A the problem is uniquely solvable and its solution gives by
formulas
u  u f wz  .
1  1 
 1  
u f w  Re 
w    
2  k z w 
 w  
(4)
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where z(w) is conformal mapping of U onto D and w(z) is its inverse function;
  min pt  and
t

w 
zw
2i



g   d
,    :   1, wU .
z    w
(4)
g    f z  z 
(ii)} If   p A , then the problem has the solution depending on one real
parameter. Solutions are given by formula
u  z   u0 w z   u f w z 
here
w a
u0 w  c Re
, a  w A ;
wa
c is arbitrary real constant; u f w is given by (4) where instead of (4') we have

z w 1
w  a 2i

w 


g    a  d
.
 
z     w
(5)
(iii) If   p A then problem is generally speaking unsolvable. The problem is
solvable only for function f for which
1
z0   l     a

1
p  A
  a
f z  

1
p  A
1
z0  l  
d
 Ll    , l    pz 
 
(6)
Here z 0 is the function from the representation of derivative of conformal mapping U
on domain with boundary   C1  A1 ,, An ;1 ,,n 
z w 
n

w  ak  k 1 z0 w ,
z0 w  exp
k 1


  d
, ak  w Ak 
 w
where  is continuous function on  . This representation is a generalization of
Warschawski result obtained by V. Paatashvili and G. Khuskivadze (see e. g. [1],
Chapter 3).
(iv) If   0 and p(t) is such function that h   pz  P  then the
problem is solvable for those f for which the condition (6) is satisfied. If condition
(6) is satisfied in both cases (iii) and (iv) problem is uniquely solvable.
Note that if   C1,L  A1 ,, An ;1 ,,n  , k  0, k  1, n in the case (iii) in
formula (6) the function z 0 may be omitted.
2.3 The case   C1  A1, , An ;1, ,n  , w  1
k  1, n ,
0  k  2 ,
Theorem 1. I. let 1)   C1  A1 ,, An ;1 ,,n  ,
~
p  P   ; 2)  k  0 , p Ak    k for all k and 3) p Ak i   k i , i  m  1, n and
 
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

p Aki   ki , i  1, m . Then the Dirichlet problem in the class e1, p D  is
solvable and general solution of the homogeneous problem is given by the equality
u0 z  
m

Ci Re
 ,
wz   wAk 
wz   w Ak i
Ci  R .
i
i 1
II. If for some k we have p Ak    k or  k =0, (in the last case we are able to
~
give the answer to considered problem when p  z    P   , then the Dirichlet
problem is, generaly speaking, unsolvable. Moreover, the problem is solvable only for
those f  Lp  for which the inclusion
1
z0   l  


i: pAki


  ak i
1
ki

1
p Aki
 
 ki or  ki  0
f  z  z 0  l  
 a

1

p Aki
 
d
 Ll   
 
i
holds. Here l    pz  .
Recently V. Kokilashvili and A. Meskhi proved that: If l  P  and
  ln   a  Ll    , a   ,
then the function

1
  a l a 
  
  a


1
l a 
d
 a
belongs to Ll    .
Based on this statement we prove the important addition to the
theorem 1:
III. If
f t ln
wt   w A  L p   
p A   or   0 (7)

k
k
i
i
ki
ki

i
then the Dirichlet problem (3) is solvable.
Remark. If   C1, L  A1 ,, An ;1 ,, n  ,  k  0 , k  1, n , then the condition
(7) may be replaced by the condition
f t  ln
t  Ak i  Lp    .

i
j
2.4 The case when   C  A1 ,, An ;1 ,, n  and  t  
1

k 1
and
7-101
t  tk
k
, tk  

1
1
pt 
 k 
, qt  
p  Ak 
q  Ak 
pt   1
(8)
For the sake of simplicity we assume that  k  0 , p Ak    k , k  1, n . In this
case the solutions we may write effectively. Let T= t1 , t 2 ,, t j . The homogeneous
problem has æ linearly independent solutions where
æ= N Ak : Ak T , p Ak   k +





p Ak 
2 p Ak  
N  Ak : Ak  T :
 k 
.
1   k p Ak 
1   k p Ak 

Here N(E) denotes the number of elements of the set E.
From here it is evident that æ depends on geometry of  (on values of  k ), on
weight function  (on values of  k ) and values of function p in angular points Ak .
3
THE NEUMANN PROBLEM
This problem we consider in the following statement: define the function u for which
u  0,
u  Re ,   E1, p  D; ,

(9)
 u  
p 
  t   f t , t  , f  L ; .
 n 
For simplicity we present only the case when   1 .
Using the N. Muskhelishvili idea we can write the boundary condition from (9)
in the following form
Re ie i t   t   f t  ,
z   z  .
where  t  is angle lying between the oriented tangent at the point t and the x axis.
Here function ie i t  is piecewise-continuous so we come to the Riemann-Hilbert


problem with piecewise-continuous coefficient. This problem in the classes E1, p D 
is studied in our works ([2], [3]). Based on this results we derive:
Theorem 2. Let Fk  be all linear independent solutions of the Riemann
problem
t  ,
F  t   ei 2 t F  t  ,
in the class of functions representable by Cauchy type integrals with density from
pt 
. Then for solvability of the Neumann problem it is necessary
Lq  , qt  
pt   1
and sufficient that
 f t F t dt  0,
k
k  1,, æ(q)+1
(10)

where
æ(q)= N  Ak :  k  q Ak .
If the conditions (10) are satisfied then the Neumann problem is solvable and its
general solution contains æ(q)+1 arbitrary real constants where
æ(p)= N  Ak :  k  p Ak .
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Note, that the functions Fk t  and solution
æ  p  1
u z   u f z  

Ck u k z 
k 1
one may write effectively.
REFERENCES
1. G. Khuskivadze, V. Kokilashvili and V. Paatashvili, Boundary value problems for
analytic and harmonic functions in domains with nonsmooth boundaries. Applications
to conformal mappings. Mem. Differential equations Math. Phys. 14(1998), 1-195.
2. V. Kokilashvili and V. Paatashvili, The Dirichlet problem for harmonic functions in
the Smirnov class with variable exponent. Georgian Math. J. 14(2007), N 2, 289-299.
3. V. Kokilashvili and V. Paatashvili, The Riemann-Hilbert problem in weighted
classes of Cauchy type integrals with density from Lp . Compex Analysis and
Operator Theory (to appear in 2008).
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