Research Journal of Applied Sciences, Engineering and Technology 4(21): 4484-4487, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: May 04, 2012
Accepted: June 08, 2012
Published: November 01, 2012
Riemann Boundary Value Problems for Koch Curve
1
Zhengshun Ruan and 2Aihua Luo
School of Science, Wuhan Institute of Technology, Wuhan, China, 4300732
2
School of Mathematics and Statistics, South-Central University for Nationalities,
HuBei Wuhan 430073
1
Abstract: In this study, when L is substituted for Koch curve, Riemann boundary value problems was
defined, but generally speaking, Cauchy-type integral is meaningless on Koch curve. When some analytic
conditions are attached to functions G (z) and g (z), through the limit function of a sequence of Cauchytype integrals, the homogeneous and non-homogeneous Riemann boundary problems on Koch curve are
introduced, some similar results was attained like the classical boundary value problems for analytic
functions.
Keywords: Holder condition, index, koch curve, riemann boundary value problem
INTRODUCTION
In the classical boundary value problems, let Lbe
an arc-wise smooth and have no sharp points in the
complex plane, G (z), g (z) satisfy Holder condition of
order a (0‹a‹1) and G (t) ≠ 0 (t L), a detailed
discussion of the Riemann boundary problems Ф+ (t) =
G (t) Ф- (t) + g (t) (t L) was presented, a complete
result was attained. But when L is substituted for Koch
curve, the problems and results will change
corresponding.
Let K0 be the curve of a equilateral triangle whose
vertices are zk – e 2/3kлi (k – 0, 1, 2), choosing the onethird of each side as the base, three equilateral triangles
are attained from the outside of K0, getting rid of these
bases (the end point is keeped), a polygonal curve K1 is
obtained. These steps are repeated all the time, the
polygonal curve Kn is obtained which consist of lines
In,j, j-1, 2, ..3×4n bumper to bumper, length of each line
√3
is |Inj | = . The Koch Curve K is obtained when n
3
approaches infinity and the Hausdorff dimension of K
is S = log3 4. Falconer (1990) have a research of the
fractal geometry: mathematics foundings and
applications. Ruan and Ai-hua (2011) study a class of
boundary value problem for analytic functions bounded
by the koch curve. Muskhelishvili (1953) study the
singular integral equations. Jian-ke (1993) have a
research of the boundary value problems for analytic
functions. Wang and Gong (2001) study the stability
about intergral curves of singular intergrals with
Cauchy Kernel. Wang and Gong (1999) analyze the
stability about intergral curves of singular intergrals and
cauchy-type integrals. Qu and Su (2000) have a
research of the estimation of the hausdorff himension of
koch curve and sierpinski mat.
For K and each Kn, oriented positively is
（or
counterclockwise, denoted by E+（orE-）and
）the interior zone and exterior zone of K and each
Kn, then
(n N).
1 , En+1
+
||
|
δ}, Kδ+ is
Let δ 0, Kδ =
defined the δ-neighborhood from the left (closed
region). From the construction of K, each
1 consist of closed region of 3×4n equilateral triangles
√3
non-intersecting to each other with side length 1. The
3
boundaries of these triangular areas are denoted by
∆n,j(j = 1, 2, …, 3×4n), ∆n,j+ is the interior zone. When
, such that
-EN0+ =
δ 0 is fixed, there exists N0
3 4n
∆ , and each ∆n,j (j = 1, 2, …, 3×4n) lies
0
in the interior of Kδ+. In the following, N0 is fixed and
δ 0and small sufficiently such that 0 Kδ+.
Definition: Assume 0
1，If φ satisfy Holder
condition of order a (0
1), which is | φ (z1)- φ
(z2)|
|z1-z2|a, ( z1, z2 Kδ+) for M 0 and φ (z) is an
(Kδ+).
analyitc function in interior of Kδ+, then φ
Lemma 1: t0 K if and only if there exists N (t0)
such that t0
n N t0 Kn .
Lemma 2: Assume:
s −1 < α < 1, s = log3 4, ϕ ∈ AH α ( Kδ+ ),
Then,
Corresponding Author: Aihua Luo, School of Mathematics and Statistics, South-Central University for Nationalities, HuBei
Wuhan 430073
4484
Res. J. Appl. Sci. Eng. Technol., 4(21): 4484-4487, 2012
Φ ( z ) = lim Φ n ( z ) = lim
n→ ∞
n→∞
1
2π i
∫
ϕ (t )
Kn
t−z
dt
when solved in
1 ; when m 0, it has the general
solution (6), when m 2, it has the unique solution (5)
if and only if m-2 conditions of solvability in (7) are
fulfilled.
(1)
Ф z and
For any t0 K, Ф+ (t0) =
Ф z Both exist, satisfied:
Ф (t0) =
+
0
−
(2)
Φ ( z ) = li m
n→ ∞
From the proof of Lemma 2, we can see that:
and
Φ ( z ) = lim
±
Φ (t0 ) = Φ n (t0 ) , (n ≥ N (t0 ))
n→ ∞
(4)
Now the Riemann boundary value problems for
Koch Curve is considered, the analyitic function Ф (z)
defined in E+ and E-, satisfying the condition of
Riemann boundary value:
Φ + (t ) = G (t )Φ − (t ) + g (t ) (t ∈ K )
where, G, g
) and Ф
(z
0
1
2π i
∫
K
n
g (t )
dt
t − z
(5 )
is the unique solution of jump problem (*)1.
By the extended Liouville theorem, the general
solution of (*)1 in Rm (m 0) is:
Φ ( z ) = Φ n ( z ) , (n ≥ N (t0 ) , z ∈ E N+(t0 ) U E − ) (3)
±
:
Proof: From (1)、(2)、(3)when solved in
Φ (t 0 ) − Φ (t 0 ) = ϕ ( t 0 )
+
（＊）
,
1
1 G (z) ≠ 0
(that is Ф (∞) is finite).
In this study, when L is substituted for Koch curve,
we define the Riemann boundary value problem.
Cauchy-type integral is meaningless on Koch curve.
Moreover, when some analytic conditions are attached
to functions G (z) and g (z), through the limit function
of a sequence of Cauchy-type integrals, the
homogeneous and non-homogeneous Riemann
boundary problems on Koch curve are introduced, some
similar results was attained like the classical boundary
value problems for analytic functions.
RIEMANN BOUNDARY VALUE PROBLEMS
AND THEIR SOLUTIONS
The simplest Riemann boundary value problem is
the jump problem: G (t) = 1 in (*), i.e., Ф+(t)- Ф-(t) = g
(t) (t K) (*)i If Ф (z) is required to have a pole of order
m 1 at most at infinity, then the problem is denoted by
, if Ф (z) is required to eaual to 0 of order m 1
Ф
.
, the problem is denoted by Ф
Theorem 1: The solution of the jump problem (*)1 is
given in following: its unique solution is given by (5)
1
2π i ∫
Kn
g (t )
dt + Pm ( z )
t−z
(6)
where, Pm (z) is an arbitrary polynomial of degree m.
From (3), for any t0
, allowing for:
Φ( z ) = Φ n ( z ) =
1
g (t )
dt ,
2π i ∫ Kn t − z
(n ≥ N(t0), z∈EN+(t0) UE−)
1
it has Ф (z) = ∑ ∞0
g tk dt z-(k+1), (n
2л
for the reigon G whose satisfied the
(t0), z
neighbourhood of z = ∞ and
. So if and only if:
lim ∫ g (t )t k dt = 0 (k = 0,1,L , m − 2)
n →∞
(7)
Kn
Ф (z) defined by (5) has a zero point at infinity of
order m 2.
If g (t) = 0 in (*), then it is called a homogeneous
Riemann boundary value problem, that is:
Φ + (t ) = G (t ) Φ − (t )
（ ＊ ）2
(t ∈ K )
Theorem 2: For the homogeneous Riemann boundary
value problem (*)2, if the solutions are in R0, it has only
the trivial solution when its index K 0 and its solution
is Ф (z) = X (z) Pk (z) when K 0, where Pm (z) is an
arbitrary polynomial of degree m.
a
Kδ ) and taking value of no
Proof: Because G
zero, there exists a integer K, such that:
1
2л
log G z
k=
1
2л
logG z
KN =
K (n
called K is the index of (*)2 or (*).
Assume G0 (z) = z-k G (z), then G0
(z) ≠ 0 (z Kδ ,
by introducing a new function ψ (z):
4485 0)
a
Kδ ), G0
Res. J. Appl. Sci. Eng. Technol., 4(21): 4484-4487, 2012
+
⎪⎧Φ ( z ) , z ∈ E
Ψ( z) = ⎨ κ
−
⎪⎩ z Φ ( z ) , z ∈ E
, therefore Γ (z)= Γn (z) =
log G0 (t )
1
dt , ( z ∉ Kn , n ≥ N0 )
∫
K
n→∞ 2π i
n
t−z
Γn ( z) = lim
+
Γ( z )
⎪⎧Χ0 ( z) = e ,
Χ( z) = ⎨ −κ
−κ Γ ( z )
⎪⎩ z Χ0 ( z) = z e ,
z∈E+
+
⎧Χ
⎪ ( z ),
Χ ( z) = ⎨
⎪⎩ Χ ( z ),
+
-
Γ (z) =
∞ Γn (z) is analystic in E E ,
which satisfied Γ+ (t)- Γ- (t) = log G0 (t) (t ) and Γ (∞)
, then Χ 0 ( z )
= 0. Introducing X0 (z) = eΓ(z) (z
+
is analystic in E
and X0 (∞) = 1，X0 (z) and X±
(t) (t
) have no zero points, also:
(t)
= G0 (t) 0 (t) = G (t) t-k 0 (t) (t
). If we
0
set:
then
g z
X z
t z
dt
z ∈ K I Kn ;
z ∈ En+且z ∉ K
(12)
holds the condition of theorem 1, we have
Theorem 3: For the non-homogeneous Riemann
boundary value problem (*), if the solutions are in R0,
its general solution is:
(8)
z∈E−
logG0 t
, N(t0)).
By the Privalov theorem, Γn (z) and Γ (t) are
,
analystic and satisfy Holder condition of order a in
since X (z) = eΓ (z) = eΓn (z)
,
(z) and
are
(z
0 ，X
.
analystic and satisfy Holder condition of order a in
Introducing n N0:
(z
then (*)2 becomes a Riemann boundary value problem
for ψ (z) ψ+ (t) =G0 (t) ψ- (t) (t
) with index 0. Since
Ф
. If we set:
0 , its solution ψ
1
2л
Χ( z)
g (t ) dt
+ Χ ( z ) Pκ ( z )
∫
K n Χ + (t ) t − z
n →∞ 2π i
Φ ( z ) = lim
and call it canonical function of (*)2 or (*), then:
•
•
•
X (z) satisfy X (z) is analystic in E+
Χ( z ) ≠ 0 ( z ∈ E + U E − )
) exists and X± (t) ≠ 0 (t
)
X± (t) (t
X (z) has finite order -K at z = ∞
and
when, k 0, where Pm (z) is an arbitrary polynomial of
degree m.. its unique solution is:
Φ( z ) = lim
n →∞
Χ + (t ) = G (t ) Χ − (t ) (t ∈ K )
lim ∫
n →∞
−
Φ (t ) Φ (t )
−
=0
Χ + (t ) Χ − (t )
（ 10）
(t ∈ K )
Let us consider the non-homogeneous Riemann
boundary value problem (*), from the index, the
canonical function X (z) of its corresponding
homogeneous problem in (8), Considering (9), (*)
becomes a jump problem:
solving
Ф t
in Rk. Since g
not a function in
g t
X t
. For any t0
, but
g t
X t
Corollary: For the non-homogeneous Riemann
boundary value problem (*), if the solutions are in 1 ,
crossponding problem (11) is solved in Rk-1, when
K 1, its general solution is:
Χ( z)
g (t ) dt
+ Χ ( z ) Pκ −1 ( z )
2π i ∫ K n Χ + (t ) t − z
its unique solution is (13) when κ = 0 ; its unique
Φ ( z ) = lim
n →∞
solution is (13) if and only if:
（ 11）
(t ∈ K )
lim ∫
n →∞
is may
, log G0
Kn
g (t )t k
dt = 0 (k = 0,1,L, −κ − 2)
Χ + (t )
is satisfied when K -2.
From the process in jump problem, the results are
Ф t
attained from solving
in Rk.
Φ + (t ) Φ − (t )
g (t )
−
= +
Χ + (t ) Χ − (t )
Χ (t )
(13)
when, K = -1, its unique solution is (13) if and only if:
(9)
Considering (9), (*)2 becomes a jump problem:
+
Χ( z )
g (t ) dt
∫
K
n
2π i
Χ + (t ) t − z
Kn
g (t )t k
dt = 0 (k = 0,1,L, −κ − 1)
Χ + (t )
is satisfied when K -1.
4486 Res. J. Appl. Sci. Eng. Technol., 4(21): 4484-4487, 2012
ACKNOWLEDGMENT
This study was supported by the Special Fund for
Basic Scientific Research of Central Colleges, SouthCentral University for Nationalities, CZQ12016.
REFERENCES
Falconer, K.J., 1990. Fractal Geometry: Mathematics
Foundings and Applications. 1st Edn., John Wiley
and Sons, New York.
Jian-Ke, L., 1993. Boundary Value Problems for
Analytic Functions. World Scientific Publishing
Co., Pte. Ltd., Singapore.
Muskhelishvili, N.I., 1953. Some Basic Problems of the
Mathematical Theory of Elasticity. Noordhoff,
Groningen, pp: 718.
Qu, C. and W. Su, 2000. Estimation of the Hausdorff
Himension of Koch Curve and Sierpinski Mat. J.
Nanjing University, pp: 397-402.
Ruan, Z. and L. Ai-hua, 2011. A class of boundary
value problem for analytic functions bounded by
the Koch curve. J. Math. Pract. Theo., 12: 15-26.
Wang, X. and Y. Gong, 1999. The stability about
integral curves of singular integrals and cauchytype integrals. Acta Math. Sinica, 42: 343-350.
Wang, X. and Y. Gong, 2001. The Stability about
Integral Curves of singular Integrals with Cauchy
Kernel. Chinese Annals of Mathematics, pp: 447452.
4487