Memoirs on Differential Equations and Mathematical Physics
Volume 33, 2004, 5–23
H. Begehr
BOUNDARY VALUE PROBLEMS
FOR THE BITSADZE EQUATION
Abstract. The Schwarz, Dirichlet, Neumann and some related boundary value problems are explicitly solved for the Bitsadze equation in the
unit disc of the complex plane. The results are obtained from iterations
of related results for the inhomogeneous Cauchy–Riemann equation. Some
generalizations for the inhomogeneous polyanalytic equation are indicated.
2000 Mathematics Subject Classification. 35J25, 35C15, 30E25.
Key words and phrases. Schwarz, Dirichlet, Neumann boundary value
problems, Bitsadze equation, inhomogeneous polyanalytic equation.
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Boundary Value Problems for the Bitsadze Equation
7
1. Introduction
From the three complex second order differential operators ∂z̄2 , ∂z ∂z̄ and
∂z2 , the Bitsadze operator ∂z̄2 being the square of the Cauchy–Riemann operator is essentially not different from ∂z2 but differs principally from the
Laplace operator ∂z ∂z̄ . This was remarked by Bitsadze [14] who showed that
the Dirichlet problem as the natural boundary condition to the Laplace operator is ill-posed for the homogeneous Bitsadze equation. Nevertheless, besides the Schwarz and the Neumann boundary value problems this Dirichlet
problem will also be treated. Under some solvability conditions it is shown
to be solvable. In fact, the solutions to all the problems considered in the
unit disc of the complex plane are given explicitly. Solutions together with
the solvability conditions are attained from the iteration of results about
related problems for the inhomogeneous Cauchy–Riemann equation. Their
theory is developed in [6]. The same boundary value problems for the Poisson equation are investigated in [7]. Related considerations are available
from [3, 9, 10, 11, 12, 13]. Basic Cauchy–Pompeiu representations are developed e.g. in [3, 5]. They originate from a hierarchy of integral operators
[2, 8] which is constructed by iterating the Pompeiu operator [20]. This
Pompeiu operator is the main tool in I.N. Vekua’s theory of generalized analytic functions and its main properties were studied by Vekua [20]. Besides
Gakhov [15], N.I. Mushelishvili [18] and Vekua [20, 21] have contributed substantially to the theory of boundary value problems for complex equations.
Bitsadze and last but not least G. Manjavidze with his extensive work on
boundary value problems with displacement [16] and on generalized analytic vectors [17] have complemented the work of Georgian mathematicians
on the theory of complex analytic methods.
The material developed here was at first presented in a mini-course on
boundary value problems in complex analysis at the University Simón Bolivar in Caracas in May 2004.
2. Cauchy–Pompeiu Representation Formulas
As from the Cauchy theorem the Cauchy formula is deduced, from the
complex Gauss theorem representation formulas can be deduced.
Cauchy–Pompeiu representation Let D ⊂ C be a regular domain and
w ∈ C 1 (D; C) ∩ C(D̄; C). Then using ζ = ξ + iη for z ∈ D,
Z
Z
1
dζ
1
dξdη
w(z) =
w(ζ)
−
wζ̄ (ζ)
(1)
2πi
ζ −z π
ζ −z
∂D
D
holds.
With respect to boundary value problems a modification of this Cauchy–
Pompeiu formula is important. In the case of the unit disc D = {z : |z| < 1}
it is as follows, see [1, 6, 8].
H. Begehr
8
Theorem 1. Any w ∈ C 1 (D; C) ∩ C(D̄; C) is representable as
Z
Z
1
ζ + z dζ
1
dζ
w(z) =
Re w(ζ)
+
Im w(ζ) −
2πi
ζ −z ζ
2π
ζ
|ζ|=1
1
−
π
Z
|ζ|<1
|ζ|=1
w (ζ) zw (ζ) ζ̄
ζ̄
+
dξdη, |z| < 1 .
ζ −z
1 − z ζ̄
(2)
Corollary. Any w ∈ C 1 (D; C) ∩ C(D̄; C) can be represented as
Z
ζ + z dζ
1
Re w(ζ)
−
w(z) =
2πi
ζ −z ζ
|ζ|=1
1
−
2π
Z wζ̄ (ζ) ζ + z
w̄ζ̄ (ζ) 1 + z ζ̄
+
dξdη+
ζ ζ −z
ζ̄ 1 − z ζ̄
|ζ|<1
+ iIm w(0),
|z| < 1.
(3)
Remark. For analytic functions, (3) is the Schwarz–Poisson formula
Z
2ζ
dζ
1
w(z) =
Re w(ζ)
−1
+ i Im w(0) .
(30 )
2πi
ζ −z
ζ
|ζ|=1
The kernel
ζ +z
2ζ
=
−1
ζ −z
ζ −z
is called the Schwarz kernel. Its real part
ζ
ζ¯
|ζ|2 − |z|2
−1=
+
ζ −z ζ −z
|ζ − z|2
is the Poisson kernel. The Schwarz operator
Z
1
ζ + z dζ
Sϕ(z) =
ϕ(ζ)
2πi
ζ −z ζ
|ζ|=1
for ϕ ∈ C(∂D; R) is known to provide an analytic function in D satisfying
Re Sϕ = ϕ on ∂D
(see [1]) in the sense
lim Sϕ(z) = ϕ(ζ),
z→ζ
ζ ∈ ∂D ,
for z in D tending to ζ. The operator
Z
dξdη
1
f (z)
,
T f (z) = −
π
ζ −z
D
on L1 (D; C) is the Pompeiu operator [19].
z ∈ C,
Boundary Value Problems for the Bitsadze Equation
9
The formula (3) is called the Cauchy–Schwarz–Poisson–Pompeiu formula.
Rewriting it according to
wz̄ = f in D,
Re w = ϕ on ∂D,
Im w(0) = c ,
we have that
w(z) =
1
2πi
Z
ϕ(ζ)
ζ + z dζ
−
ζ −z ζ
|ζ|=1
1
−
2π
Z
|ζ|<1
f (ζ) ζ + z
f (ζ) 1 + z ζ̄ +
dξdη + ic
ζ ζ −z
ζ̄ 1 − z ζ̄
(300 )
is expressed by the given data. Applying the result of Schwarz one easily
sees, taking the real part on the right-hand side and letting z tend to a
boundary point ζ, that this tends to ϕ(ζ).
Differentiating with respect to z̄, as every term on the right-hand side
is analytic besides the T -operator applied to f , one obtains f (z). Also for
z = 0 besides ic all other terms on the right-hand side are real.
Hence, (300 ) is a solution to the so-called Dirichlet problem
wz̄ = f in D,
Re w = ϕ on ∂D,
Im w(0) = c .
This shows how integral representation formulas serve to solve boundary
value problems. The method is not restricted to the unit disc but in this
case the solutions to the problems are given explicitly.
3. Iteration of Integral Representation Formulas
Integral representation formulas for solutions to first order equations can
be used to get such formulas for higher order equations via iteration, see
[1]-[3],[5, 8].
Theorem 2. Let D ⊂ C be a regular domain and w ∈ C 2 (D; C) ∩
C (D̄; C), then
Z
Z
dζ
1
ζ −z
1
w(z) =
w(ζ)
−
wζ̄ (ζ)
dζ+
2πi
ζ −z
2πi
ζ −z
1
∂D
1
+
π
Z
∂D
ζ −z
wζ̄ ζ̄ (ζ)
dξdη .
ζ −z
(4)
D
Proof. For proving (4), the formula (1) applied to wz̄ giving
Z
Z
˜
1
1
dζ̃
dξdη̃
wζ̄ (ζ) =
wζ̃¯(ζ̃)
−
wζ̃¯ζ̃¯(ζ̃)
2πi
π
ζ̃ − ζ
ζ̃ − ζ
∂D
D
H. Begehr
10
is inserted into (1), from what after having interchanged the order of integrations
Z
Z
1
dζ
1
w(z) =
w(ζ)
+
wζ̃¯(ζ̃)ψ(z, ζ̃)dζ̃−
2πi
ζ −z
2πi
∂D
∂D
Z
1
˜
wζ̃¯ζ̃¯(ζ̃)ψ(z, ζ̃)dξdη̃
(5)
−
π
D
follows with
1
ψ(z, ζ) =
π
Z
D
dξdη
1
1
=
π
(ζ − ζ̃)(ζ − z)
ζ̃ − z
Z D
1
dξdη .
−
ζ − ζ˜ ζ − z
1
The formula (1) applied to the function z̄ shows
ζ̃ − z
1
=
ζ̃ − z 2πi
Z
∂D
1
ζ̄dζ
−
(ζ − ζ̃)(ζ − z) π
Z
D
dξdη
= ψ̃(z, ζ̃) − ψ(z, ζ̃) (6)
(ζ − ζ̃)(ζ − z)
with a function ψ̃ analytic in both its variables. Hence by the complex
Gauss theorem
Z
Z
1
1
˜ =0.
wζ̃¯(ζ̃)ψ̃(z, ζ̃)dζ̃ −
wζ̃¯ζ̃¯(ζ̃)ψ̃(z, ζ̃)dξdη̃
2πi
π
∂D
D
Subtracting this from (5) and applying (6) gives (4).
Remark. There is a dual formula to (4) resulting from interchanging
the roles of z and z̄ in the preceding procedure. It can be also derived by
applying complex conjugation to (4) after replacing w by w̄. It is
Z
Z
1
dζ̄
ζ −z
1
w(z) = −
w(ζ)
+
wζ (ζ)
dζ̄+
2πi
ζ − z 2πi
ζ −z
∂D
∂D
Z
1
ζ −z
dξdη .
(40 )
+
wζζ (ζ)
π
ζ −z
D
The kernel functions (ζ − z)/(ζ − z), (ζ − z)/(ζ − z) of the second order differential operators ∂z2 , ∂z2 respectively are thus obtained from those
Cauchy and anti-Cauchy kernels 1/(ζ − z) and 1/(ζ − z) for the Cauchy–
Riemann operator ∂z̄ and its complex conjugate ∂z . The related weakly
singular integral operators are
Z
Z
1
ζ −z
1
ζ −z
f (ζ)
dξ dη, T2,0 f (z) =
f (ζ)
dξ dη
T0,2 f (z) =
π
ζ −z
π
ζ −z
D
acting on L1 (D; C).
D
Boundary Value Problems for the Bitsadze Equation
11
4. Boundary Value Problems for the Inhomogeneous Bitsadze
Equation
There are two basic second order differential operators, the Laplace operator ∂z ∂z̄ and the Bitsadze operator ∂z2 . The third one, ∂z2 is just the
complex conjugate of the Bitsadze operator and all formulas and results for
this operator can be attained by the ones for the Bitsadze operator through
complex conjugation giving dual formulas and results. Here the Bitsadze
operator will be investigated. For the problems for the Laplace operator
compare [7].
Theorem 3. The Schwarz problem for the inhomogeneous Bitsadze equation in the unit disc
wz̄ z̄ = f in D, Re w = γ0 , Re wz̄ = γ1 on ∂D, Im w(0) = c0 , Im wz̄ (0) = c1 ,
for f ∈ L1 (D; C), γ0 , γ1 ∈ C(∂D; R), c0 , c1 ∈ R is uniquely solvable through
Z
1
ζ + z dζ
w(z) = ic0 + i(z + z̄) +
γ0 (ζ)
−
2πi
ζ −z ζ
|ζ|=1
1
−
2πi
dζ
ζ +z
γ1 (ζ)
(ζ − z + ζ − z) +
ζ −z
ζ
Z
|ζ|=1
1
+
2π
f (ζ) ζ + z
f (ζ) 1 + z ζ̄ + ¯
(ζ − z + ζ − z)dξdη .
ζ ζ −z
ζ 1 − z ζ̄
Z
|ζ|<1
Proof. Rewriting the problem as the system
wz̄ = ω in D,
Re w = γ0 on ∂D,
Im w(0) = c0 ,
wz̄ = f in D,
Re ω = γ1 on ∂D,
Im ω(0) = c1 ,
and combining its solutions
1
w(z) = ic0 +
2πi
Z
γ0 (ζ)
ζ + z dζ
−
ζ −z ζ
|ζ|=1
−
1
2π
Z
ω(ζ) ζ + z
ω¯(ζ) 1 + z ζ̄
+ ¯
dξ dη ,
ζ ζ −z
ζ 1 − z ζ̄
|ζ|<1
ω(z) = ic1 +
1
2πi
Z
γ1 (ζ)
ζ + z dζ
−
ζ −z ζ
|ζ|=1
1
−
2π
Z
|ζ|<1
f (ζ) ζ + z
f (ζ) 1 + z ζ̄
+
ζ ζ −z
ζ̄ 1 − z ζ̄
dξ dη ,
(7)
H. Begehr
12
one obtains formula (7). Here the relations
Z 1
1ζ +z
1 1 + z ζ̄ −
dξ dη = −z − z̄ ,
2π
ζ ζ − z ζ̄ 1 − z ζ̄
|ζ|<1
1
2π
Z
|ζ|<1
1
2π
Z
|ζ|<1
1
2π
Z
|ζ|<1
1
2π
Z
|ζ|<1
ζ˜ + ζ 1 ζ + z
ζ˜ + z
dξ dη =
(ζ̃ − z) ,
ζ̃ − ζ ζ ζ − z
ζ̃ − z
1 + ζ¯ζ̃ 1 1 + z ζ̄
dξ dη = ζ˜ + z ,
1 − ζ¯ζ̃ ζ̄ 1 − z ζ̄
¯
¯
˜+
ζζ 1 1 + z ζ̄
1 + z ζ̃
dξ dη =
¯(ζ̃ − z) ,
1 − z ζ̃
ζ̃ − ζ ζ̄ 1 − z ζ̄
¯
¯
1 + ζ ζ˜ 1 ζ + z
1 + z ζ̃
dξdη
=
(ζ̃ − z)
˜ζ ζ − z
1 − ζ ζ¯
1 − z ζ̃¯
are used. The uniqueness of the solution follows from the unique solvability of the Schwarz problem for analytic functions and the inhomogeneous
Cauchy–Riemann equation, see [6].
It is well known that the Dirichlet problem for the Poisson equation
wzz̄ = f in D,
w = γ on ∂D ,
is well posed, i.e. it is solvable for any f ∈ L1 (D; C), γ ∈ C(∂D; C) and the
solution is unique. That the solution is unique is easily seen.
Lemma 1. The Dirichlet problem for the Laplace equation
wzz̄ = 0 in D,
w = 0 on ∂D
is only trivially solvable.
Proof. From the differential equation wz is seen to be analytic. Integrating,
one obtains w = ϕ + ψ̄, where ϕ and ψ are both analytic in D. Without
loss of generality ψ(0) = 0 may be assumed. From the boundary condition
ϕ = −ψ̄ on ∂D follows. This Dirichlet problem is solvable if and only if, see
[6],
Z
Z
Z
z̄dζ
1
dζ
1
dζ
1
ψ(ζ)
=
ψ(ζ)
−
ψ(ζ)
= ψ(z) .
0=
2πi
1 − z̄ζ
2πi
ζ − z 2πi
ζ
|ζ|=1
|ζ|=1
This also implies ϕ = 0 on D so that w = 0 in D.
|ζ|=1
As Bitsadze [14] has realized, such a result is not true for the equation
wz̄z̄ = 0.
Boundary Value Problems for the Bitsadze Equation
13
Lemma 2. The Dirichlet problem for the Bitsadze equation
wz̄z̄ = 0 in D,
w = 0 on ∂D
has infinitely many linearly independent solutions.
Proof. Here wz̄ is an analytic function in D. Integrating gives w(z) =
ϕ(z)z̄ + ψ(z) with some analytic functions in D. On the boundary we have
ϕ(z) + zψ(z) = 0. As this is an analytic function, this relation hold in D
too. Hence, w(z) = (1 − |z|2 )ψ(z) for arbitrary analytic ψ. In particular
wk (z) = (1 − |z|2 )z k is a solution of the Dirichlet problem for any k ∈ N
and these solutions are linearly independent over C.
Because of this result, the Dirichlet problem as formulated above is illposed for the inhomogeneous Bitsadze equation.
Since the Dirichlet problem formulated as for the Poisson equation is not
uniquely solvable for the Bitsadze equation, another kind Dirichlet problem
is considered which is motivated from decomposing the Bitsadze equation
into a first order system.
Theorem 4. The Dirichlet problem for the inhomogeneous Bitsadze
equation in the unit disc
wz̄ z̄ = f in D,
w = γ0 , wz̄ = γ1 on ∂D ,
for f ∈ L1 (D; C), γ0 , γ1 ∈ C(∂D; C) is solvable if and only if for |z| < 1
Z Z
ζ −z
z̄
γ0 (ζ)
γ1 (ζ)
z̄
−
dζ +
f (ζ)
dξ dη = 0
(8)
2πi
1 − z̄ζ
ζ
π
1 − z̄ζ
|ζ|=1
and
1
2πi
|ζ|<1
Z
γ1 (ζ)
z̄dζ
1
−
1 − z̄ζ
π
|ζ|=1
Z
dζ
1
γ0 (ζ)
−
ζ −z
2πi
|ζ|=1
1
π
z̄dξdη
= 0.
1 − z̄ζ
(9)
|ζ|<1
The solution then is
Z
1
w(z) =
2πi
+
f (ζ)
Z
Z
γ1 (ζ)
ζ −z
dζ+
ζ −z
|ζ|=1
f (ζ)
ζ −z
dξ dη .
ζ −z
(10)
|ζ|<1
Proof. Decomposing the problem into the system
wz̄ = ω in D,
w = γ0 on ∂D ,
ωz̄ = f in D,
ω = γ1 on ∂D ,
composing its solutions
Z
1
w(z) =
2πi
|ζ|=1
dζ
1
γ0 (ζ)
−
ζ −z π
Z
|ζ|<1
ω(ζ)
dξdη
,
ζ −z
H. Begehr
14
1
2πi
ω(z) =
Z
γ1 (ζ)
dζ
1
−
ζ −z π
|ζ|=1
Z
|ζ|=1
ω(ζ)
z̄dξdη
,
1 − z̄ζ
f (ζ)
z̄dξdη
,
1 − z̄ζ
|ζ|<1
z̄dζ
1
γ1 (ζ)
=
1 − z̄ζ
π
Z
dξdη
,
ζ −z
|ζ|<1
and using the solvability conditions
Z
Z
1
z̄dζ
1
γ0 (ζ)
=
2πi
1 − z̄ζ
π
1
2πi
f (ζ)
|ζ|=1
Z
|ζ|<1
one proves (10) together with (8) and (9). Here
Z
Z
1
dξdη
ζ̃ − z
1
ζ − z dζ
ζ̃ − z
=
−
=
π
2πi
1
−
z̄ζ
(ζ̃ − ζ)(1 − z̄ζ)
1 − z̄ ζ̃
ζ − ζ̃
1 − z̄ ζ̃
|ζ|<1
|ζ|=1
and
−
1
π
Z
|ζ|<1
1
dξdη
=−
˜ − z)
π
(ζ − ζ)(ζ
Z
|ζ|<1
1
ζ̃ − z
1
1
−
ζ − ζ˜ ζ − z
dξ dη =
are used.
ζ̃ − z
ζ̃ − z
Theorem 5. The Dirichlet–Neumann problem for the inhomogeneous
Bitsadze equation in the unit disc
wz̄ z̄ = f in D,
w = γ0 ,
∂ν wz̄ = γ1 on ∂D,
wz̄ (0) = c ,
for f ∈ L1 (D; C) ∩ C(∂D; C), γ0 , γ1 ∈ C(∂D; C), c ∈ C is solvable if and
only if for z ∈ D
Z
Z
1
dζ
1
1 − |ζ|2
c−
γ0 (ζ)
+
f (ζ)
dξ dη = 0
(11)
2πi
1 − z̄ζ
π
ζ(1 − z̄ζ)
|ζ|=1
|ζ|<1
and
1
2πi
Z
(γ1 (ζ) − ζ̄f (ζ))
dζ
1
+
ζ(1 − z̄ζ) π
|ζ|=1
Z
z̄f (ζ)
dξ dη = 0 .
(1 − z̄ζ)2
(12)
|ζ|<1
The solution then is
w(z) = cz̄ +
1
2πi
Z
γ0 (ζ)
dζ
+
ζ −z
|ζ|=1
+
1
2πi
+
1
π
Z
(γ1 (ζ) − ζ̄f (ζ))
1 − |z|2
dζ
log(1 − z ζ̄) +
z
ζ
|ζ|=1
Z
|ζ|<1
f (ζ)
|ζ|2 − |z|2
dξdη .
ζ(ζ − z)
(13)
Boundary Value Problems for the Bitsadze Equation
15
Proof. The problem is equivalent to the system
wz̄ = ω in D,
w = γ0 on ∂D ,
ωz̄ = f in D,
∂ν ω = γ1 on ∂D,
The solvability conditions are
Z
Z
1
dζ
1
γ0 (ζ)
=
2πi
1 − z̄ζ
π
|ζ|=1
ω(0) = c .
ω(ζ)
dξdη
1 − z̄ζ
|ζ|<1
and
1
2πi
Z
¯ (ζ))
(γ1 (ζ) − ζf
dζ
1
+
1 − z̄ζ)ζ
π
|ζ|=1
z̄f (ζ)
dξ dη = 0,
(1 − z̄ζ)2
Z
|ζ|<1
and the unique solutions are given by
Z
Z
1
dζ
1
γ0 (ζ)
−
w(z) =
2πi
ζ −z
π
|ζ|=1
ω(ζ)
dξdη
ζ −z
|ζ|<1
and
ω(z) = c −
1
2πi
Z
¯ (ζ)) log(1 − z ζ̄)
(γ(ζ) − ζf
dζ
1
−
ζ
π
|ζ|=1
Z
zf (ζ)
dξ dη
ζ(ζ − z)
|ζ|<1
according to the results on the Dirichlet and Neumann problems for the
inhomogeneous Cauchy–Riemann equation, see [6]. From
Z
1
dξdη
= 1,
π
1 − z̄ζ
|ζ|<1
1
π
Z
¯
˜ dξdη = 1
log(1 − ζ ζ)
1 − z̄ζ
2πi
|ζ|<1
Z
¯
˜
log(1 − ζ ζ)
dζ
=0
(1 − z̄ζ)ζ
|ζ|=1
and
1
π
Z
|ζ|<1
dξdη
ζ
|ζ̃|2 − 1
,
=
˜
ζ̃ − ζ 1 − z̄ζ
ζ̃(1 − z̄ ζ)
the condition (11) follows. Similarly (13) follows from
Z
Z
2
1
dξdη
1
¯
¯
˜ dξdη = 1 − |z| log(1 − z ζ̃)
−
= z̄,
log(1 − ζ ζ)
π
ζ −z
π
ζ −z
z
|ζ|<1
|ζ|<1
and
1
π
Z
|ζ|<1
dξdη
|ζ̃|2 − |z|2
=
.
(ζ̃ − ζ) ζ − z
(ζ̃ − z)
ζ
H. Begehr
16
Theorem 6. The boundary value problem for the inhomogeneous Bitsadze
equation in the unit disc
wz̄z̄ = f in D,
w = γ0 ,
zwzz̄ = γ1 on ∂D,
wz̄ (0) = c ,
is solvable for f ∈ L1 (D; C), γ0 , γ1 ∈ C(∂D; C), c ∈ C if and only if for
z ∈ D the condition (11) together with
Z
Z
1
dζ
z̄
dξdη
γ1 (ζ)
+
f (ζ)
=0
(14)
2πi
ζ(1 − z̄ζ) π
(1 − z̄ζ)2
|ζ|=1
|ζ|<1
holds. The solution then is uniquely given by
Z
Z
1
dζ
1
1 − |z|2
dζ
w(z) = cz̄ +
γ0 (ζ)
+
γ1 (ζ)
log(1 − z ζ̄) +
2πi
ζ −z
2πi
z
ζ
|ζ|=1
+
1
π
Z
|ζ|=1
2
f (ζ)
2
|ζ| − |z < |
dξdη .
ζ(ζ − z)
(15)
|ζ|<1
The proof is as the last one but a modification of the Neumann condition
for the inhomogeneous Cauchy–Riemann equation is involved, see [6].
Theorem 7. The Neumann problem for the inhomogeneous Bitsadze
equation in the unit disc
wz̄z̄ = f in D, ∂ν w = γ0 , ∂ν wz̄ = γ1 on ∂D, w(0) = c0 , wz̄ (0) = c1
is uniquely solvable for f ∈ C α (D; C), 0 < α < 1, γ0 , γ1 ∈ C(∂D; C),
c0 , c1 ∈ C if and only if for z ∈ ∂D
Z
Z
1
dζ̄
1
c1 z̄ +
γ0 (ζ)
−
(γ1 (ζ) − ζ̄f (ζ))(1 − z̄ζ log(1 − z ζ̄))dζ̄+
2πi
2πi
ζ −z
|ζ|=1
+
1
π
Z
|ζ|<1
|ζ|=1
f (ζ) z̄ζ(ζ − z)
1 −
dξdη = 0
ζ
(1 − z̄ζ)2
ζ −z
(16)
and
1
2πi
Z
|ζ|=1
¯ (ζ)) dζ̄ − z̄
(γ1 (ζ) − ζf
π
ζ −z
Z
f (ζ)
dξdη
= 0.
(1 − z̄ζ)2
|ζ|<1
The solution then is given as
1
w(z) = c0 + c1 z̄ −
2πi
Z
γ0 (ζ) log(1 − z ζ̄)
dζ
+
ζ
|ζ|=1
1
+
2πi
Z
|ζ|=1
(γ1 (ζ) − ζ̄f (ζ))(ζ − z) log(1 − z ζ̄)
dζ
+
ζ
(17)
Boundary Value Problems for the Bitsadze Equation
z
π
+
Z
17
f (ζ) ζ − z
dξdη .
ζ ζ −z
(18)
|ζ|<1
Proof. From the theory of the inhomogeneous Cauchy–Riemann equation
[6],
Z
Z
1
dζ
z
dξdη
(γ0 (ζ) − ζ̄ω(ζ)) log(1 − z ζ̄)
−
ω(ζ)
,
w(z) = c0 −
2πi
ζ
π
ζ(ζ − z)
|ζ|=1
1
ω(z) = c1 −
2πi
|ζ|<1
dζ
z
(γ1 (ζ) − ζ̄f (ζ)) log(1 − z ζ̄) −
ζ
π
Z
|ζ|=1
f (ζ)
dξdη
ζ(ζ − z)
|ζ|<1
if and only if
Z
Z
1
dζ
z̄
(γ0 (ζ) − ζ̄ω(ζ))
+
2πi
ζ(1 − z̄ζ) π
|ζ|=1
1
2πi
Z
ω(ζ)
dξ dη = 0 ,
(1 − z̄ζ)2
|ζ|<1
Z
(γ1 (ζ) − ζ̄f (ζ))
dζ
z̄
+
ζ(1 − z̄ζ) π
|ζ|=1
f (ζ)
dξ dη = 0 .
(1 − z̄ζ)2
Z
|ζ|<1
Inserting ω into the first condition leads to (16) on the basis of
Z
1
dζ
ω(ζ) 2
=
2πi
ζ (1 − z̄ζ)
|ζ|=1
1
= c1 z̄ −
2πi
Z
¯
˜ (ζ̃)) 1
(γ1 (ζ̃) − ζf
2πi
|ζ|=1
|ζ̃|=1
−
1
π
f (ζ̃) 1
ζ̃ 2πi
Z
|ζ̃|<1
1
= c1 z̄ +
2πi
Z
Z
Z
|ζ|=1
¯
˜ dζ dζ̃
log(1 − ζ ζ)
−
1 − z̄ζ ζ 2 ζ̃
dζ
˜ =
dξdη̃
(ζ̃ − ζ)ζ(1 − z̄ζ)
¯ (ζ))ζ̄ dζ + z̄
(γ1 (ζ) − ζf
ζ
π
|ζ|=1
Z
f (ζ)
dξdη
ζ(1 − z̄ζ)
|ζ|<1
and
z̄
π
ω(ζ)
dξdη =
(1 − z̄ζ)2
Z
|ζ|<1
=
z̄
π
Z
∂ζ̄
(ζ − z)ω(ζ)
z̄
dξdη −
2
(1 − z̄ζ)
π
|ζ|<1
z̄
=
2πi
Z
|ζ|=1
Z
ζ −z
f (ζ)dξdη =
(1 − z̄ζ)2
|ζ|<1
ζ −z
z̄
ω(ζ)dζ −
(1 − z̄ζ)2
π
Z
|ζ|<1
ζ −z
f (ζ)dξdη ,
(1 − z̄ζ)2
H. Begehr
18
where for |z| = 1
Z
Z
Z
z̄
ζ −z
ζ −z
z
1
ω(ζ) dζ
ω(ζ)dζ
=
ω(ζ)dζ
=
−
=
2πi
(1 − z̄ζ)2
2πi
(ζ − z)2
2πi
ζ −z ζ
|ζ|=1
=
1
2πi
|ζ|=1
¯ (ζ̃)) 1
(γ1 (ζ̃) − ζ̃f
2πi
Z
1
π
f (ζ̃) 1
ζ̃ 2πi
Z
|ζ̃|<1
=
1
2πi
Z
ζ(ζ − z)
dζ
|ζ|=1
|ζ̃|=1
−
|ζ|=1
¯
˜
log(1 − ζ ζ)
Z
Z
|ζ|=1
dζ
(ζ̃ − ζ)(ζ − z)
(γ1 (ζ) − ζ̄f (ζ))
dζ̃
−
ζ˜
˜ =
dξdη̃
1
dζ
log(1 − z ζ̄)
=
z
ζ
|ζ|=1
=
1
2πi
Z
(γ1 (ζ) − ζ̄f (ζ))z̄ log(1 − z ζ̄)
dζ
.
ζ
|ζ|=1
From
1
2πi
Z
1
log(1 − z ζ̄)dζ̄ = −
2πi
|ζ|=1
1
2πi
Z
Z
log(1 − z̄ζ)dζ = 0 ,
|ζ|=1
¯
˜ log(1 − z ζ̄)dζ̄ = − 1
log(1 − ζ ζ)
2πi
|ζ|=1
=
=
˜k
ζ¯
+∞
X
¯
ζ˜k
k=2
˜ log(1 − z̄ζ)dζ =
log(1 − ζ̄ ζ)
|ζ|=1
+∞
X
k=1
Z
1
k 2πi
Z
log(1 − z̄ζ)
dζ
=
ζ2
|ζ|=1
k!
∂ζk−1 log(1 − z̄ζ)|ζ=0 = −
+∞ ¯k k−1
X
ζ̃ z
=
(k − 1)k
k=2
¯
1
1 − z ζ̃
¯
¯
¯
¯
¯
¯
= ζ̃ log(1 − z ζ̃) − (log(1 − z ζ̃) + z ζ̃) = −
log(1 − z ζ̃) − ζ˜ ,
z
z
and
Z
Z
1
ζ log(1 − z ζ̄)
1
log(1 − z ζ̄)
dζ̄ =
dζ̄ = −
2πi
2πi
ζ̃ − ζ
1 − ζ̃ ζ̄
|ζ|=1
|ζ|=1
=
1
2πi
Z
|ζ|=1
the relation
Z
1
2πi
|ζ|=1
ω(ζ) log(1 − z ζ̄) dζ̄ =
log 1 − z̄ζ
dζ = 0
¯
˜
1 − ζζ
Boundary Value Problems for the Bitsadze Equation
=
1
2πi
Z
(γ1 (ζ) − ζ̄f (ζ))
|ζ|=1
follows. Similarly from
Z
Z
z
dξdη
1
=
π
ζ(ζ − z)
π
|ζ|<1
1
π
19
1 − z ζ̄
dξdη
1
−
ζ −z
π
|ζ|<1
z
dζ
log(1 − z ζ̄) + ζ¯
ζ
Z
dξdη
= −z̄ ,
ζ
(19)
|ζ|<1
¯
˜
log(1 − ζ ζ)
dξdη =
ζ −z
Z
|ζ|<1
¯ − 1
= (ζ̃ − z) log(1 − z ζ̃)
2πi
¯
˜ dζ =
(ζ̃ − ζ) log(1 − ζ ζ)
ζ −z
Z
|ζ|=1
1
¯
= (ζ̃ − z) log(1 − z ζ̃) +
2πi
Z
¯
¯
˜ log(1 − ζ ζ)
˜
(1 − ζ ζ)
dζ
=
ζ(ζ − z)
|ζ<=1
¯
1 − z ζ̃
1 − |z|2
¯
¯
¯
= (ζ̃ − z) log(1 − z ζ̃) +
log(1 − z ζ̃) =
log(1 − z ζ̃) ,
z
z
Z
Z
1
dξdη
1
ζ̄
¯
¯
˜
˜
log(1 − ζ ζ)
=
∂ζ̄ log(1 − ζ ζ)dξdη
=
π
ζ
π
ζ
|ζ|<1
=
|ζ|<1
1
2πi
¯
˜ dζ = −ζ̃¯ ,
log(1 − ζ ζ)
ζ2
Z
|ζ|=1
and
1
π
Z
|ζ|<1
1
dξdη
=
˜ − z)
(ζ − ζ)(ζ
π(ζ̃ − z)
Z
|ζ|<1
1
ζ − ζ̃
−
1
ζ̃ − z
dξ dη = −
ζ −z
ζ̃ − z
it follows
Z
z
dξdη
ω(ζ)
=
π
ζ(ζ − z)
|ζ|<1
= −c1 z̄ +
1
2πi
Z
(γ1 (ζ) − ζ̄f (ζ))
|ζ|=1
−
z
π
Z
1 − |z|2
z
log(1 − z ζ̄) + ζ̄
f (ζ) ζ − z
dξdη .
ζ ζ −z
dζ
ζ
−
(20)
|ζ|<1
From (19) and (20) the representation (18) follows.
Theorem 8. The boundary value problem for the inhomogeneous Bitsadze
equation in the unit disc
wz̄ z̄ = f in D, zwz = γ0 , zwzz̄ = γ1 on ∂D, w(0) = c0 , wz̄ (0) = c1 ,
H. Begehr
20
for f ∈ L1 (D; C), γ0 , γ1 ∈ C(∂D; C), c0 , c1 ∈ C is uniquely solvable if and
only if for |z| = 1
Z
Z
1
1
dζ
1
dζ
+
γ0 (ζ)
γ1 (ζ) log(1 − z ζ̄)
=
2πi
2πi
z
ζ
(1 − z ζ̄)ζ
|ζ|=1
|ζ|=1
z̄
π
=
Z
f (ζ)
ζ −z
dξ dη ,
(1 − z̄ζ)2
(21)
|ζ|<1
and
1
2πi
dζ
z̄
γ1 (ζ)
+
(1 − z̄ζ)ζ
π
Z
|ζ|=1
Z
f (ζ)
dξdη
= 0.
(1 − z̄ζ)2
(22)
|ζ|<1
The solution then is
w(z) = c0 + c1 z̄ −
1
2πi
Z
γ0 (ζ) log(1 − z ζ̄)
dζ
+
ζ
|ζ|=1
+
1
2πi
+
z
π
Z
γ1 (ζ)
|ζ|=1
1 − |z|2
z
log(1 − z ζ̄) + ζ̄
dζ
ζ
+
f (ζ) ζ − z
dξ dη .
ζ ζ −z
Z
(23)
|ζ|<1
Proof. The system
wz̄ = ω in D,
zwz = γ0 on ∂D,
w(0) = c0 ,
ωz̄ = f in D,
zωz = γ1 on ∂D,
ω(0) = c1 ,
is uniquely solvable if and only if
Z
Z
1
dζ
z̄
γ0 (ζ)
+
2πi
(1 − z̄ζ)ζ
π
|ζ|=1
ω(ζ)
dξdη
=0
(1 − z̄ζ)2
ω(ζ)
dξdη
= 0.
(1 − z̄ζ)2
|ζ|<1
and
1
2πi
Z
γ1 (ζ)
dζ
z̄
+
(1 − z̄ζ)ζ
π
|ζ|=1
Z
|ζ|=1
The solution then is
1
w(z) = c0 −
2πi
Z
dζ
z
γ0 (ζ) log(1 − z ζ̄) −
ζ
π
|ζ|=1
1
ω(z) = c1 −
2πi
Z
|ζ|=1
Z
ω(ζ)
dξdη
,
ζ(ζ − z)
f (ζ)
dξdη
.
ζ(ζ − z)
|ζ|<1
dζ
z
γ1 (ζ) log(1 − z ζ̄) −
ζ
π
Z
|ζ|<1
Boundary Value Problems for the Bitsadze Equation
21
Inserting the expression for ω into the first condition gives (21) because, as
in the preceding proof, on |z| = 1 one has
Z
z̄
ω(ζ)
dξdη =
π
(1 − z̄ζ)2
|ζ|<1
=
1
2πi
Z
γ1 (ζ)
1
dζ
z̄
log(1 − z ζ̄) −
z
ζ
π
|ζ|=1
Z
ζ −z
f (ζ) dξ dη .
(1 − z̄ζ)2
|ζ|<1
Also from
Z
Z
1 − |z|2
dζ
z
dξdη
1
ω(ζ)
= − c1 z̄ +
γ1 (ζ)
log(1 − z ζ̄)+ ζ̄
−
π
ζ(ζ − z)
2πi
z
ζ
|ζ|<1
|ζ|=1
z
−
π
Z
f (ζ) ζ − z
dξ dη
ζ ζ −z
|ζ|<1
the formula (23) follows.
5. The Inhomogeneous Polyanalytic Equation
As second order equations of special type were treated in the preceding
section, model equations of third, forth, fifth etc. order can also be investigated. From the material presented it is clear how to proceed and what
kind of boundary conditions can be posed. However, there is a variety of
boundary conditions possible. All kind of combinations of the three kinds,
Schwarz, Dirichlet, Neumann conditions can be posed. And there are even
others, e.g., boundary conditions of mixed type which are not investigated
here.
As a simple example, the Schwarz problem will be studied for the inhomogeneous polyanalytic equation. Another possibility is the Neumann problem
for the inhomogeneous polyharmonic equation, see [12, 13], and the Dirichlet problem, see [5]. As the results are published elsewhere [10, 11] only
statements are given.
Theorem 9. The Schwarz problem for the inhomogeneous polyanalytic
equation in the unit disc
∂z̄n w = f in D, Re ∂z̄ν w = γν on ∂D, Im ∂z̄ν w(0) = cν , 0 ≤ ν ≤ n − 1 ,
is uniquely solvable for f ∈ L1 (D; C), γν ∈ C(∂D; R), cν ∈ R, 0 ≤ ν ≤ n−1.
The solution is
n−1
n−1
X cν
X (−1)ν Z
ζ +z
dζ
w(z) = i
(z + z̄)ν +
γν (ζ)
(ζ − z + ζ − z)ν +
ν!
2πiν!
ζ −z
ζ
ν=0
ν=0
|ζ|=1
n
Z (−1)
f (ζ) ζ + z
f (ζ) 1 + z ζ̄ +
+
(ζ −z+ζ − z)n−1 dξdη . (24)
2π(n − 1)!
ζ ζ −z
ζ̄ 1 − z ζ̄
|ζ|<1
H. Begehr
22
For the proof see [11].
Theorem 10. The Dirichlet problem for the inhomogeneous polyanalytic
equation in the unit disc
∂z̄n w = f in D, ∂z̄ν w = γν on ∂D, 0 ≤ ν ≤ n − 1 ,
is uniquely solvable for f ∈ L1 (D; C), γν ∈ C(∂D; C), 0 ≤ ν ≤ n − 1, if and
only if for 0 ≤ ν ≤ n − 1
Z
n−1
X
γλ (ζ) (ζ − z)λ−ν
λ−ν z̄
(−1)
dζ−
2πi
1 − z̄ζ (λ − ν)!
λ=ν
|ζ|=1
−
z̄
π
Z
f (ζ) (z − ζ)n−1−ν
dξ dη = 0 .
1 − z̄ζ (n − 1 − ν)!
(25)
|ζ|<1
The solution then is
Z
n−1
X 1 Z γν (ζ) (z − ζ)ν
1
f (ζ) (z − ζ)n−1
dζ −
dξdη .
w(z) =
2πi
ν!
ζ −z
π
(n − 1)! ζ − z
ν=0
|ζ|=1
|ζ|<1
(26)
The proof is given in [10].
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Boundary Value Problems for the Bitsadze Equation
23
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(Received 11.10.2004)
Author’s address:
I. Math. Inst.
FU Berlin
Arnimallee 3, D-14195 Berlin
Germany
E-mail: begehr@math.fu-berlin.de