Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 202, pp. 1–7.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE AND UNIQUENESS FOR BOUNDARY-VALUE
PROBLEM WITH ADDITIONAL SINGLE POINT CONDITIONS
OF THE STOKES-BITSADZE SYSTEM
MUHAMMAD TAHIR
Abstract. This article shows the uniqueness of a solution to a Bitsadze system of equations, with a boundary-value problem that has four additional
single point conditions. It also shows how to construct the solution.
1. Introduction
The planar Stokes flow based on stream function ψ(x, y) and stress function
φ(x, y), is expressed as
φxx − φyy = −4ηψxy ,
(1.1)
−φxy = η(ψyy − ψxx ),
where η is a material constant, see for the details [4, 5, 9]. The re-scaling (2ηψ → ψ)
reduces the system (1.1) to
φxx − φyy + 2ψxy = 0,
ψxx − ψyy − 2φxy = 0,
(1.2)
which is the famous second order elliptic system called the Bitsadze system of
equations and is identified as Stokes-Bitsadze system [10]. In the literature Bitsadze
appears to have been the first to question the uniqueness and existence or even the
well-posedness of (1.2) subject to certain boundary conditions, see for reference
[2, 3, 7]. Oshorov [8] finds well-posed problems for the Cauchy-Riemann system
and extends those to the Bitsadze system (1.2). Vaitekhovich [12] discusses Dirichlet
and Schwarz problems for the inhomogeneous Bitsadze equation for a circular ring
domain. In the interior of unit disc a boundary value problem for the Bitsadze
equation is considered by Babayan [1] and is proved to be Noetherian. In his
paper Babayan also proposes solvability conditions for the inhomogeneous Bitsadze
equation. The unique solvability in a unit disc for the inhomogeneous Bitsadze
system is discussed in [6].
The Stokes-Bitsadze system (1.2) can be expressed in the matrix form as
AUxx + 2BUxy + CUyy = 0,
(1.3)
2000 Mathematics Subject Classification. 35J57.
Key words and phrases. Bitsadze system; boundary value problem; single point conditions.
c
2012
Texas State University - San Marcos.
Submitted July 24, 2012. Published November 15, 2012.
1
2
M. TAHIR
EJDE-2012/202
where
A=
1
0
0
,
1
B=
0
−1
1
,
0
C = −A,
U(x, y) =
φ
.
ψ
In a domain Ω ⊂ R2 with boundary Γ a linear boundary value problem of Poincaré
for the system (1.3) can be formulated as
p1 Ux + p2 Uy + qU = α(x, y),
(x, y) ∈ Γ
(1.4)
where p1 , p2 , q are real 2×2 matrices and α(x, y) a real vector given on the boundary
Γ. The boundary-value problems of Poincaré for the Stokes-Bitsadze system will be
discussed elsewhere. In this paper we are interested in a boundary value problem
with four additional single point conditions.
2. A boundary value problem with additional single point conditions
We consider the Stokes-Bitsadze system (1.2) in domain Ω ⊂ R2 with boundary
Γ subject to the following boundary conditions.
ψ = f,
ψn = g
on Γ,
(2.1)
at a single point P ∈ Ω̄.
(2.2)
and
φ = φP ,
∇φ = (∇φ)P ,
∆φ = (∆φ)P ,
Theorem 2.1. For f, g ∈ C(Γ), the boundary value problem (2.1)–(2.2) for the
Stokes-Bitsadze system (1.2) has a unique solution (φ, ψ) ∈ C 4 (Ω) × C 4 (Ω).
Proof. Suppose φ, ψ ∈ C 4 (Ω). If (φ, ψ) satisfies (1.2), then φ and ψ are biharmonic
in Ω, and for f, g ∈ C(Γ) the problem
∆2 ψ = 0
in Ω
ψ=f
on Γ
ψn = g
on Γ
(2.3)
4
has a unique solution ψ ∈ C (Ω), [11], that satisfies (1.2) and (2.1). Let the unique
e Now we show that for the unique ψe if there exists
solution be denoted by ψ.
φ satisfying (1.2) and (2.1)–(2.2) then that φ is unique. Assume that the pairs
e and (φ2 , ψ)
e with φ1 6= φ2 satisfy (1.2) and (2.1)–(2.2) and that δ = φ1 − φ2 .
(φ1 , ψ)
Then from (1.2) it immediately follows that
δxx − δyy = 0,
δxy = 0
on Ω.
(2.4)
But (2.2) then yields
δ = 0,
∇δ = 0,
∆δ = 0
at P,
(2.5)
and the general solution of the system (2.4) becomes,
δ = ax + by + c(x2 + y 2 ) + d,
(2.6)
which on imposing the conditions (2.5) gives δ ≡ 0 in Ω̄ and uniqueness of φ thus
follows. Hence there exists at most one pair (φ, ψ) ∈ C 4 (Ω) × C 4 (Ω) that can
satisfy (1.2) and (2.1)–(2.2). We are now in a position to assume (without proof)
e ψ)
e is a solution of (1.2) and (2.1)–(2.2).
that (φ,
Next, we suppose that P (xP , yP ) and Q(x, yP ) are the points in Ω̄, refer to the
Figure 1.
EJDE-2012/202
EXISTENCE AND UNIQUENESS
3
Figure 1. Boundary conditions and additional single point conditions
At point P the expressions (1.2)(a) and (2.2)(c) respectively take the form
P
P
φP
xx − φyy = −2ψxy ,
P
P
φP
xx + φyy = ∆φ ,
(2.7)
P
e e
from which it is obvious that φP
xx and φyy are known at P . Since (φ, ψ) satisfies
(1.2)(b), therefore
1
φexyy = [ψexxy − ψeyyy ],
(2.8)
2
and on integration along P Q we have
Z
1 x e
P
e
φyy (x, yP ) = φyy +
[ψxxy (λ, yP ) − ψeyyy (λ, yP )]dλ,
(2.9)
2 xP
Z
1 x e
[ψxx (λ, yP ) − ψeyy (λ, yP )]dλ.
(2.10)
φey (x, yP ) = φP
+
y
2 xP
Since all the terms on right hand sides of (2.9) and (2.10) are known therefore φeyy
e ψ)
e satisfies (1.2)(a), we have
and φey are known along P Q. Since (φ,
φexx = φeyy − 2ψexy ,
and using (2.9), can further be expressed as
Z
1 x e
P
e
φxx (x, yP ) = φyy +
[ψxxy (λ, yP ) − ψeyyy (λ, yP )]dλ − 2ψexy (λ, yP ).
2 xP
Further on integration along P Q, we have
Z x
Z
1 µ e
P
e
φex (x, yP ) = φP
+
φ
+
[
ψ
(λ,
y
)
−
ψ
(λ,
y
)]
dλ dµ
xxy
P
yyy
P
x
yy
2 xP
xP
Z x
−2
ψexy (λ, yP ) dλ,
xP
(2.11)
(2.12)
(2.13)
4
M. TAHIR
EJDE-2012/202
whence
e yP )
φ(x,
P
xP )φP
x
1
+ (x − xP )2 φP
yy − 2
2
Z
x
Z
µ
= φ + (x −
ψexy (λ, yP )dλ dµ
xP xP
Z Z Z
1 x ν µ e
ψxxy (λ, yP ) − ψeyyy (λ, yP ) dλ dµ dν.
+
2 xP xP xP
(2.14)
Since all the terms on right hand sides of (2.11), (2.12), (2.13) are known therefore
e ∇φe and ∆φe at Q(x, yP ).
φexx , φex and φe are known along P Q and hence we know φ,
Now from the point Q we draw the line QR where R(x, y) ∈ Ω̄ is an arbitrary
e ψ)
e satisfies (1.2)(b); therefore
point. Again, since (φ,
1
φexxy = [ψexxx − ψexyy ],
2
(2.15)
which on integration, along QR, gives
Z
1
φexx (x, y) = φexx (x, yP ) +
2
y
[ψexxx (x, λ) − ψexyy (x, λ)]dλ,
(2.16)
[ψexx (x, λ) − ψeyy (x, λ)]dλ.
(2.17)
y
Z Py
1
φex (x, y) = φex (x, yP ) +
2
yP
But the following expression from (1.2)(a)
φeyy = φexx + 2ψexy ,
(2.18)
on integration along QR gives
Z
y
φey (x, y) = φey (x, yP ) +
[φexx (x, λ) + 2ψexy (x, λ)] dλ.
(2.19)
yP
Using (2.10) and (2.16) the expression (2.19) takes the form
Z
1 x e
φey (x, y) = φP
+
[ψxx (λ, yP ) − ψeyy (λ, yP )]dλ + (y − yP )φexx (x, yP )
y
2 xP
Z Z
Z y
(2.20)
1 y µ e
+
[ψxxx (x, λ) − ψexyy (x, λ)]dλ dµ + 2
ψexy (x, λ)dλ.
2 yP yP
yP
Integrating along QR we obtain from (2.20) as follows.
e y) = φ(x,
e yP ) + (y − yP )φP + 1 (y − yP )2 φexx (x, yP )
φ(x,
y
2
Z x
1
+ (y − yP )
[ψexx (λ, yP ) − ψeyy (λ, yP )]dλ
2
xP
Z Z Z
1 y ν µ e
+
[ψxxx (x, λ) − ψexyy (x, λ)]dλ dµdν
2 yP yP yP
Z yZ µ
+2
ψexy (x, λ)dλ dµ.
yP
yP
(2.21)
EJDE-2012/202
EXISTENCE AND UNIQUENESS
5
e y) at an
Using (2.12) and (2.14) we finally obtain the following expression for φ(x,
arbitrary point (x, y) ∈ Ω̄.
e y)
φ(x,
1
2
2 P
P
= φP + (x − xP )φP
x + (y − yP )φy + [(x − xP ) + (y − yP ) ]φyy
2Z
x
1
− (y − yP )2 ψexy (x, yP ) + (y − yP )
[ψexx (λ, yP ) − ψeyy (λ, yP )] dλ
2
xP
Z x
1
[ψexxy (λ, yP ) − ψeyyy (λ, yP )]dλ
+ (y − yP )2
4
xP
Z xZ µ
Z yZ µ
−2
ψexy (λ, yP ) dλ dµ + 2
ψexy (x, λ)dλ dµ
x
x
y
(2.22)
y
P
P
ZP ZP Z
1 x ν µ e
[ψxxy (λ, yP ) − ψeyyy (λ, yP )]dλ dµ dν
+
2 xP xP xP
Z Z Z
1 y ν µ e
[ψxxx (x, λ) − ψexyy (x, λ)]dλ dµ dν.
+
2 yP yP yP
Obviously we have obtained an explicit representation for φe in terms of the point
e on the assumption that (φ,
e ψ)
e satisfies (1.2) and (2.1)–(2.2). Next
conditions and ψ,
e
e
we show that (φ, ψ) actually satisfies the Bitsadze system (1.2) and the conditions
(2.2).
e P , yP ) = φP . We use (2.13)
From expression (2.22) it is easy to verify that φ(x
in (2.17) to obtain
Z x
Z
1 µ e
P
φex (x, y) = φP
+
[φ
+
[ψxxy (λ, yP ) − ψeyyy (λ, yP )]dλ] dµ
x
yy
2 xP
xP
Z x
Z
1 y e
−2
ψexy (λ, yP ) dλ +
[ψxx (x, λ) − ψeyy (x, λ)]dλ,
2 yP
xP
and it can be easily verified that φex (xP , yP ) = φP
x . Similarly from (2.10) and (2.20)
we have
Z
Z y
1 x e
e
φey (x, y) = φP
+
[
ψ
(λ,
y
)
−
ψ
(λ,
y
)]
dλ
+
[φexx (x, λ) + 2ψexy (x, λ)] dλ,
xx
P
yy
P
y
2 xP
yP
and it follows that φey (xP , yP ) = φP
y . Again, from (2.12)and (2.16) we obtain
Z x
1
[ψexxy (λ, yP ) − ψeyyy (λ, yP )] dλ − 2ψexy (x, yP )
φexx (x, y) = φP
yy +
2 xP
Z
1 y e
+
[ψxxx (x, λ) − ψexyy (x, λ)] dλ,
2 yP
which at P yields
e
φexx (xP , yP ) = φP
yy − 2ψxy (xP , yP ),
(2.23)
and from (2.7)(a) we obtain φexx (xP , yP ) = φP
xx . Also from (2.18) it is obvious that
φeyy (xP , yP ) = φexx (xP , yP ) + 2ψexy (xP , yP ),
and (2.23)–(2.24) yield φeyy (xP , yP ) = φP
yy .
(2.24)
6
M. TAHIR
EJDE-2012/202
e y) satisfies (1.2)(a). Using (2.10) in (2.20) and then
Now we verify that φ(x,
differentiating with respect to x we obtain
1
1
φexy (x, y) = [ψexx (x, yP ) − ψeyy (x, yP )] + (y − yP )[ψexxy (x, yP ) − ψeyyy (x, yP )]
2
2
Z Z
1 y µ e
e
− 2(y − yP )ψxxy (x, yP ) +
[ψxxxx (x, λ) − ψexxyy (x, λ)] dλ dµ
2 yp yP
+ 2ψexx (x, y) − 2ψexx (x, yP ),
which, since ∆2 ψe = 0, can be simplified as
φexy (x, y)
1
1
= − [3ψexx (x, yP ) + ψeyy (x, yP )] − (y − yP )[3ψexxy (x, yP ) + ψeyyy (x, yP )]
2
2
(2.25)
1 e
1 e
e
− [3ψxx (x, yP ) + ψyy (x, y)] + [3ψxx (x, yP ) + ψeyy (x, yP )]
2
2
1
e
e
+ (y − yP )[3ψxxy (x, yP ) + ψyyy (x, yP )] + 2ψexx (x, y),
2
and we obtain
1
(2.26)
φexy (x, y) = [ψexx (x, y) − ψeyy (x, y)].
2
e y) satisfies (1.2)(b), we use (2.22) to obtain
Then, to verify that φ(x,
φexx (x, y) − φeyy (x, y)
1
= −(y − yP )2 ψexxxy (x, yP ) + (y − yP )[ψexxx (x, yP ) − ψexyy (x, yP )]
2
1
+ (y − yP )2 [ψexxxy (x, yP ) − ψexyyy (x, yP )]
4Z Z
Z
y
µ
1 x e
+2
ψexxxy (x, λ)dλ dµ +
[ψxxy (λ, yP ) − ψeyyy (λ, yP )] dλ
2 xP
yP yP
Z Z Z
1 y ν µ e
[ψxxxxx (x, λ) − ψexxxyy (x, λ)] dλ dµ dν
+
2 yP yP yP
Z
1 x e
[ψxxy (λ, yP ) − ψeyyy (λ, yP )]dλ − 2ψexy (x, y)
−
2 xP
Z
1 y e
−
[ψxxx (x, λ) − ψexyy (x, λ)] dλ,
2 yP
which can further be simplified to obtain
φexx (x, y) − φeyy (x, y)
1
= − (y − yP )2 [3ψexxxy (x, yP ) + ψexyyy (x, yP )]
4
1
− (y − yP )[3ψexxx (x, yP ) + ψexyy (x, yP )]
2Z
1 y e
1
−
[3ψxxx (x, λ) + ψexyy (x, λ)] dλ + (y − yP )[3ψexxx (x, yP ) + ψexyy (x, yP )]
2 yP
2
1
+ (y − yP )2 [3ψexxxy (x, yP ) + ψexyyy (x, yP )]
4
EJDE-2012/202
1
− 2ψexy (x, y) +
2
EXISTENCE AND UNIQUENESS
Z
7
y
[3ψexxx (x, λ) + ψexyy (x, λ)]dλ,
yP
and finally we have
φexx (x, y) − φeyy (x, y) = −2ψexy (x, y),
which completes the proof.
Conclusion. It has been proved by construction that there exists a unique solution
e ψ)
e in C 4 (Ω)×C 4 (Ω) to the Stokes-Bitsadze system (1.2) subject to the boundary
(φ,
conditions (2.1) along with additional single point conditions (2.2).
Acknowledgements. The author is grateful to Professor A. Russell Davies, Head
School of Mathematics, Cardiff University, United Kingdom for his useful suggestions.
References
[1] A. H. Babayan; A boundary value problem for Bitsadze equation in the unit disc, J. Contemp.
Math. Anal. 42(4) (2007) 177-183.
[2] A. V. Bitsadze; Some classes of partial differential equations, Gordon and Breach Science
Publishers, New York, 1988.
[3] A. V. Bitsadze; On the uniqueness of the solution of the Dirichlet problem for the elliptic
partial differential operators, Uspekhi Mat. Nauk. 3(6) (1948) 211-212.
[4] C. J. Coleman; A contour integral formulation of plane creeping Newtonian flow, Q. J. Mech.
appl. Math. XXXIV (1981) 453-464.
[5] A. R. Davies, J. Devlin; On corner flows of Oldroyd-B fluids, J. Non-Newtonian Fluid Mech.
50 (1993) 173-191.
[6] S. Hizliyel, M. Cagliyan; A boundary value problem for Bitsadze equation in matrix form.
Turkish J. Math. 35(1) (2011) 29-46.
[7] E. N. Kuzmin; On the Dirichlet problem for elliptic systems in space. Differential Equations,
3(1) (1967) 78-79.
[8] B. B. Oshorov; On boundary value problems for the Cauchy-Riemann and Bitsadze systems
of equations, Doklady Mathematics 73(2) (2006) 241-244.
[9] R. G. Owens, T. N. Phillips; Mass and momentum conserving spectral methods for Stokes
flow, J. Comput. Appl. Math. 53 (1994) 185-206.
[10] M. Tahir; The Stokes-Bitsadze system, Punjab Univ. J. Math. XXXII (1999) 173-180.
[11] A. N. Tikhonov, A. A. Samarskii; Equations of Mathematical Physics. Pergamon Press Ltd.
Oxford, 1963.
[12] T. Vaitekhovich; Boundary value problems to second order complex partial differential equations in a ring domain, Siauliai Math. Semin. 2 (10) (2007) 117-146.
Muhammad Tahir
Mathematics Department, HITEC University, Taxila Cantonment, Pakistan
E-mail address: mtahir@hitecuni.edu.pk