Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 202, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.tstate.edu or http://ejde.math.unt.edu ftp ejde.math.tstate.edu EXISTENCE AND UNIQUENESS FOR BOUNDARY-VALUE ROBLEM WITH ADDITIONAL SINGLE OINT 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(, y) and stress function φ(, y), is epressed as φ φ yy = 4ηψ y, (1.1) φ y = η(ψ yy ψ ), where η is a material constant, see for the details [4, 5, 9]. The re-scaling (2ηψ ψ) reduces the system (1.1) to φ φ yy + 2ψ y = 0, ψ ψ yy 2φ y = 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 eistence 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 etends 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 epressed in the matri form as AU + 2BU y + CU yy = 0, (1.3) 2000 Mathematics Subject Classification. 35J57. Key words and phrases. Bitsadze system; boundary value problem; single point conditions. c 2012 Teas State University - San Marcos. Submitted July 24, 2012. ublished November 15, 2012. 1
2 M. TAHIR EJDE-2012/202 where A = ( ) 1 0, B = 0 1 ( ) 0 1, C = A, U(, y) = 1 0 ( ) φ. ψ In a domain Ω R 2 with boundary Γ a linear boundary value problem of oincaré for the system (1.3) can be formulated as p 1 U + p 2 U y + qu = α(, y), (, y) Γ (1.4) where p 1, p 2, q are real 2 2 matrices and α(, y) a real vector given on the boundary Γ. The boundary-value problems of oincaré 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 R 2 with boundary Γ subject to the following boundary conditions. and ψ = f, ψ n = g on Γ, (2.1) φ = φ, φ = ( φ), φ = ( φ), at a single point Ω. (2.2) 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). roof. 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 ψ = f ψ n = g in Ω on Γ on Γ (2.3) has a unique solution ψ C 4 (Ω), [11], that satisfies (1.2) and (2.1). Let the unique solution be denoted by ψ. Now we show that for the unique ψ if there eists φ satisfying (1.2) and (2.1) (2.2) then that φ is unique. Assume that the pairs (φ 1, ψ) and (φ 2, ψ) with φ 1 φ 2 satisfy (1.2) and (2.1) (2.2) and that δ = φ 1 φ 2. Then from (1.2) it immediately follows that But (2.2) then yields δ δ yy = 0, δ y = 0 on Ω. (2.4) δ = 0, δ = 0, δ = 0 at, (2.5) and the general solution of the system (2.4) becomes, δ = a + by + c( 2 + y 2 ) + d, (2.6) which on imposing the conditions (2.5) gives δ 0 in Ω and uniqueness of φ thus follows. Hence there eists 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) that ( φ, ψ) is a solution of (1.2) and (2.1) (2.2). Net, we suppose that (, ) and Q(, ) are the points in Ω, refer to the Figure 1.
EJDE-2012/202 EXISTENCE AND UNIQUENESS 3 Figure 1. Boundary conditions and additional single point conditions At point the epressions (1.2)(a) and (2.2)(c) respectively take the form φ φ yy = 2ψ y, φ + φ yy = φ, (2.7) from which it is obvious that φ and φ yy are known at. Since ( φ, ψ) satisfies (1.2)(b), therefore and on integration along Q we have φ yy (, ) = φ yy φ y (, ) = φ y φ yy = 1 2 [ ψ y ψ yyy ], (2.8) [ ψ y (λ, ) ψ yyy (λ, )]dλ, (2.9) [ ψ (λ, ) ψ yy (λ, )]dλ. (2.10) Since all the terms on right hand sides of (2.9) and (2.10) are known therefore φ yy and φ y are known along Q. Since ( φ, ψ) satisfies (1.2)(a), we have and using (2.9), can further be epressed as φ (, ) = φ yy φ = φ yy 2 ψ y, (2.11) [ ψ y (λ, ) ψ yyy (λ, )]dλ 2 ψ y (λ, ). (2.12) Further on integration along Q, we have [ φ (, ) = φ + φ yy + 1 µ ] [ 2 ψ y (λ, ) ψ yyy (λ, )] dλ dµ 2 ψy (λ, ) dλ, (2.13)
4 M. TAHIR EJDE-2012/202 whence φ(, ) = φ + ( )φ ( ) 2 φ yy 2 µ ν µ [ ψy (λ, ) ψ yyy (λ, ) ] dλ dµ dν. ψy (λ, )dλ dµ (2.14) Since all the terms on right hand sides of (2.11), (2.12), (2.13) are known therefore φ, φ and φ are known along Q and hence we know φ, φ and φ at Q(, ). Now from the point Q we draw the line QR where R(, y) Ω is an arbitrary point. Again, since ( φ, ψ) satisfies (1.2)(b); therefore which on integration, along QR, gives φ (, y) = φ (, ) φ (, y) = φ (, ) φ y = 1 2 [ ψ ψ yy ], (2.15) [ ψ (, λ) ψ yy (, λ)]dλ, (2.16) But the following epression from (1.2)(a) on integration along QR gives φ y (, y) = φ y (, ) + [ ψ (, λ) ψ yy (, λ)]dλ. (2.17) φ yy = φ + 2 ψ y, (2.18) Using (2.10) and (2.16) the epression (2.19) takes the form [ φ (, λ) + 2 ψ y (, λ)] dλ. (2.19) φ y (, y) = φ y µ [ ψ (λ, ) ψ yy (λ, )]dλ + (y ) φ (, ) [ ψ (, λ) ψ yy (, λ)]dλ dµ + 2 ψy (, λ)dλ. (2.20) Integrating along QR we obtain from (2.20) as follows. φ(, y) = φ(, ) + (y )φ y (y ) 2 φ (, ) + 1 2 (y ) [ ψ (λ, ) ψ yy (λ, )]dλ + 2 ν µ y y µ [ ψ (, λ) ψ yy (, λ)]dλ dµdν ψy (, λ)dλ dµ. (2.21)
EJDE-2012/202 EXISTENCE AND UNIQUENESS 5 Using (2.12) and (2.14) we finally obtain the following epression for φ(, y) at an arbitrary point (, y) Ω. φ(, y) = φ + ( )φ + (y )φ y [( ) 2 + (y ) 2 ]φ yy (y ) 2 ψy (, ) (y ) + 1 4 (y ) 2 2 µ ν µ [ ψ y (λ, ) ψ yyy (λ, )]dλ ψy (λ, ) dλ dµ + 2 y ν µ µ [ ψ (λ, ) ψ yy (λ, )] dλ ψy (, λ)dλ dµ [ ψ y (λ, ) ψ yyy (λ, )]dλ dµ dν [ ψ (, λ) ψ yy (, λ)]dλ dµ dν. (2.22) Obviously we have obtained an eplicit representation for φ in terms of the point conditions and ψ, on the assumption that ( φ, ψ) satisfies (1.2) and (2.1) (2.2). Net we show that ( φ, ψ) actually satisfies the Bitsadze system (1.2) and the conditions (2.2). From epression (2.22) it is easy to verify that φ(, ) = φ. We use (2.13) in (2.17) to obtain φ (, y) = φ + 2 [φ yy µ ψy (λ, ) dλ [ ψ y (λ, ) ψ yyy (λ, )]dλ] dµ [ ψ (, λ) ψ yy (, λ)]dλ, and it can be easily verified that φ (, ) = φ. Similarly from (2.10) and (2.20) we have φ y (, y) = φ y [ ψ (λ, ) ψ yy (λ, )] dλ + [ φ (, λ) + 2 ψ y (, λ)] dλ, and it follows that φ y (, ) = φ y. Again, from (2.12)and (2.16) we obtain φ (, y) = φ yy which at yields [ ψ y (λ, ) ψ yyy (λ, )] dλ 2 ψ y (, ) [ ψ (, λ) ψ yy (, λ)] dλ, φ (, ) = φ yy 2 ψ y (, ), (2.23) and from (2.7)(a) we obtain φ (, ) = φ. Also from (2.18) it is obvious that φ yy (, ) = φ (, ) + 2 ψ y (, ), (2.24) and (2.23) (2.24) yield φ yy (, ) = φ yy.
6 M. TAHIR EJDE-2012/202 Now we verify that φ(, y) satisfies (1.2)(a). differentiating with respect to we obtain Using (2.10) in (2.20) and then φ y (, y) = 1 2 [ ψ (, ) ψ yy (, )] (y )[ ψ y (, ) ψ yyy (, )] 2(y ) ψ y (, ) + 2 ψ (, y) 2 ψ (, ), µ which, since 2 ψ = 0, can be simplified as φ y (, y) y p [ ψ (, λ) ψ yy (, λ)] dλ dµ = 1 2 [3 ψ (, ) + ψ yy (, )] 1 2 (y )[3 ψ y (, ) + ψ yyy (, )] 1 2 [3 ψ (, ) + ψ yy (, y)] [3 ψ (, ) + ψ yy (, )] (2.25) (y )[3 ψ y (, ) + ψ yyy (, )] + 2 ψ (, y), and we obtain φ y (, y) = 1 2 [ ψ (, y) ψ yy (, y)]. (2.26) Then, to verify that φ(, y) satisfies (1.2)(b), we use (2.22) to obtain φ (, y) φ yy (, y) = (y ) 2 ψy (, ) (y )[ ψ (, ) ψ yy (, )] + 1 4 (y ) 2 [ ψ y (, ) ψ yyy (, )] + 2 1 2 1 2 µ y y ν µ y ψy (, λ)dλ dµ [ ψ y (λ, ) ψ yyy (λ, )] dλ [ ψ (, λ) ψ yy (, λ)] dλ dµ dν [ ψ y (λ, ) ψ yyy (λ, )]dλ 2 ψ y (, y) [ ψ (, λ) ψ yy (, λ)] dλ, which can further be simplified to obtain φ (, y) φ yy (, y) = 1 4 (y ) 2 [3 ψ y (, ) + ψ yyy (, )] 1 2 (y )[3 ψ (, ) + ψ yy (, )] 1 2 [3 ψ (, λ) + ψ yy (, λ)] dλ (y )[3 ψ (, ) + ψ yy (, )] + 1 4 (y ) 2 [3 ψ y (, ) + ψ yyy (, )]
EJDE-2012/202 EXISTENCE AND UNIQUENESS 7 2 ψ y (, y) and finally we have which completes the proof. [3 ψ (, λ) + ψ yy (, λ)]dλ, φ (, y) φ yy (, y) = 2 ψ y (, y), Conclusion. It has been proved by construction that there eists a unique solution ( φ, ψ) in C 4 (Ω) C 4 (Ω) to the Stokes-Bitsadze system (1.2) subject to the boundary conditions (2.1) along with additional single point conditions (2.2). Acknowledgements. The author is grateful to rofessor 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 ublishers, 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 matri 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. hillips; Mass and momentum conserving spectral methods for Stokes flow, J. Comput. Appl. Math. 53 (1994) 185-206. [10] M. Tahir; The Stokes-Bitsadze system, unjab Univ. J. Math. XXXII (1999) 173-180. [11] A. N. Tikhonov, A. A. Samarskii; Equations of Mathematical hysics. ergamon ress Ltd. Oford, 1963. [12] T. Vaitekhovich; Boundary value problems to second order comple partial differential equations in a ring domain, Siauliai Math. Semin. 2 (10) (2007) 117-146. Muhammad Tahir Mathematics Department, HITEC University, Taila Cantonment, akistan E-mail address: mtahir@hitecuni.edu.pk