ABOUT SOME TYPES OF BOUNDARY VALUE PROBLEMS WITH INTERFACES

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1 13 Kragujevac J. Math. 3 27) ABOUT SOME TYPES OF BOUNDARY VALUE PROBLEMS WITH INTERFACES Boško S. Jovanović University of Belgrade, Faculty of Mathematics, Studentski trg 16, 11 Belgrade, Serbia bosko@matf.bg.ac.yu) Received December 17, 26) Abstract. We report results concerning different types of boundary value problems with interfaces. A broad class of such problems is defined. The corresponding abstract problems are investigated and some a priori estimates are presented. Intrinsic function spaces containing solutions of considered problems are introduced. 1. INTRODUCTION Interface problems occur in many applications in science and engineering [16], [18]. Let us mention, for example, heat transfer in presence of concentrated capacity, oscillations with concentrated mass, Pupin s induction coils etc. Such kind of problems can be modelled by partial differential equations with discontinuous or singular input data. First partial derivatives of their solutions have discontinuities across one or several interfaces, which have lower dimension than the domain where the problem is defined. Numerical methods designed for the solution of problems with smooth solutions do not work efficiently for the interface problems.

2 14 In the present work, we report results concerning different types of boundary value problems with interfaces [5], [7]-[13], [21]. In particular, we analyze different ways to set an interface problem strong form of equation with singular coefficients, problem with conjugation conditions), abstract models, a priori estimates, intrinsic function spaces containing the solution etc. 2. PARTIAL DIFFERENTIAL EQUATIONS WITH SINGULAR COEFFICIENTS Let Ω be a bounded domain in R n with the boundary Γ = Ω. Let S Ω be a hypersurface of dimension n 1 interface) dividing Ω into two disjoint parts Ω 1 and Ω 2. As a model problem we consider the following elliptic boundary value problem a ij x) u ) + [cx) + kx)δ S x)]u = fx), x Ω, x i x j 1) where a ij = a ji, ux) =, x Γ, a ij y i y j c yi 2, c >, i=1 cx) c 1 >, kx) k >, and δ S x) is the Dirac distribution [19] concentrated on S. We also consider analogous initial boundary value problems of parabolic [cx) + kx)δ S x)] u t a ij x) u ) = fx, t), x, t) Q, x i x j and hyperbolic types [cx) + kx)δ S x)] 2 u t 2 where Q = Ω, T ). ux, t) =, x, t) Γ, T ), ux, ) = u x), x Ω a ij x) u ) = fx, t), x, t) Q, x i x j ux, t) =, x, t) Γ, T ), ux, ) = u x), u t x, ) = u 1x), x Ω, 2) 3)

3 15 3. CONJUGATION CONDITIONS The considered interface problems can be formulated in an alternative manner, without explicit use of the Dirac distribution. It is well known that for a piecewise smooth function ϕ C 1 [a, ξ] C 1 [ξ, b] the derivative in distributional sense can be expressed in the following way [19]: ϕ x) = {ϕ x)} + [ϕ] ξ δx ξ) 4) where {ϕ x)} is the derivative in the classic sense and [ϕ] ξ = ϕξ + ) ϕξ ) is the jump of the function ϕ in the point ξ. Because of δ S x) = for x S from 1) immediately follows a ij x) u ) + cx)u = fx), x Ω 1 Ω 2, x i x j ux) =, x Γ, Taking into account 4), we conclude that on the interface S the following conjugation conditions are satisfied where u ν [u] S =, denote the co-normal derivative: u ν = 5) [ ] u = ku, x S, 6) ν S a ij x) u x j cosν, x i ). Analogously, the parabolic initial boundary value problem 2) can be represented in an equivalent form cx) u t a ij x) u ) = fx, t), x, t) Q 1 Q 2, x i x j ux, t) =, x, t) Γ, T ), [ ] u [u] S =, = k u, x S, t, T ), ν t S ux, ) = u x), x Ω, 7)

4 16 where Q i = Ω i, T ), while the hyperbolic problem 3) reduces to cx) 2 u t 2 a ij x) u ) = fx, t), x, t) Q 1 Q 2, x i x j ux, t) =, x, t) Γ, T ), [ ] u [u] S =, = k 2 u, x S, t, T ), ν S t2 u ux, ) = u x), t x, ) = u 1x), x Ω. 8) Our aim is to investigate the relation between differentiable properties of the solutions of the boundary value problems 1)-3) and the smoothness of its coefficients and right hand sides. It will be shown that this task can be resolved using methods of functional analysis to construct a priori estimates for the solutions of linear equations and the Cauchy problems for linear differential equations in Hilbert space. 4. ABSTRACT MODELS Let H be a real separable Hilbert space endowed with the inner product, ) and norm. Let A be an unbounded, selfadjoint, positive definite linear operator acting in H, with domain DA) dense in H. The product u, v) A = Au, v) u, v DA)) satisfies the inner product axioms. Reinforcing DA) in the norm u A = u, u) 1/2 A we obtain so called energy space H A H. The inner product u, v) continuously extends to H A H A, where H A = H A 1 H A, H and H A 1 is the adjoint space for H A. The spaces form Gel fand triple: H A H H A 1. Operator A extends to a mapping A : H A H A 1. There exists unbounded selfadjoint positive definite linear operator A 1/2 such that DA 1/2 ) = H A and u, v) A = Au, v) = A 1/2 u, A 1/2 v) see [15], [17]). For 1 p we define the Lebesgue space L p a, b), H) of functions mapping real interval a, b) into H, with the norm

5 17 b 1/p, u Lpa,b),H) = ut) dt) p 1 p < a u L a,b),h) = supess ut) t a,b) We also introduce the spaces of continuous functions C[a, b], H) and C k [a, b], H) with the norms: u C[a,b],H) = max t [a,b] ut), u C k [a,b],h) = max j k uj) C[a,b],H), where u j) t) = d j u/dt j, and the Sobolev spaces W k p a, b), H) with the norms [22]: k 1/p, u W k p a,b),h) = u j) p L pa,b),h)) k =, 1, 2,..., 1 p <, j= u W k+α p a,b),h) = u p + Wp k a,b),h) uk) p Wp α a,b),h) )1/p, < α < 1, b u W α p a,b),h) = with standard modification for p =. a b a ut) ut ) p t t 1+αp dt dt ) 1/p, Let B be another unbounded, selfadjoint, positive definite linear operator acting in H, such that DA) DB) H. In general, A and B are noncommutative. We assume that the quotient u A / u B is unbounded on DA). Under these assumptions there exists a countable set of eigenvalues {λ i } of the spectral problem Au = λbu. 9) All eigenvalues are positive and λ i when i. Further, u A λ 1 u B, u AB 1 A λ 1 u A, u AB 1 ) n A λ 1 u AB 1 ) n 1 A, n = 2, 3,... where λ 1 is the first minimal) eigenvalue of the spectral problem 9). We consider the following linear equation of the first kind in H A + B)u = f, 1)

6 18 which can be treated as an abstract model for the boundary value problem 1). Its solution satisfies the following a priori estimates: u B f A 1 BA 1, 11) u A f A 1, 12) u AB 1 A f B 1, 13) etc. Note that 11) and 12) also hold even for nonnegative B. We also consider the following abstract Cauchy problem Bu + Au = ft), < t < T ; u) = u 14) where ut) is an unknown function from,t) into H. Setting B 1/2 u = v equation 14) reduces to more simple v + Λv = gt), < t < T ; v) = v 15) where Λ = B 1/2 AB 1/2, g = B 1/2 f and v = B 1/2 u. Using the known a priori estimates for the solution of the problem 15) see [15], [14], [6]) T T vt) 2 Λ dt + T T ) vt) 2 dt M v 2Λ + Λ 1 gt) 2 dt, 1 T T vt) vt ) 2 t t 2 dt dt M Λvt) 2 + v t) 2 ) dt M v 2 Λ + T v 2 + gt) 2 Λ ), dt 1 T ) gt) 2 dt, etc. we immediately obtain the corresponding a priori estimates for the problem 14): u 2 L 2,T ),H B ) M u 2 BA 1 B + f 2 L 2,T ),H A 1 BA 1 )), 16) u 2 L 2,T ),H A ) + u 2 W 1/2 2,T ),H B ) M u 2 B + f 2 L 2,T ),H A 1 )), 17) u 2 L 2,T ),H AB 1 A ) + u 2 W 1 2,T ),H B) M u 2 A + f 2 L 2,T ),H B 1 )), 18)

7 19 etc. Here and in the sequel M denotes positive generic constant which can take different values in different formulas. Note that obtained a priori estimates contain whole information about the relation between the differentiability of the solution of Cauchy problem 14) and the smoothness properties of input data. For example, from 18) immediately follows: if u H A and f L 2, T ; H B 1) then u L 2, T ), H AB 1 A) W2 1, T ), H B ). Estimates 16) and 17) are satisfied also for nonnegative B. Finally, let us consider the Cauchy problem for the second order abstract differential equation Bu + Au = ft), < t < T ; u) = u ; u ) = u 1. 19) Similarly as in the previous case one obtains the following a priori estimates for its solution u C[,T ],HB ) M u B + u 1 BA 1 B + f L1,T ),H A 1 )), 2) u C[,T ],HA ) + u C[,T ],HB ) M u A + u 1 B + f L1,T ),H B 1 )), 21) etc. Estimate 16) also holds for nonnegative B. 5. IDENTIFICATION OF FUNCTION SPACES AND NORMS Let us choose H = L 2 Ω). Then the boundary value problem 1) reduces to the abstract form 1), where Au = a ij x) u ), Bu = [cx) + kx)δ S x)]u. x i x j Analogously, the initial boundary value problem 2) reduces to the abstract Cauchy problem 14), while the problem 3) reduces to 19). For u DA) = W2 2 Ω) W 2Ω) 1 using partial integration we get u 2 A = Ω a ij x) u x i u x j dx.

8 2 For a ij L Ω) under the assumption of strong ellipticity one immediately obtains u A u W 1 2 Ω) i.e. M 1 u W 1 2 Ω) u A M 2 u W 1 2 Ω)... In such a way, we conclude that H A = W 1 2 Ω). Consequently, H A 1 = H A = W 1 2 Ω)) = W 1 2 Ω) and u A 1 u W 1 2 Ω). If c L Ω), k L S), cx) c 1 > and kx) k > then u 2 B = Ω cx) u 2 x) dx + S kx) u 2 x) ds u 2 L 2 Ω) + u 2 L 2 S). In such a way H B = L 2 Ω) L 2 S), H B 1 = L 2 Ω) L 2 S)) and u B 1 sup v L 2 Ω) L 2 S) HB 1 u, v HB v 2 L 2 Ω) + v 2 L 2 S) )1/2. Here HB 1 u, v HB denotes the duality pairing in H B 1 H B. In addition to the previous assumptions, let a ij W 1 Ω). Then u 2 AB 1 A u 2 W 2 2 Ω 1) + u 2 W 2 2 Ω 2) + u 2 W 1 2 Ω). In such a way H AB 1 A = W2 2 Ω 1 ) W2 2 Ω 2 ) W2 1 Ω), H A 1 BA 1 W2 2 Ω 2 ) W2Ω)) 1 and = W 2 2 Ω 1 ) u A 1 BA 1 sup v W2 2Ω 1) W2 2 Ω 2) W2 1Ω) Finally, the norm u BA 1 B can be estimated as HA 1 BA 1 u, v HAB 1 A u 2 W 2 2 Ω 1) + u 2 W 2 2 Ω 2) + u 2 W 1 2 Ω))1/2. u BA 1 B = Bu A 1 u + u δ S W 1 2 Ω).

9 21 6. DEFINITION OF INTRINSIC FUNCTION SPACES A priori estimates 11)-13), 16)-18) and 2)-21) implicitly define intrinsic function spaces containing the solutions of considered interface problems. In the elliptic case, such spaces are Sobolev-like spaces W k 2 Ω) defined in the following manner: W 2 Ω) = L 2 Ω) = L 2 Ω) L 2 S), W 2 1 Ω) = W 2Ω), 1 W 2 k Ω) = W2 k Ω 1 ) W2 k Ω 2 ) W 2Ω), 1 k = 2, 3,... In the parabolic case we define anisotropic Sobolev-like spaces W k,k/2 2 Q) = L 2, T ), W k 2 Ω)) W k/2 2, T ), L 2 Ω)), k =, 1, 2,... The a priori estimates 11)-13) can now be rewritten as u L2 Ω) M f W 2 2 Ω)), 22) ) u W 1 2 Ω) i.e. M f u W 2 1Ω)) W2 1Ω) W 1 2 Ω), 23) Analogously, the estimates 16)-18) reduce to u W 2 2 Ω) M f L 2 Ω)). 24) u 2 W M u, 2 Q) + u δ S 2 + W 1 2 Ω) f 2 ), 25) L 2,T ), W 2 2Ω)) ) u 2 W M u 1,1/2 2 Q) 2 L2 + Ω) f 2 L 2 ), 26),T ),W 1 2 Ω)) u 2 W M u 2,1 2 Q) 2 + f 2 W2 1Ω) L 2,T ), L 2 ), 27) Ω)) ) while the estimates 2)-21) reduce to u C[,T ], L2 Ω)) M u L2 Ω) + u 1 + u 1 δ S W 1 2 Ω) + f L 1,T ),W 1 2 Ω)) ), 28) u u C[,T ], W2 1Ω)) t C[,T ], L2 Ω)) M u W2 1Ω) 1 L2 Ω) + f L 1,T ), L 2 Ω)) ) ). 29)

10 22 7. SOME OTHER INTERFACE PROBLEMS Initial boundary value problems with dynamical boundary conditions of the following type dx) k u t k = u ν, x Γ 1 Γ, t, T ), k = 1, 2 have similar properties as the previously investigated interface problems. Let us consider, for the sake of simplicity, the simplest one-dimensional parabolic problem with dynamical boundary condition cx) u t ax) u ) = fx, t), x, t) Q =, 1), T ), x x K u u, t) = a), t), u1, t) =, t x ux, ) = u x), x, 1) The problem 3) can be reduced to a problem of the type 2) using even extension of the input data: cx) = c x), ax) = a x), u x) = u x), and fx, t) = f x, t) for x 1, ). It easily follows that the solution ux, t) can also be evenly extended on 1, ), T ) and it satisfies the conditions [cx) + 2Kδx)] u t ax) u ) = fx, t), x, t) 1, 1), T ), x x u 1, t) =, u1, t) =, ux, ) = u x), x 1, 1). Analogous results hold for the corresponding hyperbolic problem with dynamical boundary condition. Finally, let us mention so called weakly evolution problems, i.e. 3) 31) the initial boundary value problems of the form 2) and 3) where cx) =. In this case the operator B reduces to Bu = kx)δ S x)ux) and it is only nonnegative. Consequently, the inverse operator B 1 is not defined and the a priori estimates 18) and 21) are meaningless. We have u B u L2 S)

11 23 and H B = L 2 S), therefore the a priori estimate 26) reduces to u 2 W 1,1/2 2 Q) u 2 + u 2 L 2,T ), W2 1Ω)) W 1/2 2,T ),L 2 S)) M u 2 L 2 S) + f 2 ), 32) L 2,T ),W 1 2 Ω)) Note that in this case the initial conditions 2) and 3) are determined only on interface S. 8. FINITE DIFFERENCE APPROXIMATION For the finite difference analogues of 1), 14) and 19) similar results hold. For example, the discrete analogues of the a priori estimates 16)-18) are obtained in [7], [8], while the discrete analogues of the a priori estimates 2)-21) are obtained in [9]. In particular, convergence of finite difference schemes approximating interface problems in a natural way can be proved in the discrete version of norms appearing in 22)-29) see [7]-[13], [21]). Numerical methods for the solution of different interface problems are also investigated in [1], [2], [3], [2], [4], [23] etc. Acknowledgements: This research was supported by MSEP of Republic of Serbia project 1445A). References [1] I. Braianov, Convergence of a Crank Nicolson difference scheme for heat equation with interface in the heat flow and concentrated heat capacity, Lect. Notes Comput. Sci ), [2] I. Braianov, L. Vulkov, Finite Difference Schemes with Variable Weights for Parabolic Equations with Concentrated Capacity, Notes on Numerical Fluid Dynamics, Vieweg, ),

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