On the solvability of the initial boundary value problem for schrodinger systems in conical domains

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2010, Vol. 55, No. 6, pp. 82-89 ON THE SOLVABILITY OF THE INITIAL BOUNDARY VALUE PROBLEM FOR SCHRO¨DINGER SYSTEMS IN CONICAL DOMAINS Nguyen Thi Lien Hanoi National University of Education E-mail: Lienhnue@gmail.com Abstract. In this paper, we consider the initial boundary value problem for Schro¨dinger systems in the cylinders with base containing the conical point. The existence and the uniqueness of the generanized solution of this problem are given. Keywords: Initial boundary value problem, generalized solution, cylinders with conical base. 1. Introduction The initial boundary value problems for Schro¨dinger equations in the cylinders with base containing conical points were established in [2,3]. Such problems for parabolic systems have been studied in Sobolev spaces with weights [4,5]. We are concerned with initial boundary value problems for Schro¨dinger sys- tems in cylinders with base containing conical point. The paper is organized is the following way. In Section 2 we define the prob- lem. In Section 3 we establish the unique existence of the generalized solution of the problem. Finally, in Section 4 we apply the obtained results to a problem of mathematical physics. 2. Notations and formulation of the problem Let Ω be a bounded domain in Rn (n ≥ 2) with the boundary ∂Ω. We suppose that S = ∂Ω \ {0} is a smooth manifold and Ω is in a neighbourhood U of the origin 0 coincides with the cone K = {x : x/ | x |∈ G}, where G is a smooth domain on the unit sphere Sn−1 in Rn. Let T be a positive real number or T =∞. Set Ωt = Ω× (0, t), St = S × (0, t). For each multi-index α = (α1, . . . , αn) ∈ Nn, |α| = α1 + · · ·+ αn, the symbol Dα = ∂|α|/∂xα11 ...∂x αn n denotes the generalized derivative of order α with respect to x = (x1, ..., xn); ∂ k/∂tk is the generalized derivative of order k with respect to t. Let u = (u1, ..., us) be a complex-valued vector function defined on ΩT . We use notation: Dαu = (Dαu1, ..., D αus); utj = ∂ ku/∂tk = (∂ju1/∂t j , .., ∂jus/∂t j). Let us introduce some functional spaces used in this paper (see [1]): 82 On the solvability of the initial boundary value problem for Schro¨dinger systems... We use H l(Ω) be the space of functions defined in Ω with the norm ‖u‖Hl(Ω) = ( l∑ |α|=0 ∫ Ω |Dαu|2dx) 12 Let X, Y be Banach spaces. Denote by L2((0, T );X) the space consisting of all measureable functions u : (0, T ) −→ X with the norm ‖u‖L2((0,T );X) = ( ∫ T 0 ‖u(t)‖2Xdt ) 1 2 and by H((0, T );X, Y ) the space consisting of all functions u ∈ L2((0, T );X) such that the generalized derivative ut exists and belongs to L2((0, T );Y ). The norm in H((0, T );X, Y ) is defined by ‖u‖H((0,T );X,Y ) = (‖u‖2L2((0,T );X) + ‖ut‖2L2((0,T );Y )) 12 Now we introduce a differential operator of order 2m L(x, t,D) = m∑ |p|,|q|=0 (−1)|p|Dp(apq(x, t)Dq), where apq are s×s matrices smooth elements of which are in ΩT , apq = (−1)|p|+|q|a?pq (a∗qp is the transportated conjugate matrix to apq).We assume there exists a constant c0 > 0 independing on t such that: ∀ξ ∈ Rn \ {0}, ∀η ∈ Cs \ {0} :∑ |p|=|q|=m apq(x, t)ξ pξqηη ≥ c0|ξ|2m|η|2, (2.1) where ξp = ξp11 ...ξ pn n , ξ q = ξq11 ...ξ qn n . We introduce also a system of boundary operators Bj = Bj(x, t,D) = ∑ |p|≤µj bj,p(x, t)D p, j = 1, ..., m, on S. Suppose that bj,p(x, t) are s× s matrices smooth elements of which are in ΩT and ordBj = µj ≤ m− 1 for j = 1, ..., χ, m ≤ ordBj = µj ≤ 2m− 1 for j = χ+ 1, ..., m. Assume that coefficients ofBj are independent of t if ordBj < m and {Bj(x, t,D)}mj=1 is a normal system on S for all t ∈ [0, T ], i.e., the two following conditions are satisfied: 83 Nguyen Thi Lien (i) µj 6= µk for j 6= k, (ii) Boj (x, t, ν(x)) 6= 0 for all (x, t) ∈ ST , j = 1, ..., m. Here ν(x) is the unit outer normal to S at point x and Boj (x, t,D) is the principal part of Bj(x, t,D), Boj = B o j (x, t,D) = ∑ |p|=µj bj,p(x, t)D p, j = 1, ..., m. Furthermore, Boj (0, t, ν(x)) 6= 0 for all x ∈ S closed enough to the origin 0. We set HmB (Ω) = { u ∈ Hm(Ω) : Bju = 0 on S for j = 1, .., χ } with the same norm in Hm(Ω) and B(t, u, v) = m∑ |p|,|q|=0 ∫ Ω apqD quDpvdx, t ∈ [0, T ]. Doing the same in the Garding’s inequality, we have: Lemma 2.1. Suppose that coefficients of the operator L(x,t,D) satisfy condition (2.1). Then there exists two constant µ0 and λ0 such that (−1)mB(t, u, u) ≥ µ0‖u(x, t)‖2Hm(Ω) − λ0‖u(x, t)‖2L2(Ω) for all functions u(x, t) ∈ H((0, T );HmB (Ω), H−mB (Ω)). Thus, set u = eiλ0tv if necessary, we can suppose that (−1)mB(t, u, u) ≥ µ0‖u(x, t)‖2Hm(Ω) (2.2) for all u ∈ HmB (Ω) and t ∈ [0, T ]. Applying Green’s formula, we can assume that it can be choose boundary operators Φj on ST , j = 1, ..., m such that B(t, u, v) = ∫ Ω Luv + χ∑ j=1 ∫ S ΦjBjvds+ m∑ j=χ+1 ∫ S BjΦjvds. (∗) Denote H−mB (Ω) the dual space to H m B (Ω). We write 〈 ., . 〉 to stand for the pair- ing between HmB (Ω) and H −m B (Ω), and (., .) to define the inner product in L2(Ω). We then have the continuous imbeddings HmB (Ω) ↪→ L2(Ω) ↪→ H−mB (Ω) with the equation 〈 f, v 〉 = (f, v) for f ∈ L2(Ω) ⊂ H−mB (Ω), v ∈ HmB (Ω). 84 On the solvability of the initial boundary value problem for Schro¨dinger systems... We study the following problem in the cylinder ΩT : (−1)m−1iL(x, t,D)u− ut = f(x, t) in ΩT , (2.3) Bju = 0, on ST , j = 1, ..., m, (2.4) u |t=0= φ, on Ω, (2.5) where f ∈ L2((0, T );HmB (Ω)) and φ ∈ L2(Ω) are given functions. The solution u(x, t) is searched in the generalized sense. That means u ∈ H((0, T );HmB (Ω), H−mB (Ω)) is called a generalized solution of the problem (2.3)- (2.5) if u(., 0) = φ and the equality (−1)m−1iB(t, u, v)− 〈ut, v〉 = 〈f(t), v〉 (2.6) holds for a.e. t ∈ (0, T ) and all v ∈ HmB (Ω). 3. The unique solvability of the problem Theorem 3.1. Suppose that coefficients of the operator L(x,t,D) satisfy condition (2.2). Then problem (2.3)-(2.5) has at most one generalized solution in the space generalized solution u ∈ H((0, T );HmB (Ω), H−mB (Ω)). Proof. Suppose u1(x, t), u2(x, t) are two generalized solutions of problem (2.3)-(2.5) inH((0, T );HmB (Ω), H −m B (Ω)). Denote u(x, t) = u 1(x, t)−u2(x, t). Arccording to the denifition of generalized solutions, substituting v := u into (2.6), then integrating both sides of the obtained equality with respect to t from 0 to b (b > 0), we arrive at (−1)m−1i ∫ b 0 B(t, u(., t), u(., t))dt− ∫ b 0 〈 ut, u(., t) 〉 dt = 0. Thus (−1)m ∫ b 0 B(t, u(., t), u(., t))dt− i ∫ b 0 〈 ut, u(., t) 〉 dt = 0. (3.1) Since ∫ b 0 〈 ut, u(., t) 〉 dt = ‖u(b)‖2L2(Ω) − ∫ b 0 〈 u, ut(., t) 〉 dt, we get ∫ b 0 〈 ut, u(., t) 〉 dt = 1 2 ‖u(b)‖2L2(Ω). Adding (3.1) with its complex conjugate, we discover∫ b 0 B(t, u(., t), u(., t))dt = 0 Using the inequality (2.2), we have∫ b 0 ‖u‖2Hm B (Ω)dt ≤ ∫ b 0 B(t, u(., t), u(., t))dt = 0, 85 Nguyen Thi Lien so ‖u‖2L2((0,b);HmB (Ω)) = ∫ b 0 ‖u‖2L2((0,T );HmB (Ω))dt = 0. This implies u ≡ 0 on [0, b]. Therefore, u ≡ 0 on ΩT . The proof of the uniquence of generalized solution is completed. Theorem 3.2. Suppose that f ∈ L2((0, T );H−mB (Ω)), φ ∈ L2(Ω) and the conditions of Theorem 3.1 is fulfilled. Then there exists a generalized solution in generalized solution u ∈ H((0, T );HmB (Ω), H−mB (Ω)) of the problem (2.3)-(2.5) which satisfies ‖u‖2 H((0,T );Hm B (Ω),H−m B (Ω)) ≤ C(‖φ‖2L2(Ω) + ‖f‖2L2((0,T );H−mB (Ω))), where C is a constant independent of φ, f and u. Proof. Suppose {ψk(x)}∞k=1 be a system functions in HmB (Ω), which is orthonormal in L2(Ω) and its linear closure is just H m B (Ω). We look for u N(x, t) in the form: uN(x, t) = N∑ k=1 CNk (t)ψk(x), where {CNk (t)}Nk=1 is the solution of the ordinary differ- ential system: (−1)m−1i m∑ |p|,|q|=0 ∫ Ω apqD quNDpψldx− ∫ Ω uNt ψldx = 〈 f, ψl 〉 , l = 1, ..., N (3.2) CNk (0) = Ck = (φ, ψk), k = 1, ..., N. (3.3) After multiplying both sides of (3.2) by CNl (t), taking sum with respect to l from 1 to N and integrating with respect to t from 0 to τ (τ > 0), we get (−1)m−1i τ∫ 0 B(t, uN , uN)dt− τ∫ 0 (uNt , u N)dt = τ∫ 0 〈 f, uN 〉 dt. From this equality we obtain (−1)m τ∫ 0 B(t, uN , uN)dt− 1 2 i (‖u(τ)‖2L2(Ω) − ‖u(0)‖2L2(Ω)) = i τ∫ 0 〈 f, uN 〉 dt. (3.4) Adding (3.4) with its complex conjugate, we have (−1)m−1 τ∫ 0 B(t, uN , uN)dt = Im τ∫ 0 〈 f, uN 〉 dt 1 2 (‖u(τ)‖2L2(Ω) − ‖u(0)‖2L2(Ω)) = −Re τ∫ 0 〈 f, uN 〉 dt. 86 On the solvability of the initial boundary value problem for Schro¨dinger systems... Noting that |(−1)m−1 τ∫ 0 B(t, uN , uN)dt| ≥ µ‖uN‖2L2((0,τ);HmB (Ω)) ‖uN(0)‖2L2(Ω) = ‖ N∑ k=1 (φ, ψk)ψk‖2L2(Ω) ≤ ‖φ‖2L2(Ω) and |Im τ∫ 0 〈 f, uN 〉 dt| − |Re τ∫ 0 〈 f, uN 〉 dt| ≤ 2 τ∫ 0 ‖f‖H−m B (Ω)‖uN‖HmB (Ω) ≤ ‖uN‖2L2((0,τ);HmB (Ω)) + 1  ‖f‖2 L2((0,τ);H −m B (Ω)) , So we have ‖uN‖2L2((0,τ);HmB (Ω)) ≤ C (‖f‖2 L2((0,τ);H −m B (Ω)) + ‖φ‖2L2(Ω) ) . Letting τ tend to T , we get ‖uN‖2L2((0,T );HmB (Ω)) ≤ C (‖f‖2 L2((0,T );H −m B (Ω)) + ‖φ‖2L2(Ω) ) . (3.5) Now, fix any v ∈ HmB (Ω) with ‖v‖HmB (Ω) ≤ 1 and write v = v1 + v2, where v1 ∈ span{ψl}Nl=1, (v2, ψl)L2(Ω) = 0, l = 1, ..., N. Since ‖v‖HmB (Ω) ≤ 1, ‖v1‖HmB (Ω) ≤ 1. We obtain from (3.2) that −(uNt , v1) + (−1)m−1iB(t, uN , v1) = 〈 f, v1 〉 . Thus, 〈 uNt , v 〉 = (uNt , v) = (u N t , v1) = (−1)m−1iB(t, uN , v1)− 〈 f, v1 〉 . Since ‖v1‖Hm B (Ω) ≤ 1, ‖uNt ‖H−m B (Ω) ≤ | 〈 uNt , v 〉| ≤ |B(t, uN , v1)|+ |〈f, v1〉| ≤ C(‖f‖H−m B (Ω) + ‖uN‖HmB (Ω) ) . Therefore, by (3.5), ‖uNt ‖2L2((0,T );H−mB (Ω)) ≤ C (‖f‖2 L2((0,T );H −m B (Ω)) + ‖uN‖2L2((0,T );HmB (Ω)) ) ≤ C(‖f‖2 L2((0,T );H −m B (Ω)) + ‖φ‖2L2(Ω) ) . From this inequality and (3.5) we get ‖uNt ‖2H((0,T );Hm B (Ω),H−m B (Ω)) ≤ C(‖f‖2 L2((0,T );H −m B (Ω)) + ‖φ‖2L2(Ω) ) , (3.6) 87 Nguyen Thi Lien where C is the constant independent of φ, f and N . Because {uN} is bounded in Hilbert space H((0, T );HmB (Ω), H−mB (Ω)), we can choose a subsequence weakly convergent to u(x, t) ∈ H((0, T );HmB (Ω), H−mB (Ω)). We will prove that u(x, t) is a generalized solution of problem (2.3)-(2.5). Fix a positive real number τ, τ ≤ T and a positive integer h. Take a function η ∈ L2((0, τ);H m B (Ω)) in the form η(x, t) = h∑ l=1 dl(t)ψl(x), (3.7) where dl(t) are smooth functions defined on [0, τ ]. Multiplying both sides of (3.2) with N ≥ h by dl(t), taking sum with respect to l from 1 to h and integrating with respect to t from 0 to τ , we have (−1)m−1i τ∫ 0 B(t, uN , η)dt− τ∫ 0 〈 uNt , η) 〉 dt = τ∫ 0 〈 f, η 〉 dt. Letting N tend to ∞, we have (−1)m−1i τ∫ 0 B(t, u, η)dt− τ∫ 0 (ut, η)dt = τ∫ 0 〈 f, η 〉 dt. (3.8) Since the set of functions of the form (3.7) is dense in L2((0, τ);H m B (Ω)), the equality (3.8) holds for all η ∈ L2((0, τ);HmB (Ω)). This implies (−1)m−1iB(t, u, v)− 〈ut, v〉 = 〈f(t), v〉 holds for a.e. t ∈ (0,+∞) and all v ∈ HmB (Ω).The inequality in the theorem is followed from (3.6). Now, we will prove that u(., 0) = φ. An intergration by parts from (3.8) yields (−1)m−1i τ∫ 0 B(t, u, η)dt− τ∫ 0 (u, ηt)dt+ (u(., 0), η(., 0)) = τ∫ 0 〈 f, η 〉 dt (3.9) holds for all η ∈ C1([0, τ ], HmB (Ω)) satisfying η(., τ) = 0. We have (−1)m−1i τ∫ 0 B(t, uN , η)dt− τ∫ 0 (uN , ηt)dt+ (u N(., 0), η(., 0)) = τ∫ 0 〈 f, η 〉 dt. Passing to the limit as N →∞ with noting that uN(., 0)→ φ in L2(Ω), we get (−1)m−1i τ∫ 0 B(t, u, η)dt− τ∫ 0 (u, ηt)dt+ (φ, η(., 0)) = τ∫ 0 〈 f, η 〉 dt. (3.10) 88 On the solvability of the initial boundary value problem for Schro¨dinger systems... Comparing (3.9) and (3.10),we obtain (u(., 0), η(., 0)) = (φ, η(., 0)). Since η(., 0) ∈ HmB (Ω) is arbitrary u(., 0) = φ.Theorem 3.1 is proved. 4. An example In this section we apply the previous results to the Cauchy-Dirichlet problem for the wave equation. We consider the following problem: 4u− utt = f(x, t), (x, t) ∈ ΩT , (4.1) u|t=0 = ut|t=0 = 0, x ∈ Ω, (4.2) u|ST = 0, (4.3) where 4 is the Laplace operator. By o H1(Ω) we denote the completion of o C∞(Ω) in the norm of the spaceH1(Ω). ThenH((0, T );H1B(Ω), H −1 B (Ω) = H((0, T ); o H1(Ω), o H−1(Ω)). From this fact and Theorem 3.1 and 3.2 we obtain following results. Theorem 4.1. Suppose that f ∈ L2((0, T ); o H−1(Ω)), φ ∈ L2(Ω). Then problem (4.1)-(4.3) has a unique generalized solution u in the spaceH((0, T ); o H1(Ω), o H−1(Ω)) and ‖u‖2 H((0,T ); o H1(Ω), o H−1(Ω)) ≤ C ( ‖φ‖2L2(Ω) + ‖f‖2 L2((0,T ); o H−1(Ω)) ) , where C is a constant independent of φ, f and u. REFERENCES [1] R. A. Adams, 1975. Sobolev Spaces, Academic Press. [2] Nguyen Manh Hung and Cung The Anh, 2010. Asymtotic expansions of solutions of the first initial boundary value problem for the Schrodinger system near conical points of the boundary. Differentsial’nye Uravneniya, Vol. 46, No. 2, pp. 285-289. [3] Nguyen Manh Hung and Nguyen Thi Kim Son, 2009.On the regularity of solution of the second initial boundary value problem for Schrodinger systems in domains with conical points. Taiwanese journal of mathematics. Vol. 13, No. 6, pp. 1885- 1907. [4] Nguyen Manh Hung and Nguyen Thanh Anh, 2008. Regularity of solutions of initial-boundary value problems for parabolic equations in domains with conical points. Journal of Differential Equations, Vol. 245, Issue 7, pp. 1801-1818. [5] Solonnikov V. A., 1983. On the solvability of classical initial-boundary value problem for the heat equation in a dihedral angle. Zap. Nachn. Sem. Leningr. Otd. Math. Inst., 127, pp. 7-48. [6] Fichera. G., 1972. Existense theorems in elasticity. Springer, New York-Berlin. 89