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

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In this paper, we consider the initial boundary value problem for Schrodinger systems in the cylinders with base containing the conical point. The existence and the uniqueness of the generanized solution of this problem are given.

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Nội dung Text: 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 SCHRODINGER 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 Schr¨ odinger 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 Schr¨odinger 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 Schr¨odinger 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 S n−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 D = ∂ |α| /∂xα1 1 ...∂xαnn denotes the generalized derivative of order α with respect α 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: D α u = (D α u1 , ..., D α us ); utj = ∂ k u/∂tk = (∂ j u1 /∂tj , .., ∂ j us /∂tj ). Let us introduce some functional spaces used in this paper (see [1]): 82
On the solvability of the initial boundary value problem for Schr¨ odinger systems... We use H l (Ω) be the space of functions defined in Ω with the norm l Z X 1 kukH l(Ω) = |D α u|2dx 2 |α|=0 Ω 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 Z T 1 kukL2 ((0,T );X) = ku(t)k2X dt 2 0 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 1 kukH((0,T );X,Y ) = kuk2L2 ((0,T );X) + kut k2L2 ((0,T );Y ) 2 Now we introduce a differential operator of order 2m m X (−1)|p| D p apq (x, t)D q , L(x, t, D) = |p|,|q|=0 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} : X apq (x, t)ξ p ξ q ηη ≥ c0 |ξ|2m |η|2, (2.1) |p|=|q|=m where ξ p = ξ1p1 ...ξnpn , ξ q = ξ1q1 ...ξnqn . We introduce also a system of boundary operators X Bj = Bj (x, t, D) = bj,p (x, t)D p , j = 1, ..., m, |p|≤µj 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 of Bj are independent of t if ordBj < m and {Bj (x, t, D)}m j=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) Bjo (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 Bjo (x, t, D) is the principal part of Bj (x, t, D), X Bjo = Bjo (x, t, D) = bj,p (x, t)D p , j = 1, ..., m. |p|=µj Furthermore, Bjo (0, t, ν(x)) 6= 0 for all x ∈ S closed enough to the origin 0. We set HBm (Ω) = u ∈ H m (Ω) : Bj u = 0 on S for j = 1, .., χ with the same norm in H m (Ω) and m X Z B(t, u, v) = apq D q uD p vdx, t ∈ [0, T ]. |p|,|q|=0 Ω 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)m B(t, u, u) ≥ µ0 ku(x, t)k2H m (Ω) − λ0 ku(x, t)k2L2 (Ω) for all functions u(x, t) ∈ H((0, T ); HBm(Ω), HB−m (Ω)). Thus, set u = eiλ0 t v if necessary, we can suppose that (−1)m B(t, u, u) ≥ µ0 ku(x, t)k2H m (Ω) (2.2) for all u ∈ HBm (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 Z χ Z m Z X X B(t, u, v) = Luv + Φj Bj vds + Bj Φj vds. (∗) Ω j=1 S j=χ+1 S Denote HB−m (Ω) the dual space to HBm (Ω). We write ., . to stand for the pair- ing between HBm (Ω) and HB−m (Ω), and (., .) to define the inner product in L2 (Ω). We then have the continuous imbeddings HBm (Ω) ,→ L2 (Ω) ,→ HB−m (Ω) with the equation f, v = (f, v) for f ∈ L2 (Ω) ⊂ HB−m (Ω), v ∈ HBm (Ω). 84
On the solvability of the initial boundary value problem for Schr¨ odinger systems... We study the following problem in the cylinder ΩT : (−1)m−1 iL(x, t, D)u − ut = f (x, t) in ΩT , (2.3) Bj u = 0, on ST , j = 1, ..., m, (2.4) u |t=0 = φ, on Ω, (2.5) where f ∈ L2 ((0, T ); HBm(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 ); HBm(Ω), HB−m (Ω)) is called a generalized solution of the problem (2.3)- (2.5) if u(., 0) = φ and the equality (−1)m−1 iB(t, u, v) − ut , v = f (t), v (2.6) holds for a.e. t ∈ (0, T ) and all v ∈ HBm (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 ); HBm(Ω), HB−m (Ω)). Proof. Suppose u1 (x, t), u2 (x, t) are two generalized solutions of problem (2.3)-(2.5) in H((0, T ); HBm(Ω), HB−m (Ω)). Denote u(x, t) = u1 (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 Z b Z b m−1 (−1) i B(t, u(., t), u(., t))dt − ut , u(., t) dt = 0. 0 0 Thus Z b Z b m (−1) B(t, u(., t), u(., t))dt − i ut , u(., t) dt = 0. (3.1) 0 0 Since Z b Z b ut , u(., t) dt = ku(b)k2L2 (Ω) − u, ut (., t) dt, 0 0 we get b 1 Z ut , u(., t) dt = ku(b)k2L2 (Ω) . 0 2 Adding (3.1) with its complex conjugate, we discover Z b B(t, u(., t), u(., t))dt = 0 0 Using the inequality (2.2), we have Z b Z b 2 kukHBm(Ω) dt ≤ B(t, u(., t), u(., t))dt = 0, 0 0 85
Nguyen Thi Lien so Z b kuk2L2 ((0,b);HBm (Ω)) = kuk2L2((0,T );HBm (Ω)) dt = 0. 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 ); HB−m(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 ); HBm(Ω), HB−m (Ω)) of the problem (2.3)-(2.5) which satisfies kuk2H((0,T );H m (Ω),H −m (Ω)) ≤ C(kφk2L2 (Ω) + kf k2L2 ((0,T );H −m (Ω)) ), B B B where C is a constant independent of φ, f and u. Proof. Suppose {ψk (x)}∞ m k=1 be a system functions in HB (Ω), which is orthonormal in L2 (Ω) and its linear closure is just HBm (Ω). We look for uN (x, t) in the form: N uN (x, t) = CkN (t)ψk (x), where {CkN (t)}N P k=1 is the solution of the ordinary differ- k=1 ential system: m X Z Z m−1 q N uN (−1) i apq D u D p ψl dx − t ψl dx = f, ψl , l = 1, ..., N (3.2) |p|,|q|=0 Ω Ω CkN (0) = Ck = (φ, ψk ), k = 1, ..., N. (3.3) After multiplying both sides of (3.2) by ClN (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 Zτ Zτ Zτ (−1)m−1 i B(t, uN , uN )dt − (uN N f, uN dt. t , u )dt = 0 0 0 From this equality we obtain Zτ Zτ m 1 N N B(t, u , u )dt − i ku(τ )k2L2 (Ω) − ku(0)k2L2(Ω) = i (−1) f, uN dt. (3.4) 2 0 0 Adding (3.4) with its complex conjugate, we have Zτ Zτ (−1)m−1 B(t, uN , uN )dt = Im f, uN dt 0 0 Zτ 1 ku(τ )k2L2 (Ω) − ku(0)k2L2 (Ω) = −Re f, uN dt. 2 0 86
On the solvability of the initial boundary value problem for Schr¨ odinger systems... Noting that Zτ |(−1)m−1 B(t, uN , uN )dt| ≥ µkuN k2L2 ((0,τ );HBm (Ω)) 0 N X N ku (0)k2L2 (Ω) =k (φ, ψk )ψk k2L2 (Ω) ≤ kφk2L2 (Ω) k=1 and Zτ Zτ Zτ kf kH −m (Ω) kuN kHBm (Ω) |Im f, uN dt| − |Re f, uN dt| ≤ 2 B 0 0 0 1 ≤ kuN k2L2 ((0,τ );HBm (Ω)) + kf k2L2 ((0,τ );H −m (Ω)) , B So we have kuN k2L2 ((0,τ );HBm (Ω)) ≤ C kf k2L2 ((0,τ );H −m (Ω)) + kφk2L2 (Ω) . B Letting τ tend to T , we get kuN k2L2 ((0,T );HBm (Ω)) ≤ C kf k2L2 ((0,T );H −m (Ω)) + kφk2L2 (Ω) . (3.5) B Now, fix any v ∈ HBm (Ω) with kvkHBm (Ω) ≤ 1 and write v = v1 + v2 , where v1 ∈ span{ψl }N m (Ω) ≤ 1, kv1 kH m (Ω) ≤ 1. l=1 , (v2 , ψl )L2 (Ω) = 0, l = 1, ..., N. Since kvkHB B We obtain from (3.2) that −(uN m−1 iB(t, uN , v1 ) = f, v1 . t , v1 ) + (−1) Thus, N ut , v = (uN N m−1 t , v) = (ut , v1 ) = (−1) iB(t, uN , v1 ) − f, v1 . Since kv1 kHBm (Ω) ≤ 1, kuN N N t kH −m (Ω) ≤ | ut , v | ≤ |B(t, u , v1 )| + | f, v1 | B ≤ C kf kH −m (Ω) + kuN kHBm (Ω) . B Therefore, by (3.5), kuN 2 2 N 2 t kL2 ((0,T );H −m (Ω)) ≤ C kf kL2 ((0,T );H −m (Ω)) + ku kL2 ((0,T );HB m (Ω)) B B ≤ C kf k2L2 ((0,T );H −m (Ω)) + kφk2L2 (Ω) . B From this inequality and (3.5) we get kuN 2 2 2 t kH((0,T );H m (Ω),H −m (Ω)) ≤ C kf kL2 ((0,T );H −m (Ω)) + kφkL2 (Ω) , (3.6) B B B 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 ); HBm(Ω), HB−m (Ω)), we can choose a subsequence weakly convergent to u(x, t) ∈ H((0, T ); HBm(Ω), HB−m (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 ); HBm(Ω)) in the form h X η(x, t) = dl (t)ψl (x), (3.7) l=1 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 Zτ Zτ Zτ m−1 N N (−1) i B(t, u , η)dt − ut , η) dt = f, η dt. 0 0 0 Letting N tend to ∞, we have Zτ Zτ Zτ m−1 (−1) i B(t, u, η)dt − (ut , η)dt = f, η dt. (3.8) 0 0 0 Since the set of functions of the form (3.7) is dense in L2 ((0, τ ); HBm(Ω)), the equality (3.8) holds for all η ∈ L2 ((0, τ ); HBm (Ω)). This implies (−1)m−1 iB(t, u, v) − ut , v = f (t), v holds for a.e. t ∈ (0, +∞) and all v ∈ HBm (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 Zτ Zτ Zτ (−1)m−1 i B(t, u, η)dt − (u, ηt )dt + (u(., 0), η(., 0)) = f, η dt (3.9) 0 0 0 holds for all η ∈ C 1 ([0, τ ], HBm (Ω)) satisfying η(., τ ) = 0. We have Zτ Zτ Zτ m−1 N N N (−1) i B(t, u , η)dt − (u , ηt )dt + (u (., 0), η(., 0)) = f, η dt. 0 0 0 Passing to the limit as N → ∞ with noting that uN (., 0) → φ in L2 (Ω), we get Zτ Zτ Zτ m−1 (−1) i B(t, u, η)dt − (u, ηt )dt + (φ, η(., 0)) = f, η dt. (3.10) 0 0 0 88
On the solvability of the initial boundary value problem for Schr¨ odinger systems... Comparing (3.9) and (3.10),we obtain (u(., 0), η(., 0)) = (φ, η(., 0)). Since η(., 0) ∈ HBm (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) o o where 4 is the Laplace operator. By H 1 (Ω) we denote the completion of C ∞ (Ω) in o o the norm of the space H 1 (Ω). Then H((0, T ); HB1 (Ω), HB−1 (Ω) = H((0, T ); H 1 (Ω), H −1 (Ω)). From this fact and Theorem 3.1 and 3.2 we obtain following results. o Theorem 4.1. Suppose that f ∈ L2 ((0, T ); H −1 (Ω)), φ ∈ L2 (Ω). Then problem o o (4.1)-(4.3) has a unique generalized solution u in the space H((0, T ); H 1 (Ω), H −1 (Ω)) and 2 2 2 kuk o o ≤ C kφkL2 (Ω) + kf k o , H((0,T ); H 1 (Ω), H −1 (Ω)) L2 ((0,T ); 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