On the regularity of solution of the boundary value problem without initial condition for Schrodinger systems in conical domains

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The purpose of this paper is to establish the existence, the uniqueness and regularity with respect to time variable of solution of the boundary value problems without initial condition for Schrodinger systems in cylinders with base containing conical points.

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Nội dung Text: On the regularity of solution of the boundary value problem without initial condition for Schrodinger systems in conical domains

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2011, Vol. 56, No. 7, pp. 14-17 ON THE REGULARITY OF SOLUTION OF THE BOUNDARY VALUE PROBLEM WITHOUT INITIAL CONDITION ¨ FOR SCHRODINGER SYSTEMS IN CONICAL DOMAINS Nguyen Manh Hung and Nguyen Thi Lien(∗) Hanoi National University of Education (∗) E-mail: Lienhnue@gmail.com Abstract. The purpose of this paper is to establish the existence, the uniqueness and regularity with respect to time variable of solution of the boundary value problems without initial condition for Schr¨ odinger systems in cylinders with base containing conical points. Keywords: Regularity, generalized solution, problems without initial con- dition, conical domain. 1. Introduction Schr¨odinger systems plays important role in quantum physics. The unique solvability and the regularity of the general boundary value problems for Schr¨odinger systems in domains with conical point are completed in [2, 3]. In this paper, we are concerned with the existence, the uniqueness and the regularity with respect to time variable of solution of the boundary value problems without initial condition for Schr¨odinger systems in cylinders with base containing conical points. 2. Statement 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 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 . Set Ω∞ = Ω × R, S∞ = S × R. We use notations and functional spaces in [3]. Now we introduce a differential operator of order 2m m X L(x, t, D) = (−1)|p| D p apq (x, t)D q , |p|,|q|=0 where apq are s × s matrices whose smooth elements in Ω∞ , apq = a∗pq (a∗qp is the transportated conjugate matrix to apq ). We introduce also a system of boundary 14
On the regularity of solution of the boundary value problem... 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 whose smooth elements in Ω∞ . All properties of Bj are given in [3]. It is known that there is a test function χ(t) which is equal to 1 on [1, +∞), is equal to 0 on (−∞, 0] and assumes value in [0, 1] on otherwises (see [2, Theorem 5.5] for more details). Let h ∈ [−T, 0] be an integer. Setting f h (x, t) = χ(t−h+1)f (x, t) then h f if t ≥ h f = 0 if t < h − 1 Moreover, if f ∈ L2 (R; HB−m (Ω)), then f h ∈ L2 (R; HB−m (Ω)) and kfh k2L2 (R;H −m (Ω)) ≤ kf k2L2 (R;H −m (Ω)) , (2.1) B B Fixing f ∈ L2 (R; HB−m(Ω)), we consider the following problem in the cylinder Ω∞ : (−1)m−1 iL(x, t, D)u − ut = f h (x, t) in Ω∞ , (2.2) Bj u = 0 on S∞ , j = 1, ..., m, (2.3) u |t=h−1 = 0 on Ω. (2.4) It is easily seen that the problem (2.2) - (2.4) has an unique generalized so- lution called uh . This solution belongs to H(R; HBm(Ω), HB−m (Ω)) (see [3]). Clearly, uh (x, t) = 0 for all t < h − 1 and the following estimate is satisfied: kuh k2H(R;H m (Ω),H −m (Ω)) ≤ Ckf h k2L2 (R;H −m (Ω)) . B B B From (2.1), we get kuh k2H(R;H m (Ω),H −m (Ω)) ≤ Ckf k2L2 (R;H −m (Ω)) . (2.5) B B B Let k be an integer less than h, denote uk a generalized solution of the problem (2.2) - (2.4) when we replaced h by k. Putting v = uh − uk , f ∗ = f h − f k , so v is the generalized solution of the following problem: (−1)m−1 iL(x, t, D)v − vt = f ∗ (x, t) in Ω∞ , (2.6) Bj v = 0 on S∞ , j = 1, ..., m, (2.7) v |t=k−1 = uh (k − 1) − uk (k − 1) = 0 on Ω. (2.8) We have kvk2H(R;H m (Ω),H −m (Ω)) ≤ Ckfh − fk k2L2 (R;H −m (Ω)) , (2.9) B B B 15
Nguyen Manh Hung and Nguyen Thi Lien and Zh kf h − f k k2L2 (R;H −m (Ω)) = kf h − f k k2H −m (Ω) dt. (2.10) B B k−1 Rh Since f h , f k ∈ L2 (R; HB−m (Ω)), lim kf h − f k k2H −m (Ω) dt = 0 when h, k → −∞. B k−1 From (2.9) and (2.10) it follows that {uh }−∞ is a Cauchy sequence. So uh is con- h=0 vergent to u in H(R; HBm(Ω), HB−m (Ω)). We call u is the generalized solution of the following problem without initial condition: (−1)m−1 iL(x, t, D)u − ut = f in Ω∞ , (2.11) Bj u = 0 on S∞ , j = 1, ..., m. (2.12) Using (2.5), letting h tend to −∞, we get: kuk2H(R;H m (Ω),H −m (Ω)) ≤ Ckf k2L2 (R;H −m (Ω)) . (2.13) B B B 3. The regularity with respect to time variable Assume that s is a natural number and f is a function in L2 (Ω∞ ) such that ftl ∈ L2 (Ω∞ ) for l = 0, ..., s. We have : Xl l fthl (x, t) = χtr (t − h + 1)ftl−r (x, t). (3.1) r=0 r From (3.1) and because of the fact that ftl ∈ L2 (Ω∞ ) so fthl ∈ L2 (Ω∞ ) for all l = 0, ..., s, and l X kfthl k2L2 (Ω∞ ) ≤ C kftk k2L2 (Ω∞ ) , k=0 where constant C is independent of f, h, l, t. Now we have the following theorem: Theorem 3.1. Let s be a nonnegative integer. Suppose that f in L2 (Ω∞ ) satisfying ftl ∈ L2 (Ω∞ ) for l = 0, ..., s. Then the generalized solution u ∈ H(R; HBm(Ω), HB−m (Ω)) of problem (2.11) - (2.12) satisfies utl ∈ H(R; HBm(Ω), L2 (Ω)) for l = 0, ..., s, (3.2) and s s X X kutl k2H(R;HBm (Ω),L2 (Ω)) ≤ C kftl k2L2 (Ω∞ ) , (3.3) l=0 l=0 where C is a constant independent of u, f, φ. 16
On the regularity of solution of the boundary value problem... Proof. Fixing k < h < 0, v = uh − uk is the generalized solution of the problem (2.6) - (2.8). Since ftl ∈ L2 (Ω∞ ), l = 0, ..., s, ft∗l is in L2 (Ω∞ ) too. In [4] it is known that the problem (2.6) - (2.8) has a unique solution v satisfying vtl ∈ H(R; HBm(Ω), L2 (Ω)) for l = 0, ..., s, and s s X X kvtl k2H(R;HBm (Ω),L2 (Ω)) ≤C kft∗l k2L2 (Ω∞ ) . l=0 l=0 So {uhtl }−∞ h=0 are a Cauchy sequences for all l = 0, ..., s. Thus, for any l = 0, ..., s fixed, the sequence {uhtl } is convergent. Then there exists utl and utl = limh→−∞ uhtl , for l = 0, ..., s. Because uhtl ∈ H(R; HBm(Ω), L2 (Ω)) for l = 0, ..., s, we obtain utl ∈ H(R; HBm(Ω), L2 (Ω)) for l = 0, ..., s. And s s X X kuhtl k2H(R;HBm (Ω),L2 (Ω)) ≤C kfthl k2L2 (Ω∞ ) , l=0 l=0 s X s X kuhtl k2H(R;HBm (Ω),L2 (Ω)) ≤C kftl k2L2 (Ω∞ ) . l=0 l=0 Let h tend to −∞, we have (3.3). The proof of this theorem is completed. Acknowledgement. This work was supported by the National Foundation for Science and Technology Development (NAFOSTED:101.01.58.09), Vietnam. REFERENCES [1] R. A. Adams, 1975. Sobolev Spaces. Academic Press. [2] Micheal Renardy, Robert C. Rogers, 2004. An Introduction to Partial Differential Equations. Springer. [3] Nguyen Thi Lien, 2010. On the solvability of the initial boundary value problem for Schr¨odinger systems in conical domains. Journal of Science of HNUE, Vol. 55, No. 6, pp. 82-89. [4] Nguyen Thi Lien, 2011. On the regularity of solution of the initial boundary value problem for Schr¨odinger systems in conical domains. Journal of Science of HNUE, Vol. 56, No. 3, pp. 3-12. 17