Báo cáo toán học: On the Smoothness of Solutions of the First Initial Boundary Value Problem for Schr¨dinger o Systems in Domains with Conical Points

Một số kết quả trên suốt các giải pháp tổng quát ban đầu đầu tiêngiá trị vấn đề biên giới mạnh mẽ Schr ¨ hệ thống dinger trong các lĩnh vực với hình nón o điểm trên ranh giới nhất định.
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Báo cáo toán học: "On the Smoothness of Solutions of the First Initial Boundary Value Problem for Schr¨dinger o Systems in Domains with Conical Points" 9LHWQDP -RXUQDOVietnam Journal of Mathematics 33:2 (2005) 135–147 RI 0$7+(0$7,&6 ‹ 9$67 On the Smoothness of Solutions of the FirstInitial Boundary Value Problem for Schr¨dinger o Systems in Domains with Conical Points Nguyen Manh Hung and Cung The Anh Department of Mathematics, Hanoi University of Education, 136 Xuan Thuy Road, Hanoi, Vietnam Received March 12, 2004 Revised March 14, 2005Abstract. Some results on the smoothness of generalized solutions of the first initialboundary value problem for strongly Schr¨dinger systems in domains with conical opoints on boundary are given.1. IntroductionBoundary value problems for Schr¨dinger equations and Schr¨dinger systems in o oa finite cylinder ΩT = Ω × (0, T ) have been studied by many authors [4,8,9].The unique solvability of the first initial boundary value problem for stronglySchr¨dinger systems in an infinite cylinder Ω∞ = Ω × (0, ∞) was given in [5]. The oaim of this paper is to establish some theorems on the smoothness of generalizedsolutions of the problem in domains with conical points on boundary. Let Ω be a bounded domain in Rn . Its boundary ∂ Ω is assumed to be aninfinitely differentiable surface everywhere, except for the coordinate origin, ina neighborhood of which Ω coincides with the cone K = x : x/|x| ∈ G , whereG is a smooth domain on the unit sphere S n−1 . We introduce some notations:ΩT = Ω × (0, T ), ST = ∂ Ω × (0, T ), Ω∞ = Ω × (0, ∞), S∞ = ∂ Ω × (0, ∞), x =(x1 , . . . , xn ) ∈ Ω, u(x, t) = (u1 (x, t), . . . , us (x, t)) is a vector complex function, s s 2|Dα u|2 = |Dα ui |2 , utj = ∂ j u1 /∂tj , . . . , ∂ j us /∂tj , |utj |2 = ∂ j ui /∂tj , i=1 i=1dx = dx1 . . . dxn , r = |x| = x2 + · · · + x2 . n 1 In this paper we use frequently the following functional spaces:136 Nguyen Manh Hung and Cung The Anh l• Hβ (Ω) - the space of all functions u(x) = (u1 (x), . . . , us (x)) which have gen-eralized derivatives Dα ui , |α| ≤ l, 1 ≤ i ≤ s, satisfying l 2 r2(β +|α|−l) |Dα u|2 dx < +∞. u = l Hβ (Ω) |α|=0 Ω• H l,k (e−γt , Ω∞ ) - the space of all functions u(x, t) which have generalized deriva- ∂ j uitives Dα ui , , |α| ≤ l, 1 ≤ j ≤ k , 1 ≤ i ≤ s, satisfying ∂tj l k 2 |Dα u|2 + |utj |2 e−2γt dxdt < +∞. u = H l,k (e−γt ,Ω∞ ) j =1 |α|=0 Ω∞In particular l 2 |Dα u|2 e−2γt dxdt. u = H l,0 (e−γt ,Ω∞ ) |α|=0Ω ∞ ◦• H l,k (e−γt , Ω∞ ) - the closure in H l,k (e−γt , Ω∞ ) of the set of all infinitely dif-ferentiable in Ω∞ functions which belong to H l,k (e−γt , Ω∞ ) and vanish nearS∞ . l,k• Hβ (e−γt , Ω∞ ) - the space of all functions u(x, t) which have generalized deriva- ∂ j uitives Dα ui , , |α| ≤ l, 1 ≤ j ≤ k , 1 ≤ i ≤ s, satisfying ∂tj l k 2 ...