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. | 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@ 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 .
