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

The purpose of this paper is to establish the regularity with respect to time variable of solution of the initial boundary value problems for Schrodinger systems in the cylinders with base containing the conical point. | JOURNAL OF SCIENCE OF HNUE Natural Sci. 2011 Vol. 56 No. 3 pp. 3-12 ON THE REGULARITY OF SOLUTION 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. The purpose of this paper is to establish the regularity with respect to time variable of solution of the initial boundary value problems for Schr odinger systems in the cylinders with base containing the conical point. Keywords Regularity generalized solution conical domain. 1. Introduction The first and second initial boundary value problem for Schr odinger in conical domains were researched in 2 3 . The unique solvability of the general boundary value problems for Schr odinger systems in domains with conical point is completed in 4 . In this paper we are concerned with the regularity with respect to time vari- ables of solutions of the problem in 4 . This paper includes following sections In the first section we define the prob- lem. The regularity with respect to time variable is dealt with in sections 2. Finally in section 3 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 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 . We use notations and the functional spaces in 4 . 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 3 Nguyen Thi Lien where apq are s s matrices whose smooth elements in ΩT apq a pq a qp is the transportated conjugate matrix to apq . 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 whose smooth elements in ΩT and ordBj µj m 1 for j 1 . χ m ordBj