Spectral and Initial-Boundary Conjugation Problems
- PDF / 507,151 Bytes
- 23 Pages / 594 x 792 pts Page_size
- 50 Downloads / 209 Views
SPECTRAL AND INITIAL-BOUNDARY CONJUGATION PROBLEMS UDC 517.95+517.98
K. A. Radomirskaya
Abstract. The approach to abstract conjugation boundary-value problems developed in [18] is applied to one-domain and two-domain spectral conjugation problems. An operator pencil with self-adjoint operator coefficients acting in a Hilbert space and depending on two parameters arises; we study it in detail. Both possible cases (where one parameter is the spectral one, while the other one is fixed) are considered; we deduce the corresponding properties of the solutions. Also, we study the initialvalue problems of mathematical physics generating conjugation problems. We obtain existence and uniqueness theorems for strong solutions valued in the correspondent Hilbert spaces.
CONTENTS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Spectral Problems Generated by Mixed Boundary-Value Problems and Conjugation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. One-domain spectral problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Spectral conjugation problems for two docked domains . . . . . . . . . . . . . . 2. On Properties of Solutions of Spectral Problems . . . . . . . . . . . . . . . . . . . . 2.1. The case of spectral parameter μ: properties of solutions . . . . . . . . . . . . . 2.2. The case of spectral parameter λ: properties of solutions . . . . . . . . . . . . . 3. Initial-Boundary Value Problems Generating Spectral Conjugation Problems . . . . 3.1. The first problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The second problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. The third problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. The fourth problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 660 . . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
661 661 663 666 666 670 673 673 675 677 678 680
Introduction In this paper, we study mixed boundary-value conjugation problems basing on the generalized Green formula for the Laplace operators. In [18], a general approach to the investigation of mixed boundary-value conjugation problems is developed and used to investigate spectral conjugation problems and to obtain the spectral problem for the corresponding operator pencil. In this paper, we investigate the dependence of properties of solutions of this pencil on parameters of the problem. In the first section, we study spectral problems for mixed boundary-value problems in one domain and in two docked domains. We find that, in both cases, the original spectral problems of mathematical physics are reduced to the investigation of the same operator pencil with self-adjoint operator coefficients. The pencil depends on two complex parameters λ and μ such that one of them is assumed to be fi
Data Loading...
