In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operatorAu（x） = -△△u（x） ＋ V（x）u（x）,for all x ∈ R^n, in the Hilbert space H = L2（R^n,H1） with the operator po...In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operatorAu（x） = -△△u（x） ＋ V（x）u（x）,for all x ∈ R^n, in the Hilbert space H = L2（R^n,H1） with the operator potential V（x） ∈ C^1 （R^n, L （H1） ）, where L （H1 ） is the space of all bounded linear operators on the Hilbert space H1, while AAu is the biharmonic differential operator and△u=-∑i,j=1^n 1/√detg δ/δxi[√detgg-1（x）δu/δxj]is the Laplace-Beltrami differential operator in R^n. Here g（x） = （gij（x）） is the Riemannian matrix, while g^-1 （x） is the inverse of the matrix g（x）. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation Au = - △△u ＋ V（x） u （x） = f（x） in the Hilbert space H where f（x） ∈ H as an application of the separation approach.展开更多
The linear equilibrium theory of thermoelasticity with microtemperatures for microstretch solids is considered.The basic internal and external boundary value problems(BVPs)are formulated and uniqueness theorems are gi...The linear equilibrium theory of thermoelasticity with microtemperatures for microstretch solids is considered.The basic internal and external boundary value problems(BVPs)are formulated and uniqueness theorems are given.The single-layer and double-layer thermoelastic potentials are constructed and their basic properties are established.The integral representation of general solutions is obtained.The existence of regular solutions of the BVPs is proved by means of the potential method(boundary integral method)and the theory of singular integral equations.展开更多
文摘In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operatorAu（x） = -△△u（x） ＋ V（x）u（x）,for all x ∈ R^n, in the Hilbert space H = L2（R^n,H1） with the operator potential V（x） ∈ C^1 （R^n, L （H1） ）, where L （H1 ） is the space of all bounded linear operators on the Hilbert space H1, while AAu is the biharmonic differential operator and△u=-∑i,j=1^n 1/√detg δ/δxi[√detgg-1（x）δu/δxj]is the Laplace-Beltrami differential operator in R^n. Here g（x） = （gij（x）） is the Riemannian matrix, while g^-1 （x） is the inverse of the matrix g（x）. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation Au = - △△u ＋ V（x） u （x） = f（x） in the Hilbert space H where f（x） ∈ H as an application of the separation approach.
文摘The linear equilibrium theory of thermoelasticity with microtemperatures for microstretch solids is considered.The basic internal and external boundary value problems(BVPs)are formulated and uniqueness theorems are given.The single-layer and double-layer thermoelastic potentials are constructed and their basic properties are established.The integral representation of general solutions is obtained.The existence of regular solutions of the BVPs is proved by means of the potential method(boundary integral method)and the theory of singular integral equations.
