Abstract
We study properties of generalized solutions of the Dirichlet–Robin problem for an elasticity system in the exterior of a compact, as well as the asymptotic behavior of solutions of this mixed problem at infinity, with the condition that the energy integral with the weight |x|a is finite. Depending on the value of the parameter a, we have proved uniqueness (or non-uniqueness) theorems for the mixed Dirichlet–Robin problem, and also given exact formulas for the dimension of the space of solutions. The main method for studying the problem under consideration is the variational principle, which assumes the minimization of the corresponding functional in the class of admissible functions.
Highlights
Dedicated to the blessed memory of my parents who went to heaven this year.Let Ω be an unbounded domain in Rn, n ≥ 2, Ω = Rn \ G with the boundary ∂Ω ∈ C1, where G is a bounded connected domain in Rn, Ω ∪ ∂Ω = Ω is the closure of Ω, x = ( x1, . . . , xn ), | x | = x12 + · · · + xn2, u = (u1, . . . , un ).In the domain Ω, we consider the linear system of elasticity theory n Lu ≡ ( Lu)i = ∂ ∑ ∂xk j,k,h=1 ij akh ∂u j ∂xh= 0, i = 1, . . . , n. (1)
We study properties of generalized solutions of the Dirichlet–Robin problem for an elasticity system in the exterior of a compact, as well as the asymptotic behavior of solutions of this mixed problem at infinity, with the condition that the energy integral with the weight | x | a is finite
In this paper we study the properties of generalized solutions of the mixed Dirichlet–Robin problem for the elasticity system in an unbounded domain Ω with a finiteness condition of the weighted energy integral: Ea (u, Ω) ≡
Summary
Dedicated to the blessed memory of my parents who went to heaven this year. Let Ω be an unbounded domain in Rn , n ≥ 2, Ω = Rn \ G with the boundary ∂Ω ∈ C1 , where G is a bounded connected domain (or aqunion of finitely many such domains) in Rn , Ω ∪ ∂Ω = Ω is the closure of Ω, x = ( x1 , . . . , xn ), | x | = x12 + · · · + xn , u = (u1 , . . . , un ). The paper [10] uses Korn’s inequality and Hardy’s inequality to study the uniqueness and stability of generalized solutions of mixed boundary value problems for the elasticity system in an unbounded domain provided that E(u, Ω) is finite. Imposing the same constraint on the behavior of the solution at infinity in various classes of unbounded domains, the author [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37] studied the uniqueness (non–uniqueness) problem and found the dimensions of the spaces of solutions of boundary value problems for the elasticity system and the biharmonic (polyharmonic) equation.
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