Abstract
We study the properties of generalized solutions in unbounded domains and the asymptotic behavior of solutions of elliptic boundary value problems at infinity. Moreover, we study the unique solvability of the mixed Dirichlet--Farwig biharmonic problem in the exterior of a compact set under the assumption that generalized solutions of these problems has a bounded Dirichlet integral with weight $|x|^a$. Admitting different boundary conditions, we used the variation principle and depending on the value of the parameter $a$, we obtained uniqueness (non-uniqueness) theorems of the problem or present exact formulas for the dimension of the space of solutions.
Highlights
Let Ω be an unbounded domain in Rn, n ≥ 2, Ω = Rn \ G with the boundary ∂Ω ∈ C2, where G is a bounded connected domain in Rn, 0 ∈ G, Ω = Ω ∪ ∂Ω is the closure of Ω, x = (x1, . . . , xn) ∈ Rn and |x| = x21 + · · · + x2n
The proof of Theorem 3.3 is based on Lemma 2.2 about the asymptotic expansion of the solution of the biharmonic equation and the Hardy type inequalities for unbounded domains
In case (iv), we need to determine the number of linearly independent solutions of the biharmonic equation (1), the degree of which not exceed the fixed number
Summary
In Ω we consider the following problems for the biharmonic equation The behavior of solutions of the Dirichlet problem for the biharmonic equation as |x| → ∞ In various classes of unbounded domains with finite weighted Dirichlet (energy) integral, one of the author [14]– [24] studied 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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