On the Mixed Dirichlet–Steklov-Type and Steklov-Type Biharmonic Problems in Weighted Spaces

Hovik Matevossian

doi:10.3390/mca24010025

Abstract

We studied the properties of generalized solutions in unbounded domains and the asymptotic behavior of solutions of elliptic boundary value problems at infinity. Moreover, we studied the unique solvability of the mixed Dirichlet–Steklov-type and Steklov-type biharmonic problems 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 . Depending on the value of the parameter a, we obtained uniqueness (non-uniqueness) theorems of these problems 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| = x12 + · · · + xn2
By developing an approach based on the use of Hardy type inequalities [1,2,3,30], in the present note, we obtain a uniqueness criterion for a solution of the mixed Dirichlet–Steklovtype and Steklov-type problems for the biharmonic equation
The proof of Theorem 2 is based on Lemma 1 about the asymptotic expansion of the solution of the biharmonic equation and the Hardy type inequalities for unbounded domains [1,2,3]

Summary

Introduction

The behavior of solutions of the Dirichlet problem for the biharmonic equation as |x| → ∞ is considered in [4,5], where estimates for |u(x)| and |∇u(x)| as |x| → ∞ are obtained under certain geometric conditions on the domain boundary. In various classes of unbounded domains with finite weighted Dirichlet (energy) integral, one of the authors [16,17,18,19,20,21,22,23,24,25,26,27,28,29] 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.

Results

Conclusion