On the Biharmonic Problem with the Steklov-Type and Farwig Boundary Conditions
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On the Biharmonic Problem with the Steklov-Type and Farwig Boundary Conditions H. A. Matevossian1, 2* (Submitted by A. V. Lapin) 1
Federal Research Center “Computer Science and Control,” Russian Academy of Sciences, Moscow, 119333 Russia 2 Moscow Aviation Institute (National Research University), Moscow, 125993 Russia Received March 22, 2020; revised April 5, 2020; accepted April 17, 2020
Abstract—We study the unique solvability of the mixed biharmonic problem with the Steklov-type and Farwig conditions on the boundary in the exterior of a compact set under the assumption that generalized solutions of this problem has a bounded Dirichlet integral with weight |x|a . Depending on the value of the parameter a, we obtained uniqueness (non-uniqueness) theorems of this problem or present exact formulas for the dimension of the space of solutions. DOI: 10.1134/S1995080220100133 Keywords and phrases: biharmonic operator, Steklov-type and Farwig boundary conditions, Dirichlet integral, weighted spaces.
1. INTRODUCTION Let Ω be an unbounded domain in Rn , n ≥ 2, Ω = Rn \ G with the boundary ∂Ω ∈ C 2 , where G is a bounded simply connected domain (or a union of finitely many such domains) in Rn , 0 ∈ G, x = (x1 , . . . , xn ), and |x| = x21 + · · · + x2n . In Ω we consider the following mixed problem for the biharmonic equation Δ2 u = 0
(1)
with the Steklov-type condition on Γ1 and the Farwig condition on Γ2 ∂Δu ∂u ∂Δu ∂u + τ u = = 0, = = 0, ∂ν ∂ν ∂ν ∂ν Γ2 Γ1
Γ1
(2)
Γ2
where Γ1 ∪ Γ2 = ∂Ω, Γ1 ∩ Γ2 = ∅, mesn−1 Γ1 = 0, ν = (ν1 , . . . , νn ) is the outer unit normal vector to ∂Ω, τ ∈ C(∂Ω), τ ≥ 0, τ ≡ 0, and τ > 0 on a set of positive (n − 1)-dimensional measure on ∂Ω, Ω = Ω ∪ ∂Ω is the closure of Ω. Elliptic problems with parameters in the boundary conditions have been called Steklov or Steklovtype problems since their first appearance in [29]. For the biharmonic operator, these conditions were first considered in [1, 10] and [27], whose authors the isoperimetric properties of the first eigenvalue were studied. The standard elliptic regularity results are available in [4]. The monograph covers higher order linear and nonlinear elliptic boundary value problems, mainly with the biharmonic or polyharmonic operator as leading principal part. The underlying models and, in particular, the role of different boundary conditions are explained in detail. As for linear problems, after a brief summary of the existence theory and Lp and Schauder estimates, the focus is on positivity. The required kernel estimates are also presented in detail. *
E-mail: [email protected]
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MATEVOSSIAN
In [3] and [4], the spectral and positivity preserving properties for the inverse of the biharmonic operator under Steklov and Navier boundary conditions are studied. These are connected with the first Steklov eigenvalue. It is shown that the positivity preserving property is quite sensitive to the parameter involved in the boundary condition. Moreover, positivity of the Steklov boundary value problem is
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