The Dirichlet problems for the Stokes resolvent equations in bounded and exterior domains in ℝⁿ

Abstract

The Dirichlet problems for the Stokes resolvent equations are studied from the point of view of the theory of hydrodynamic potentials. Existence and uniqueness results as well as boundary integral representations of classical solutions are given for domains having compact but not connected boundaries of class C 1,α (0 < α ≤ 1) in both the bounded and exterior cases.

Asymptotic Behaviors for Eigenvalues and Eigenfunctions Associated to Stokes Operator in the Presence of Small Boundary Perturbations

“…Define, B δ (λ)ϕ(x) = (W(λ)ϕ)(Ψ −1 δ )(Ψ δ (x)), where ϕ ∈ L 2 (∂Ω) and Ψ δ given by Section 1, and W(λ) is the operator associated to hydrodynamic double layer potential [10,11,17]. Then, the following main results hold.…”

“…Let the operator A δ be defined by (10). Then, there exists a small positive number δ 0 such that the eigenvalue λ(δ) is analytic in ] − δ 0 , δ 0 [ and satisfy:…”

“…The main objective of this paper is to present a schematic way to derive high-order asymptotic expansions for both eigenvalues and eigenfunctions for the Stokes operator caused by small perturbations of the boundary. The properties of eigenvalue problems under shape deformation have been a subject of comprehensive studies [1,2,5,13] and the area continues to carry great importance [8,10,12,14,9]. A substantial portion of these investigations discusses the properties of smoothness and analyticity of eigenvalues and eigenfunctions with respect to perturbations.…”

“…From [10,12] we deduce that there exists a continuous function ϕ(t, δ), which is analytic in R 2 ×] − δ 0 , δ 0 [ and such that v δ (x) = S(λ δ )ϕ + W(λ δ )ϕ, x ∈ Ω (59) solves the eigenvalue problem (4). Moreover, the function given by…”

Integral representations for solutions of some BVPs for the Lamé system in multiply connected domains

“…Recently this approach was considered in multiply connected domains for several partial differential equations (see, e.g., [2][3][4][5][6][7]). …”