Linear independence of boundary traces of eigenfunctions of elliptic and Stokes operators and applications

Roberto Triggiani

Applicationes Mathematicae, Tome 35 (2008), p. 481-512 / Harvested from The Polish Digital Mathematics Library

Résumé

This paper is divided into two parts and focuses on the linear independence of boundary traces of eigenfunctions of boundary value problems. Part I deals with second-order elliptic operators, and Part II with Stokes (and Oseen) operators. Part I: Let $λ_i$ be an eigenvalue of a second-order elliptic operator defined on an open, sufficiently smooth, bounded domain Ω in ℝⁿ, with Neumann homogeneous boundary conditions on Γ = tial Ω. Let ${φ_{ij}}^{ℓ_i}_{j=1}$ be the corresponding linearly independent (normalized) eigenfunctions in L₂(Ω), so that $ℓ_i$ is the geometric multiplicity of $λ_i$. We prove that the Dirichlet boundary traces ${φ_{ij}|_{Γ₁}}^{ℓ_i}_{j=1}$ are linearly independent in L₂(Γ₁). Here Γ₁ is an arbitrary open, connected portion of Γ, of positive surface measure. The same conclusion holds true if the setting Neumann B.C., Dirichlet boundary traces is replaced by the setting Dirichlet B.C., Neumann boundary traces. These results are motivated by boundary feedback stabilization problems for parabolic equations [L-T.2]. Part II: The same problem is posed for the Stokes operator with motivation coming from the boundary stabilization problems in [B-L-T.1]- [B-L-T.3] (with tangential boundary control), and [R] (with just boundary control), where we take Γ₁ = Γ. The aforementioned property of boundary traces of eigenfunctions critically hinges on a unique continuation result from the boundary of corresponding over-determined problems. This is well known in the case of second-order elliptic operators of Part I; but needs to be established in the case of Stokes operators. A few proofs are given here.

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Publié le : 2008-01-01

Zbl 1152.35420

EUDML-ID : urn:eudml:doc:280029