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2008 | 35 | 4 | 481-512
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Linear independence of boundary traces of eigenfunctions of elliptic and Stokes operators and applications

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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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  • Department of Mathematics, University of Virginia, Charlottesville, VA 22903, U.S.A.
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