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

Abstract

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 λi 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 = ∂Ω. Let {φij} j=1 be the corresponding linearly independent (normalized) eigenfunctions in L2(Ω), so that `i is the geometric multiplicity of λi. We prove that the Dirichlet boundary traces {φij |Γ1} `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. 2000 Mathematics Subject Classification: 35L20, 47, 49K20, 76N25, 76Q05, 93B29, 93C20.