Linear independence of boundary traces of eigenfunctions of elliptic and Stokes operators and applications
Volume 35 / 2008
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 $\lambda_i$ be an eigenvalue of a second-order elliptic operator defined on an open, sufficiently smooth, bounded domain ${\mit\Omega}$ in $\mathbb{R}^n$, with Neumann homogeneous boundary conditions on ${\mit\Gamma} = \partial {\mit\Omega}$. Let $\{\varphi_{ij}\}^{\ell_i}_{j=1}$ be the corresponding linearly independent (normalized) eigenfunctions in $L_2({\mit\Omega})$, so that $\ell_i$ is the geometric multiplicity of $\lambda_i$. We prove that the Dirichlet boundary traces $\{\varphi_{ij}|_{{\mit\Gamma}_{1}}\}^{\ell_i}_{j=1}$ are linearly independent in $L_2({\mit\Gamma}_1)$. Here ${\mit\Gamma}_1$ is an arbitrary open, connected portion of ${\mit\Gamma}$, 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 ${\mit\Gamma}_1 = {\mit\Gamma}$. 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.
