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

Roberto Triggiani¹

¹ Department of Mathematics University of Virginia Charlottesville, VA 22903, U.S.A.

Applicationes Mathematicae, Tome 35 (2008) no. 4, pp. 481-512.

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 $\lambda_i$ 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.

Détail

DOI : 10.4064/am35-4-6

Keywords: paper divided parts focuses linear independence boundary traces eigenfunctions boundary value problems part deals second order elliptic operators part stokes oseen operators part lambda eigenvalue second order elliptic operator defined sufficiently smooth bounded domain mit omega mathbb neumann homogeneous boundary conditions mit gamma partial mit omega varphi ell corresponding linearly independent normalized eigenfunctions mit omega ell geometric multiplicity lambda prove dirichlet boundary traces varphi mit gamma ell linearly independent mit gamma here mit gamma arbitrary connected portion mit gamma positive surface measure conclusion holds setting neumann dirichlet boundary traces replaced setting dirichlet neumann boundary traces these results motivated boundary feedback stabilization problems parabolic equations l t part problem posed stokes operator motivation coming boundary stabilization problems b l t b l t tangential boundary control just boundary control where mit gamma mit gamma aforementioned property boundary traces eigenfunctions critically hinges unique continuation result boundary corresponding over determined problems known second order elliptic operators part needs established stokes operators few proofs given here