Elliptic problems involving an indefinite weight

Abstract:

We consider a selfadjoint elliptic eigenvalue problem, which is derived formally from a variational problem, of the form $Lu = \lambda \omega (x)u$ in $\Omega$, ${B_j}u = 0$ on $\Gamma$, $j = 1, \ldots ,m$, where $L$ is a linear elliptic operator of order $2m$ defined in a bounded open set $\Omega \subset {{\mathbf {R}}^n}\quad (n \geq 2)$ with boundary $\Gamma$, the ${B_j}$ are linear differential operators defined on $\Gamma$, and $\omega$ is a real-valued function assuming both positive and negative values. For our problem we prove the completeness of the eigenvectors and associated vectors in two function spaces which arise naturally in such an indefinite problem. We also establish some results concerning the eigenvalues of the problem which complement the known results and investigate the structure of the principal subspaces.