Eigenvalues of elliptic boundary value problems with an indefinite weight function

Abstract:

We consider general selfadjoint elliptic eigenvalue problems (P) \[ \mathcal {A}u = \lambda r(x)u,\] in an open set $\Omega \subset {{\mathbf {R}}^k}$. Here, the operator $\mathcal {A}$ is positive and of order $2m$ and the "weight" $r$ is a function which changes sign in $\Omega$ and is allowed to be discontinuous. A scalar $\lambda$ is said to be an eigenvalue of $({\text {P}})$ if $\mathcal {A}u = \lambda ru$—in the variational sense—for some nonzero $u$ satisfying the appropriate growth and boundary conditions. We determine the asymptotic behavior of the eigenvalues of $({\text {P}})$, under suitable assumptions. In the case when $\Omega$ is bounded, we assumed Dirichlet or Neumann boundary conditions. When $\Omega$ is unbounded, we work with operators of "Schrödinger type"; if we set $r \pm = \max ( \pm r,0)$, two cases appear naturally: First, if $\Omega$ is of "weighted finite measure" (i.e., $\int _\Omega {{{({r_ + })}^{k/2m}} < + \infty \;} {\text {or}}\;\int _\Omega {{{({r_ - })}^{k/2m}} < + \infty }$), we obtain an extension of the well-known Weyl asymptotic formula; secondly, if $\Omega$ is of "weighted infinite measure" (i.e., $\int _\Omega {{{({r_ + })}^{k/2m}} = + \infty \;{\text {or}}\;\int _\Omega {{{({r_ - })}^{k/2m}} = + \infty } }$), our results extend the de Wet-Mandl formula (which is classical for Schrödinger operators with weight $r \equiv 1$). When $\Omega$ is bounded, we also give lower bounds for the eigenvalues of the Dirichlet problem for the Laplacian.