Eigenvalues of elliptic boundary value problems with an indefinite weight function
Jacqueline Fleckinger, Michel L. Lapidus · Transactions of the American Mathematical Society · 1986
We consider general selfadjoint elliptic eigenvalue problems (P) \[ A u = λ r ( x ) u , \mathcal {A}u = \lambda r(x)u, \] in an open set Ω ⊂ R k \Omega \subset {{\mathbf {R}}^k} . Here, the operator A \mathcal {A} is positive and of order 2 m 2m and the "weight" r r is a function which changes sign in Ω \Omega and is allowed to be discontinuous. A scalar λ \lambda is said to be an eigenvalue of ( P ) ({\text {P}}) if A u = λ r u \mathcal {A}u = \lambda ru —in the variational sense—for some nonzero u u satisfying the appropriate growth and boundary conditions. We determine the asymptotic behavior of the eigenvalues of ( P ) ({\text {P}}) , under suitable assumptions. In the case when Ω \Omega is bounded, we assumed Dirichlet or Neumann boundary conditi