Eigenvalues of elliptic boundary value problems with an indefinite weight function

Jacqueline Fleckinger, Michel L. Lapidus
1986 Transactions of the American Mathematical Society  
We consider general selfadjoint elliptic eigenvalue problems (P) Au = Xr(x)u, in an open set O C R*1. Here, the operator A is positive and of order 2m and the "weight" r is a function which changes sign in O and is allowed to be discontinuous. A scalar A is said to be an eigenvalue of (P) if Au = Xru-in the variational sense-for some nonzero u satisfying the appropriate growth and boundary conditions. We determine the asymptotic behavior of the eigenvalues of (P), under suitable assumptions. In
more » ... the case when Ü is bounded, we assumed Dirichlet or Neumann boundary conditions. When Q is unbounded, we work with operators of "Schrodinger type"; if we set r± = max(±r,0), two cases appear naturally: First, if f! is of "weighted finite measure" (i.e., fc¡(r+)k/2rn < +oo or fn(r-)k/2m < +oo), we obtain an extension of the well-known Weyl asymptotic formula; secondly, if f! is of "weighted infinite measure" (i.e., /n(r+)'c/2m = +co or /n(r_)fc/2m = +oo), our results extend the de Wet-Mandl formula (which is classical for Schrodinger operators with weight r = 1). When 0 is bounded, we also give lower bounds for the eigenvalues of the Dirichlet problem for the Laplacian. Fetnassi [F1F 1, F1F 2]. Key words and phrases. Elliptic boundary value problems, indefinite weight function, asymptotic behavior of eigenvalues, lower bounds of eigenvalues, spectral theory, variational methods, operators of Schrodinger type.
doi:10.1090/s0002-9947-1986-0831201-3 fatcat:kfo4vomccvgeplwjc24rd5pqoi