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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Eigenvalues of elliptic boundary value problems with an indefinite weight function

Authors: Jacqueline Fleckinger and Michel L. Lapidus
Journal: Trans. Amer. Math. Soc. 295 (1986), 305-324
MSC: Primary 35P20; Secondary 35J10
MathSciNet review: 831201
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Abstract: We consider general selfadjoint elliptic eigenvalue problems (P)

$\displaystyle \mathcal{A}u = \lambda r(x)u,$

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.

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Keywords: Elliptic boundary value problems, indefinite weight function, asymptotic behavior of eigenvalues, lower bounds of eigenvalues, spectral theory, variational methods, operators of Schrödinger type
Article copyright: © Copyright 1986 American Mathematical Society