Eigenvalue Asymptotics Under a Non-dissipative Eigenvalue Dependent Boundary Condition for Second-order Elliptic Operators

  • Joachim von Below
  • Gilles François

Abstract

The asymptotic behavior of the eigenvalue sequence of the eigenvalue problem
$$ - \Delta \phi + q\left( x \right)\phi = \lambda \phi $$
in a bounded Lipschitz domain D ⊂ ℝN under the eigenvalue dependent boundary condition
$$ \varphi n = \sigma \lambda \varphi $$
with a continuous function Σ is investigated in the case Σ ≢ 0, the dissipative one Σ ≥ 0 having been settled in [6]. For N = 1 the eigenvalues grow like k2 with leading asymptotic coefficient equal to the Weyl constant. For N ≥ 2 the positive eigenvalues grow like k2/N , while the negative eigenvalues grow in absolute value like |k|1/(N−1). Moreover, asymptotic bounds in dependence on the dynamical coefficient function Σ are derived, firstly in the constant case, secondly for Σ of constant sign, and finally for a function Σ changing sign.

Keywords

Laplacian eigenvalue problems eigenvalue dependent boundary conditions asymptotic behavior of eigenvalues dynamical boundary conditions for parabolic problems 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Joachim von Below
    • 1
  • Gilles François
    • 1
  1. 1.LMPA J. Liouville, FR 2956 CNRSUniversité du Littoral Côte d’OpaleCalais CedexFrance

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