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

Joachim von Below () and Gilles François ()
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Joachim von Below: Université du Littoral Côte d’Opale, LMPA J. Liouville, FR 2956 CNRS
Gilles François: Université du Littoral Côte d’Opale, LMPA J. Liouville, FR 2956 CNRS

A chapter in Functional Analysis and Evolution Equations, 2007, pp 67-81 from Springer

Abstract: 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 ⊂ ℝ N 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 k 2 with leading asymptotic coefficient equal to the Weyl constant. For N ≥ 2 the positive eigenvalues grow like k 2/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 (search for similar items in EconPapers)
Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-7643-7794-6_5

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DOI: 10.1007/978-3-7643-7794-6_5

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