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December 2005 Spectral asymptotics for the Laplacian under an eigenvalue dependent boundary condition
Gilles François, Joachim von Below
Bull. Belg. Math. Soc. Simon Stevin 12(4): 505-519 (December 2005). DOI: 10.36045/bbms/1133793338


This paper deals with a spectral problem for the Laplacian stemming from a parabolic problem in a bounded domain under a dynamical boundary condition. As a distinctive feature the eigenvalue parameter appears here also in the boundary condition: $$ \begin{cases} \,-\Delta u=\lambda u&\text{ in }\Omega,\\ \,\partial_\nu u=\lambda\sigma u&\text{ on }\partial\Omega. \end{cases} $$ By variational techniques the resulting eigenvalue sequence can be compared with the spectra under Dirichlet or Neumann boundary conditions and with the spectrum of the Steklov problem in order to get upper bounds for the spectral growth. For continuous positive $\sigma$, the growth order is determined and upper and lower bounds for the leading asymptotic coefficient are obtained. Moreover, the exact asymptotic behavior of the eigenvalue sequence is determined in the one--dimensional case.


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Gilles François. Joachim von Below. "Spectral asymptotics for the Laplacian under an eigenvalue dependent boundary condition." Bull. Belg. Math. Soc. Simon Stevin 12 (4) 505 - 519, December 2005.


Published: December 2005
First available in Project Euclid: 5 December 2005

zbMATH: 1132.35423
MathSciNet: MR2205994
Digital Object Identifier: 10.36045/bbms/1133793338

Primary: 35P15, 35P20
Secondary: 35J05, 35J25, 35K20, 47A75

Rights: Copyright © 2005 The Belgian Mathematical Society


Vol.12 • No. 4 • December 2005
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