Bulletin of the Belgian Mathematical Society - Simon Stevin

Spectral asymptotics for the Laplacian under an eigenvalue dependent boundary condition

Gilles François and Joachim von Below

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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.

Article information

Bull. Belg. Math. Soc. Simon Stevin Volume 12, Number 4 (2005), 505-519.

First available in Project Euclid: 5 December 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35P15: Estimation of eigenvalues, upper and lower bounds 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Secondary: 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx] 35J25: Boundary value problems for second-order elliptic equations 47A75: Eigenvalue problems [See also 47J10, 49R05] 35K20: Initial-boundary value problems for second-order parabolic equations

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


von Below, Joachim; François, Gilles. Spectral asymptotics for the Laplacian under an eigenvalue dependent boundary condition. Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 4, 505--519. https://projecteuclid.org/euclid.bbms/1133793338.

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