## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

### Spectral asymptotics for the Laplacian under an eigenvalue dependent boundary condition

Gilles François and Joachim von Below

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 5 December 2005

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1133793338

**Digital Object Identifier**

doi:10.36045/bbms/1133793338

**Mathematical Reviews number (MathSciNet)**

MR2205994

**Zentralblatt MATH identifier**

1132.35423

**Subjects**

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

**Keywords**

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

#### Citation

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. doi:10.36045/bbms/1133793338. https://projecteuclid.org/euclid.bbms/1133793338