Spectral asymptotics for the Laplacian under an eigenvalue dependent boundary condition

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.