Spectral asymptotics stemming from parabolic equations under dynamical boundary conditions

Abstract

This paper deals with a spectral problem for a second-order elliptic operator A stemming from a parabolic problem under a dynamical boundary condition. The discrete character and the convergence to infinity of the eigenvalue sequence of the problem $\[\cases{Au=\lambda u& \mbox{in }\varOmega,\cr \noalign{\vskip3pt}\curpartial_{\nu_{A}}u=\lambda\sigma u& \mbox{on }\curpartial\varOmega,}\]$ are shown. By means of min–max formulae, a comparison of the eigenvalue sequence with the spectra of the Dirichlet and Neumann problem is obtained and yields an upper bound for λ_k. On the other hand, comparing with the Steklov problem leads to a lower bound. In the two-dimensional case, this yields the exact growth order of the eigenvalue sequence and leads to inequalities about the constant of the leading term in its asymptotic behavior. For higher dimensions, the same arguments hold for a sufficiently smooth domain, while for a Lipschitz boundary, lower and upper bounds for the growth exponent are obtained. Finally, the variational characterization of the eigenvalues yields an upper bound for the number of nodal domains of the k-th eigenfunction. The one-dimensional case is also discussed.