Abstract
A weighted Laplace-Beltrami operator on a Riemannian manifold M is an operator of the form
where σ is a positive, locally bounded function defined on M. We obtain upper and lower bounds on the eigenvalue counting function of H for a class of incomplete manifolds with locally bounded geometry and for certain weights σ. Our method relies upon Dirichlet-Neumann bracketing and so no smoothness assumption on the metric or on σ is needed. Our results apply, in particular, to the Dirichlet Laplacian on unbounded domains in Rn satisfying Hardy’s inequality and to certain elliptic operators with degenerate coefficients.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ancona, A. On strong barriers and an inequality of Hardy for domains inR n,J. London Math. Soc.,34, 274–290, (1986).
van den Berg, M. Dirichlet-Neumann bracketing for Horn-Shaped regionsJ. Functional Anal,104, 110–120, (1992).
van den Berg, M. On the spectral counting function for the Dirichlet Laplacian,J. Functional Anal.,107, 352–361, (1992).
Coifman, R. and Weiss, G.Analyse Harmonique Non-Commutative sur Certains Espaces Homogè nes,242, Springer-Verlag, Berlin.
Davies, E.B. Trace properties of the Dirichlet Laplacian,Math. Z.,188, 245–251, (1985).
Davies, E.B. Heat kernel bounds for second order elliptic operators on Riemannian manifolds,A. J. Math.,109, 545–570, (1987).
Davies, E.B.Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.
Davies, E.B. Heat kernel bounds, conservation of probability and the Feller property,J. Analyse Math.,58, 99–119, (1992).
Davies, E.B. and Lianantonakis, M. Heat kernel and Hardy estimates for locally Euclidean manifolds with fractal boundaries, GAFA,3, 527–563, (1993).
Donnelly, H. Spectrum of domains in Riemannian manifolds,Ill. J. Math.,31, 692–698, (1987).
Kufner, A. and Opic, B.Hardy-Type Inequalities. Longman Scientific and Technical, 1990.
Lapidus, M. Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture,Trans. Am. Math. Soc.,325, 465–529, (1991).
Levendorskii, S.Z. Non-classical spectral asymptotics,Russian Math. Surveys,43, 149–192, (1988).
Lianantonakis, M. Ultracontractive heat kernel bounds for singular second order elliptic operators,J. London Math. Soc.,47, 358–384, (1993).
Lianantonakis, M. Non-classical spectral asymptotics for weighted Laplace-Beltrami operators,Duke Math. J., (IMRN),71, 241–244, (1993).
Pang, M.M.H. L1 properties of singular elliptic operators,J. London Math. Soc.,38, 525–543, (1988).
Simon, B. Non-classical eigenvalue asymptotics,J. Functional Anal,53, 84–98, (1983).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lianantonakis, M. On the eigenvalue counting function for weighted Laplace-Beltrami operators. J Geom Anal 10, 299–322 (2000). https://doi.org/10.1007/BF02921827
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02921827