\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup

Abstract Related Papers Cited by
  • If $\Omega$ is any compact Lipschitz domain, possibly in a Riemannian manifold, with boundary $\Gamma = \partial \Omega$, the Dirichlet-to-Neumann operator $\mathcal{D}_\lambda$ is defined on $L^2(\Gamma)$ for any real $\lambda$. We prove a close relationship between the eigenvalues of $\mathcal{D}_\lambda$ and those of the Robin Laplacian $\Delta_\mu$, i.e. the Laplacian with Robin boundary conditions $\partial_\nu u =\mu u$. This is used to give another proof of the Friedlander inequalities between Neumann and Dirichlet eigenvalues, $\lambda^N_{k+1} \leq \lambda^D_k$, $k \in N$, and to sharpen the inequality to be strict, whenever $\Omega$ is a Lipschitz domain in $R^d$. We give new counterexamples to these inequalities in the general Riemannian setting. Finally, we prove that the semigroup generated by $-\mathcal{D}_\lambda$, for $\lambda$ sufficiently small or negative, is irreducible.
    Mathematics Subject Classification: Primary: 35P15; Secondary: 47D06.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    W. Arendt and C. Batty, Domination and ergodicity for positive semigroups, Proc. Amer. Math. Soc., 114 (1992), 743-747.doi: 10.1090/S0002-9939-1992-1072082-3.

    [2]

    W. Arendt and C. Batty, Exponential stability of diffusion equations with absorption, Diff. Int. Equ., 6 (1993), 1009-1024.

    [3]

    W. Arendt and R. Mazzeo, Spectral properties of the Dirichlet-to-Neumann operator on Lipschitz domains, Ulmer Seminare, Heft 12 (2007), 28-38.

    [4]

    W. Arendt and T. ter Elst, The Dirichlet to Neumann operator on rough domains, J. Differential Equations, 251 (2011), 2100-2124. arXiv:1005.0875

    [5]

    W. Arendt and M. Warma, Dirichlet and Neumann boundary conditions: What is between? J. Evolution Equ., 3 (2003), 119-136.doi: 10.1007/s000280300005.

    [6]

    E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge University Press 1990.

    [7]

    H. Emamirad and I. Laadnani, An approximating family for the Dirichlet-to-Neumann semigroup, Adv. Diff. Equ., 11 (2006), 241-257.

    [8]

    N. Filinov, On an inequality between Dirichlet and Neumann eigenvalues for the Laplace operator, St. Petersburg Math. J., 16 (2005), 413-416.doi: 10.1090/S1061-0022-05-00857-5.

    [9]

    L. Friedlander, Some inequalities between Dirichlet and Neumann eigenvalues, Arch. Rational Mech. Anal., 116 (1991), 153-160.doi: 10.1007/BF00375590.

    [10]

    F. Gesztesy and M. Mitrea, Nonlocal Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities, J. Diff. Eq., 247 (2009), 2871-2896.doi: 10.1016/j.jde.2009.07.007.

    [11]

    J. P. Grégoire, J. C. Nédélec and J. Planchard, A method of finding the eigenvalues and eigenfunctions of self-adjoint elliptic operators, Comp. Methods in Appl. Mech. and Eng., 8 (1976), 201-214.

    [12]

    E. Hebey, "Sobolev Spaces on Riemannian Manifolds," Springer, LN 1635 (1993).

    [13]

    E. Hebey, "Nonlinear Analysis on Manifolds Sobolev Spaces and Inequalities," Courant Lecture Notes in Mathematics 5, New York, 1999.

    [14]

    A. Henrot, "Extremum Problems for Eigenvalues of Elliptic Operators," Birkhäuser, Basel, 2006.

    [15]

    A. Henrot and M. Pierre, "Variation et Optimisation de Formes," Springer, Berlin, 2005.

    [16]

    T. Kato, "Perturbation Theory," Springer, Berlin, 1966.

    [17]

    D. Mitrea, M. Mitrea and M. Taylor, "Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds," Memoires of Amer. Math. Soc., 713, Vol. 150 (2001).

    [18]

    M. Mitrea and M. Taylor, Boundary layer methods for Lipschitz domains in Riemannian manifolds, Journal of Functional Anal., 163 (1999), 181-251.doi: 10.1006/jfan.1998.3383.

    [19]

    R. Mazzeo, Remarks on a paper of L. Friedlander concerning inequalities between Neumann and Dirichlet eigenvalues, Internat. Math. Res. Notices, 4 (1991), 41-48.doi: 10.1155/S1073792891000065.

    [20]

    R. Nagel ed., "One-parameter Semigroups of Positive Operators," Springer LN, 1184 (1986).

    [21]

    J. Necaš, "Les Méthodes Directes en Théorie des Equations Elliptiques," Masson, Paris, 1967.

    [22]

    E. M. Ouhabaz, "Analysis of the Heat Equation on Domains," London Mathematical Society Monographs 31, Princeton University Press, 2005.

    [23]

    J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains, J. Funct. Anal., 18 (1975), 27-59.

    [24]

    Y. Safarov, On the comparison of the Dirichlet and Neumann counting functions, In "Spectral Theory of Differential Operators: M.Sh. Birman 80th Anniversary Collection" (T. Suslina, D. Yafaev eds.), Amer. Math. Soc. Transl. Ser. 2, vol. 225, Providence, RI (2008), 191-204.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(297) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return