November  2012, 11(6): 2201-2212. doi: 10.3934/cpaa.2012.11.2201

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

1. 

Abteilung Angewante Analysis, Universität Ulm, 89069 Ulm

2. 

Department of Mathematics, Stanford University, Stanford, CA 94305, United States

Received  February 2011 Revised  February 2011 Published  April 2012

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.
Citation: Wolfgang Arendt, Rafe Mazzeo. Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2201-2212. doi: 10.3934/cpaa.2012.11.2201
References:
[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.

show all references

References:
[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.

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