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

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

Wolfgang Arendt 1, and  Rafe Mazzeo 2,

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.
Keywords: Eigenvalue inequalities, Dirichlet-to-Neumann operator.
Mathematics Subject Classification: Primary: 35P15; Secondary: 47D06.
Citation: Wolfgang Arendt, Rafe Mazzeo. Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup. Communications on Pure & 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.  Google Scholar

[2]

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

[3]

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

[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 Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

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.  Google Scholar

[2]

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

[3]

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

[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 Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

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