# American Institute of Mathematical Sciences

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 & 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. 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. [3] W. Arendt and R. Mazzeo, Spectral properties of the Dirichlet-to-Neumann operator on Lipschitz domains,, Ulmer Seminare, Heft 12 (2007), 28. [4] W. Arendt and T. ter Elst, The Dirichlet to Neumann operator on rough domains,, J. Differential Equations, 251 (2011), 2100. [5] W. Arendt and M. Warma, Dirichlet and Neumann boundary conditions: What is between?, J. Evolution Equ., 3 (2003), 119. doi: 10.1007/s000280300005. [6] E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge University Press 1990., (1990). [7] H. Emamirad and I. Laadnani, An approximating family for the Dirichlet-to-Neumann semigroup,, Adv. Diff. Equ., 11 (2006), 241. [8] N. Filinov, On an inequality between Dirichlet and Neumann eigenvalues for the Laplace operator,, St. Petersburg Math. J., 16 (2005), 413. 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. 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. 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. [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, (1999). [14] A. Henrot, "Extremum Problems for Eigenvalues of Elliptic Operators,", Birkh\, (2006). [15] A. Henrot and M. Pierre, "Variation et Optimisation de Formes,", Springer, (2005). [16] T. Kato, "Perturbation Theory,", Springer, (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., 150 (2001). [18] M. Mitrea and M. Taylor, Boundary layer methods for Lipschitz domains in Riemannian manifolds,, Journal of Functional Anal., 163 (1999), 181. 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. 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, (1967). [22] E. M. Ouhabaz, "Analysis of the Heat Equation on Domains,", London Mathematical Society Monographs {\bf 31}, 31 (2005). [23] J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains,, J. Funct. Anal., 18 (1975), 27. [24] Y. Safarov, On the comparison of the Dirichlet and Neumann counting functions,, In, (2008), 191.

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##### References:
 [1] W. Arendt and C. Batty, Domination and ergodicity for positive semigroups,, Proc. Amer. Math. Soc., 114 (1992), 743. 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. [3] W. Arendt and R. Mazzeo, Spectral properties of the Dirichlet-to-Neumann operator on Lipschitz domains,, Ulmer Seminare, Heft 12 (2007), 28. [4] W. Arendt and T. ter Elst, The Dirichlet to Neumann operator on rough domains,, J. Differential Equations, 251 (2011), 2100. [5] W. Arendt and M. Warma, Dirichlet and Neumann boundary conditions: What is between?, J. Evolution Equ., 3 (2003), 119. doi: 10.1007/s000280300005. [6] E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge University Press 1990., (1990). [7] H. Emamirad and I. Laadnani, An approximating family for the Dirichlet-to-Neumann semigroup,, Adv. Diff. Equ., 11 (2006), 241. [8] N. Filinov, On an inequality between Dirichlet and Neumann eigenvalues for the Laplace operator,, St. Petersburg Math. J., 16 (2005), 413. 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. 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. 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. [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, (1999). [14] A. Henrot, "Extremum Problems for Eigenvalues of Elliptic Operators,", Birkh\, (2006). [15] A. Henrot and M. Pierre, "Variation et Optimisation de Formes,", Springer, (2005). [16] T. Kato, "Perturbation Theory,", Springer, (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., 150 (2001). [18] M. Mitrea and M. Taylor, Boundary layer methods for Lipschitz domains in Riemannian manifolds,, Journal of Functional Anal., 163 (1999), 181. 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. 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, (1967). [22] E. M. Ouhabaz, "Analysis of the Heat Equation on Domains,", London Mathematical Society Monographs {\bf 31}, 31 (2005). [23] J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains,, J. Funct. Anal., 18 (1975), 27. [24] Y. Safarov, On the comparison of the Dirichlet and Neumann counting functions,, In, (2008), 191.
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