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

Previous Article
The maximal regularity operator on tent spaces
CPAA Home
This Issue
Next Article
Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems

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

Abstract
Related Papers

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 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.

[1]	Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure and Applied Analysis, 2015, 14 (5) : 2043-2067. doi: 10.3934/cpaa.2015.14.2043
[2]	Kevin Arfi, Anna Rozanova-Pierrat. Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets. Discrete and Continuous Dynamical Systems - S, 2019, 12 (1) : 1-26. doi: 10.3934/dcdss.2019001
[3]	Jussi Behrndt, A. F. M. ter Elst. The Dirichlet-to-Neumann map for Schrödinger operators with complex potentials. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 661-671. doi: 10.3934/dcdss.2017033
[4]	Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems and Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745
[5]	Victor Isakov, Jenn-Nan Wang. Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map. Inverse Problems and Imaging, 2014, 8 (4) : 1139-1150. doi: 10.3934/ipi.2014.8.1139
[6]	Oleg Yu. Imanuvilov, Masahiro Yamamoto. Stability for determination of Riemannian metrics by spectral data and Dirichlet-to-Neumann map limited on arbitrary subboundary. Inverse Problems and Imaging, 2019, 13 (6) : 1213-1258. doi: 10.3934/ipi.2019054
[7]	Mourad Bellassoued, Zouhour Rezig. Recovery of transversal metric tensor in the Schrödinger equation from the Dirichlet-to-Neumann map. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1061-1084. doi: 10.3934/dcdss.2021158
[8]	Mei Ming. Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264
[9]	Wei-Ming Ni, Xuefeng Wang. On the first positive Neumann eigenvalue. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 1-19. doi: 10.3934/dcds.2007.17.1
[10]	Leszek Gasiński, Nikolaos S. Papageorgiou. Multiple solutions for a class of nonlinear Neumann eigenvalue problems. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1491-1512. doi: 10.3934/cpaa.2014.13.1491
[11]	Isabeau Birindelli, Stefania Patrizi. A Neumann eigenvalue problem for fully nonlinear operators. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845
[12]	Robert Carlson. Dirichlet to Neumann maps for infinite quantum graphs. Networks and Heterogeneous Media, 2012, 7 (3) : 483-501. doi: 10.3934/nhm.2012.7.483
[13]	Shouchuan Hu, Nikolaos S. Papageorgiou. Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2889-2922. doi: 10.3934/cpaa.2013.12.2889
[14]	Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Neumann equations driven by a nonhomogeneous differential operator. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1055-1078. doi: 10.3934/cpaa.2011.10.1055
[15]	Manli Song, Jinggang Tan. Hardy inequalities for the fractional powers of the Grushin operator. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4699-4726. doi: 10.3934/cpaa.2020192
[16]	Rola Kiwan, Ahmad El Soufi. Where to place a spherical obstacle so as to maximize the second Dirichlet eigenvalue. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1193-1201. doi: 10.3934/cpaa.2008.7.1193
[17]	Victor Isakov. Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 631-640. doi: 10.3934/dcdss.2011.4.631
[18]	Ihsane Bikri, Ronald B. Guenther, Enrique A. Thomann. The Dirichlet to Neumann map - An application to the Stokes problem in half space. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 221-230. doi: 10.3934/dcdss.2010.3.221
[19]	Carmen Calvo-Jurado, Juan Casado-Díaz, Manuel Luna-Laynez. Parabolic problems with varying operators and Dirichlet and Neumann boundary conditions on varying sets. Conference Publications, 2007, 2007 (Special) : 181-190. doi: 10.3934/proc.2007.2007.181
[20]	Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems and Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Wolfgang Arendt Rafe Mazzeo
on Google Scholar
Wolfgang Arendt Rafe Mazzeo

[Back to Top]