# 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:

show all references

##### References:
 [1] Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure & 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 & 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 & 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 & 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 & Imaging, 2014, 8 (4) : 1139-1150. doi: 10.3934/ipi.2014.8.1139 [6] Mei Ming. Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264 [7] Wei-Ming Ni, Xuefeng Wang. On the first positive Neumann eigenvalue. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 1-19. doi: 10.3934/dcds.2007.17.1 [8] Leszek Gasiński, Nikolaos S. Papageorgiou. Multiple solutions for a class of nonlinear Neumann eigenvalue problems. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1491-1512. doi: 10.3934/cpaa.2014.13.1491 [9] Isabeau Birindelli, Stefania Patrizi. A Neumann eigenvalue problem for fully nonlinear operators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845 [10] Robert Carlson. Dirichlet to Neumann maps for infinite quantum graphs. Networks & Heterogeneous Media, 2012, 7 (3) : 483-501. doi: 10.3934/nhm.2012.7.483 [11] Shouchuan Hu, Nikolaos S. Papageorgiou. Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2889-2922. doi: 10.3934/cpaa.2013.12.2889 [12] Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Neumann equations driven by a nonhomogeneous differential operator. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1055-1078. doi: 10.3934/cpaa.2011.10.1055 [13] Rola Kiwan, Ahmad El Soufi. Where to place a spherical obstacle so as to maximize the second Dirichlet eigenvalue. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1193-1201. doi: 10.3934/cpaa.2008.7.1193 [14] Ihsane Bikri, Ronald B. Guenther, Enrique A. Thomann. The Dirichlet to Neumann map - An application to the Stokes problem in half space. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 221-230. doi: 10.3934/dcdss.2010.3.221 [15] Victor Isakov. Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 631-640. doi: 10.3934/dcdss.2011.4.631 [16] 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 [17] Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems & Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959 [18] Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377 [19] Xing-Bin Pan. An eigenvalue variation problem of magnetic Schrödinger operator in three dimensions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 933-978. doi: 10.3934/dcds.2009.24.933 [20] Jun He, Guangjun Xu, Yanmin Liu. Some inequalities for the minimum M-eigenvalue of elasticity M-tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-11. doi: 10.3934/jimo.2019092

2018 Impact Factor: 0.925