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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 |
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] | |
[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] | |
[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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