Nonlocal Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities

Journal of Differential Equations

Volumn 247, Issue 10, 2009, Pages 2871-2896

  Gesztesy, Fritz ^a   Mitrea, Marius ^a  

^a UNIVERSITY OF MISSOURI   (United States)

Author keywords

Eigenvalue inequalities; Lipschitz domains; Nonlocal Robin Laplacians; Spectral analysis

Indexed keywords

EID: 70249086895     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2009.07.007     Document Type: Article

References (54)

1. Adams R.A., and Fournier J.J.F. Sobolev Spaces. second ed. (2003), Academic Press
2. W. Arendt, R. Mazzeo, Spectral properties of the Dirichlet-to-Neumann operator on Lipschitz domains, preprint, Ulmer Seminare über Funktionalanalysis und Differentialgleichungen, Heft 12, 2007
3. Arendt W., and Warma M. Dirichlet and Neumann boundary conditions: What is in between?. J. Evol. Equ. 3 (2003) 119-135
4. Arendt W., and Warma M. The Laplacian with Robin boundary conditions on arbitrary domains. Potential Anal. 19 (2003) 341-363
5. Ashbaugh M.S., and Levine H.A. Inequalities for the Dirichlet and Neumann eigenvalues of the Laplacian for domains on spheres. Journées "Équations aux Dérivées Partielles", Exp. No. I. Saint-Jean-de-Monts, 1997 (1997), École Polytech., Palaiseau 15 pp
6. Auchmuty G. Steklov eigenproblems and the representation of solutions of elliptic boundary value problems. Numer. Funct. Anal. Optim. 25 (2004) 321-348
7. s (∂ Ω). SIAM J. Math. Anal. 38 (2006) 894-905
8. Aviles P. Symmetry theorems related to Pompeiu's problem. Amer. J. Math. 108 (1986) 1023-1036
9. Benguria R., Levitin M., and Parnovski L. Fourier transform, null variety, and Laplacian's eigenvalues. preprint (2008). arXiv:0801.1617 J. Funct. Anal., doi:10.1016/j.jfa.2009.06.022, in press
10. Berry M.V., and Dennis M.R. Boundary-condition-varying circle billiards and gratings: The Dirichlet singularity. J. Phys. A 41 (2008) 135203
11. Courant R., and Hilbert D. Methods of Mathematical Physics. Vol. I. Wiley Classics Lib. (1989), Wiley, New York
12. Dancer E.N., and Daners D. Domain perturbation for elliptic equations subject to Robin boundary conditions. J. Differential Equations 138 (1997) 86-132
13. Daners D. Robin boundary value problems on arbitrary domains. Trans. Amer. Math. Soc. 352 (2000) 4207-4236
14. Daners D. A Faber-Krahn inequality for Robin problems in any space dimension. Math. Ann. 335 (2006) 767-785
15. Daners D., and Kennedy J. Uniqueness in the Faber-Krahn inequality for Robin problems. SIAM J. Math. Anal. 39 (2007) 1191-1207
16. Dautray R., and Lions J.-L. Mathematical Analysis and Numerical Methods for Science and Technology, vol. 2, Functional and Variational Methods (2000), Springer, Berlin
17. Dávila J., Kowalczyk M., and Montenegro M. Critical points of the regular part of the harmonic Green function with Robin boundary condition. J. Funct. Anal. 255 (2008) 1057-1101
18. Edmunds D.E., and Evans W.D. Spectral Theory and Differential Operators (1989), Clarendon Press, Oxford
19. Faris W.G. Self-Adjoint Operators. Lecture Notes in Math. vol. 433 (1975), Springer, Berlin
20. Filonov N. On an inequality between Dirichlet and Neumann eigenvalues for the Laplace operator. St. Petersburg Math. J. 16 (2004) 413-416
21. Friedlander L. Some inequalities between Dirichlet and Neumann eigenvalues. Arch. Ration. Mech. Anal. 116 (1991) 153-160
22. Friedlander L. Remarks on Dirichlet and Neumann eigenvalues. Amer. J. Math. 117 (1995) 257-262
23. Friedlander L. An inequality between Dirichlet and Neumann eigenvalues in a centrally symmetric domain. Proc. Amer. Math. Soc. 129 (2000) 2057-2060
24. Gesztesy F., Latushkin Y., Mitrea M., and Zinchenko M. Nonselfadjoint operators, infinite determinants, and some applications. Russ. J. Math. Phys. 12 (2005) 443-471
25. Gesztesy F., and Mitrea M. Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains. In: Mitrea D., and Mitrea M. (Eds). Perspectives in Partial Differential Equations, Harmonic Analysis and Applications. Proc. Sympos. Pure Math. vol. 79 (2008), Amer. Math. Soc., Providence, RI 105-173
26. Gesztesy F., and Mitrea M. Robin-to-Robin maps and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains. In: Adamyan V., Berezansky Yu., Gohberg I., Gorbachuk M., Gorbachuk V., Kochubei A., Langer H., and Popov G. (Eds). Modern Analysis and Applications. The Mark Krein Centenary Conference. Oper. Theory Adv. Appl. vol. 191 (2009), Birkhäuser, Basel 81-113
27. Gesztesy F., Mitrea M., and Zinchenko M. Variations on a theme of Jost and Pais. J. Funct. Anal. 253 (2007) 399-448
28. Giorgi T., and Smits R. Monotonicity results for the principal eigenvalue of the generalized Robin problem. Illinois J. Math. 49 (2005) 1133-1143
29. Giorgi T., and Smits R. Bounds and monotonicity for the generalized Robin problem. Z. Angew. Math. Phys. 59 (2008) 600-618
30. Grisvard P. Elliptic Problems in Nonsmooth Domains (1985), Pitman, Boston
31. Hansson A.M. An inequality between Dirichlet and Neumann eigenvalues of the Heisenberg Laplacian. Comm. Partial Differential Equations 33 (2008) 2157-2163
32. Hsu Y.-J., and Wang T.-H. Inequalities between Dirichlet and Neumann eigenvalues for domains in spheres. Taiwanese J. Math. 5 (2001) 755-766
33. Jerison D., and Kenig C. The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130 (1995) 161-219
34. Kato T. Perturbation Theory for Linear Operators. corr. printing of the 2nd ed. (1980), Springer, Berlin
35. Kennedy J. An isoperimetric inequality for the second eigenvalue of the Laplacian with Robin boundary conditions. Proc. Amer. Math. Soc. 137 (2009) 627-633
36. 2 u + λ u = 0 and its application. J. Math. Phys. 31 (1952) 45-54
37. Lanzani L., and Shen Z. On the Robin boundary condition for Laplace's equation in Lipschitz domains. Comm. Partial Differential Equations 29 (2004) 91-109
38. Levine H.A. Some remarks on inequalities between Dirichlet and Neumann eigenvalues. In: Schaefer P.W. (Ed). Maximum Principles and Eigenvalue Problems in Partial Differential Equations (1988), Longman Group, UK
39. Levine H.A., and Weinberger H.F. Inequalities between Dirichlet and Neumann eigenvalues. Arch. Ration. Mech. Anal. 94 (1986) 193-208
40. Levitin M., and Parnovski L. On the principal eigenvalue of a Robin problem with a large parameter. Math. Nachr. 281 (2008) 272-281
41. Lions J.L. Espaces d'interpolation et domaines de puissances fractionnaires d'opérateurs. J. Math. Soc. Japan 14 (1962) 233-241
42. Maz'ja V.G. Sobolev Spaces (1985), Springer, Berlin
43. Mazzeo R. Remarks on a paper of Friedlander concerning inequalities between Neumann and Dirichlet eigenvalues. Int. Math. Res. Not. 4 (1991) 41-48
44. McLean W. Strongly Elliptic Systems and Boundary Integral Equations (2000), Cambridge Univ. Press, Cambridge
45. Megginson R.E. An Introduction to Banach Space Theory. Grad. Texts in Math. vol. 183 (1998), Springer, New York
46. Payne L.E. Inequalities for eigenvalues of membranes and plates. J. Ration. Mech. Anal. 4 (1955) 517-529
47. Pólya G. Remarks on the foregoing paper. J. Math. Phys. 31 (1952) 55-57
48. Reed M., and Simon B. Methods of Modern Mathematical Physics. IV: Analysis of Operators (1978), Academic Press, New York
49. Safarov Y. On the comparison of the Dirichlet and Neumann counting functions. In: Suslina T., and Yafaev D. (Eds). Spectral Theory of Differential Operators: M.Sh. Birman 80th Anniversary Collection. Amer. Math. Soc. Transl. Ser. 2 vol. 225 (2008), Amer. Math. Soc., Providence, RI 191-204
50. Simon B. Classical boundary conditions as a tool in quantum physics. Adv. Math. 30 (1978) 268-281
51. Triebel H. Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers. Rev. Mat. Complut. 15 (2002) 475-524
52. Warma M. The Robin and Wentzell-Robin Laplacians on Lipschitz domains. Semigroup Forum 73 (2006) 10-30
53. Weidmann J. Linear Operators in Hilbert Spaces. Grad. Texts in Math. vol. 68 (1980), Springer, New York
54. Wloka J. Partial Differential Equations (1987), Cambridge Univ. Press, Cambridge