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Séminaire Équations aux dérivées partielles (Polytechnique)

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Victor Ivrii
Eigenvalue asymptotics for Neumann Laplacian in domains with ultra-thin cusps
Séminaire Équations aux dérivées partielles (Polytechnique) (1998-1999), Exp. No. 10, 6 p.
Article PDF | Analyses MR 1721328 | Zbl 1061.35504

Résumé - Abstract

Asymptotics with sharp remainder estimates are recovered for number ${\mathbf{N}}(\tau )$ of eigenvalues of the generalized Maxwell problem and for related Laplacians which are similar to Neumann Laplacian. We consider domains with ultra-thin cusps (with $\exp (-|x| ^{m+1}$) width ; $m>0$) and recover eigenvalue asymptotics with sharp remainder estimates.

Bibliographie

[Bir1] M. Sh. Birman. The Maxwell operator in domains with edges. J. Sov. Math., 37 (1987), 793–797.  Zbl 0642.35071
[Bir2] M. Sh. Birman. The Maxwell operator for a resonator with inward edges. Vestn. Leningr. Univ., Math., 19, no. 3 (1986), 1–8.  MR 867387 |  Zbl 0621.35016
[BS1] M.Sh.Birman, M.Z.Solomyak. The Maxwell operator in domains with a nonsmooth boundary. Sib. Math. J., 28 (1987), 12–24.  MR 886850 |  Zbl 0655.35067
[BS2] M.Sh.Birman, M.Z.Solomyak. Weyl asymptotics of the spectrum of the Maxwell operator for domains with a Lipschitz boundary. Vestn. Leningr. Univ., Math., 20, no. 3 (1987), 15–21.  MR 928156 |  Zbl 0639.35062
[BS3] M.Sh.Birman, M.Z.Solomyak. $L_2$-theory of the Maxwell operator in arbitrary domains. Russ. Math. Surv., 42, no. 6 (1987), 75–96.  MR 933995 |  Zbl 0653.35075
[BS4] M.Sh.Birman, M.Z.Solomyak. The self-adjoint Maxwell operator in arbitrary domains. Leningr. Math. J., 1, no. 1 (1990), 99–115.  MR 1015335 |  Zbl 0733.35099
[DS] E.B.Davies and B.Simon. Spectral properties of Neumann Laplacian of horns. Geom. and Func. Anal., 2, (1992), pp. 105–117. Article |  MR 1143665 |  Zbl 0749.35024
[Ivr1] V.Ivrii. Microlocal analysis and precise spectral asymptotics. Springer-Verlag, SMM, 1998.  MR 1631419 |  Zbl 0906.35003
[Ivr2] V.Ivrii. Accurate spectral asymptotics for Neumann Laplacian in domains with cusps. Applicable Analysis, 71, (to appear)  Zbl 1031.35113
[IF] V.Ivrii, S. Fedorova. Dilatations and the asymptotics of the eigenvalues of spectral problems with singularities. Funct. Anal. Appl., 20, (1986), pp. 277–281".  MR 916536 |  Zbl 0628.35077
[JMS] V.Jakšić, S.Molčanov and B.Simon. Eigenvalue asymptotics of the Neumann Laplacian of regions and manifolds with cusps. J. Func. Anal., 106, (1992), pp. 59–79.  MR 1163464 |  Zbl 0783.35040
[Sol1] M.Solomyak. On the negative discrete spectrum of the operator $-\Delta _N -\alpha V$ for a class of unbounded domains in $\mathbb{R} ^d$, CRM Proceedings and Lecture Notes, Centre de Recherches Mathematiques, 12, (1997), pp. 283–296.  MR 1479254 |  Zbl 0888.35075
[Sol2] M.Solomyak. On the discrete spectrum of a class of problems involving the Neumann Laplacian in unbounded domains Advances in Mathematics, AMS (volume dedicated to 80-th birthday of S.G.Krein (P. Kuchment and V.Lin, Editors) - in press.  MR 1729937 |  Zbl 0906.35065
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