Eigenvalue asymptotics for Neumann Laplacian in domains with ultra-thin cusps
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1998-1999), Exposé no. 10, 6 p.

Asymptotics with sharp remainder estimates are recovered for number N(τ) 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.

Ivrii, Victor 1

1 Department of Mathematics, University of Toronto, 100, St.George Str., Toronto, Ontario M5S 3G3, CANADA
@article{SEDP_1998-1999____A10_0,
     author = {Ivrii, Victor},
     title = {Eigenvalue asymptotics for {Neumann} {Laplacian} in domains with ultra-thin cusps},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
     note = {talk:10},
     pages = {1--6},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {1998-1999},
     zbl = {1061.35504},
     mrnumber = {1721328},
     language = {fr},
     url = {http://archive.numdam.org/item/SEDP_1998-1999____A10_0/}
}
TY  - JOUR
AU  - Ivrii, Victor
TI  - Eigenvalue asymptotics for Neumann Laplacian in domains with ultra-thin cusps
JO  - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
N1  - talk:10
PY  - 1998-1999
SP  - 1
EP  - 6
PB  - Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://archive.numdam.org/item/SEDP_1998-1999____A10_0/
LA  - fr
ID  - SEDP_1998-1999____A10_0
ER  - 
%0 Journal Article
%A Ivrii, Victor
%T Eigenvalue asymptotics for Neumann Laplacian in domains with ultra-thin cusps
%J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
%Z talk:10
%D 1998-1999
%P 1-6
%I Centre de mathématiques Laurent Schwartz, École polytechnique
%U http://archive.numdam.org/item/SEDP_1998-1999____A10_0/
%G fr
%F SEDP_1998-1999____A10_0
Ivrii, Victor. Eigenvalue asymptotics for Neumann Laplacian in domains with ultra-thin cusps. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1998-1999), Exposé no. 10, 6 p. http://archive.numdam.org/item/SEDP_1998-1999____A10_0/

[Bir1] M. Sh. Birman. The Maxwell operator in domains with edges. J. Sov. Math., 37 (1987), 793–797. | Zbl

[Bir2] M. Sh. Birman. The Maxwell operator for a resonator with inward edges. Vestn. Leningr. Univ., Math., 19, no. 3 (1986), 1–8. | MR | Zbl

[BS1] M.Sh.Birman, M.Z.Solomyak. The Maxwell operator in domains with a nonsmooth boundary. Sib. Math. J., 28 (1987), 12–24. | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[DS] E.B.Davies and B.Simon. Spectral properties of Neumann Laplacian of horns. Geom. and Func. Anal., 2, (1992), pp. 105–117. | MR | Zbl

[Ivr1] V.Ivrii. Microlocal analysis and precise spectral asymptotics. Springer-Verlag, SMM, 1998. | MR | Zbl

[Ivr2] V.Ivrii. Accurate spectral asymptotics for Neumann Laplacian in domains with cusps. Applicable Analysis, 71, (to appear) | Zbl

[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 | Zbl

[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 | Zbl

[Sol1] M.Solomyak. On the negative discrete spectrum of the operator -Δ N -αV for a class of unbounded domains in d , CRM Proceedings and Lecture Notes, Centre de Recherches Mathematiques, 12, (1997), pp. 283–296. | MR | Zbl

[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 | Zbl