Eigenvalue asymptotics for Neumann Laplacian in domains with ultra-thin cusps

Ivrii, Victor

Ivrii, Victor ¹

¹ Department of Mathematics, University of Toronto, 100, St.George Str., Toronto, Ontario M5S 3G3, CANADA

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1998-1999), Exposé no. 10, 6 p.

Résumé

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.

MR Zbl

Affiliations des auteurs :

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://www.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://www.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://www.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://www.numdam.org/item/SEDP_1998-1999____A10_0/

Bibliographie
Cité par

[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 $- Δ_{N} - α V$ for a class of unbounded domains in $ℝ^{d}$ , 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