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

Victor Ivrii[1]

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

Séminaire Équations aux dérivées partielles (1998-1999)

  • Volume: 1998-1999, page 1-6

Abstract

top
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.

How to cite

top

Ivrii, Victor. "Eigenvalue asymptotics for Neumann Laplacian in domains with ultra-thin cusps." Séminaire Équations aux dérivées partielles 1998-1999 (1998-1999): 1-6. <http://eudml.org/doc/10960>.

@article{Ivrii1998-1999,
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 ; $m&gt;0$) and recover eigenvalue asymptotics with sharp remainder estimates.},
affiliation = {Department of Mathematics, University of Toronto, 100, St.George Str., Toronto, Ontario M5S 3G3, CANADA},
author = {Ivrii, Victor},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {domains with cusps; Maxwell operator; Laplacians; eigenvalue asymptotics with sharp remainder estimates},
language = {fre},
pages = {1-6},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Eigenvalue asymptotics for Neumann Laplacian in domains with ultra-thin cusps},
url = {http://eudml.org/doc/10960},
volume = {1998-1999},
year = {1998-1999},
}

TY - JOUR
AU - Ivrii, Victor
TI - Eigenvalue asymptotics for Neumann Laplacian in domains with ultra-thin cusps
JO - Séminaire Équations aux dérivées partielles
PY - 1998-1999
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 1998-1999
SP - 1
EP - 6
AB - 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 ; $m&gt;0$) and recover eigenvalue asymptotics with sharp remainder estimates.
LA - fre
KW - domains with cusps; Maxwell operator; Laplacians; eigenvalue asymptotics with sharp remainder estimates
UR - http://eudml.org/doc/10960
ER -

References

top
  1. M. Sh. Birman. The Maxwell operator in domains with edges. J. Sov. Math., 37 (1987), 793–797. Zbl0642.35071
  2. M. Sh. Birman. The Maxwell operator for a resonator with inward edges. Vestn. Leningr. Univ., Math., 19, no. 3 (1986), 1–8. Zbl0621.35016MR867387
  3. M.Sh.Birman, M.Z.Solomyak. The Maxwell operator in domains with a nonsmooth boundary. Sib. Math. J., 28 (1987), 12–24. Zbl0655.35067MR886850
  4. 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. Zbl0639.35062MR928156
  5. 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. Zbl0653.35075MR933995
  6. M.Sh.Birman, M.Z.Solomyak. The self-adjoint Maxwell operator in arbitrary domains. Leningr. Math. J., 1, no. 1 (1990), 99–115. Zbl0733.35099MR1015335
  7. E.B.Davies and B.Simon. Spectral properties of Neumann Laplacian of horns. Geom. and Func. Anal., 2, (1992), pp. 105–117. Zbl0749.35024MR1143665
  8. V.Ivrii. Microlocal analysis and precise spectral asymptotics. Springer-Verlag, SMM, 1998. Zbl0906.35003MR1631419
  9. V.Ivrii. Accurate spectral asymptotics for Neumann Laplacian in domains with cusps. Applicable Analysis, 71, (to appear) Zbl1031.35113
  10. V.Ivrii, S. Fedorova. Dilatations and the asymptotics of the eigenvalues of spectral problems with singularities. Funct. Anal. Appl., 20, (1986), pp. 277–281". Zbl0628.35077MR916536
  11. 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. Zbl0783.35040MR1163464
  12. 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. Zbl0888.35075MR1479254
  13. 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. Zbl0906.35065MR1729937

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.