Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals

Pedro Freitas; Batłomiej Siudeja

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 16, Issue: 3, page 648-676
  • ISSN: 1292-8119

Abstract

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We prove some new upper and lower bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals. In particular, we improve Pólya and Szegö's [Annals of Mathematical Studies 27 (1951)] lower bound for quadrilaterals and extend Hersch's [Z. Angew. Math. Phys. 17 (1966) 457–460] upper bound for parallelograms to general quadrilaterals.

How to cite

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Freitas, Pedro, and Siudeja, Batłomiej. "Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals." ESAIM: Control, Optimisation and Calculus of Variations 16.3 (2010): 648-676. <http://eudml.org/doc/250741>.

@article{Freitas2010,
abstract = { We prove some new upper and lower bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals. In particular, we improve Pólya and Szegö's [Annals of Mathematical Studies 27 (1951)] lower bound for quadrilaterals and extend Hersch's [Z. Angew. Math. Phys. 17 (1966) 457–460] upper bound for parallelograms to general quadrilaterals. },
author = {Freitas, Pedro, Siudeja, Batłomiej},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Dirichlet eigenvalues; polygons; variational bounds},
language = {eng},
month = {7},
number = {3},
pages = {648-676},
publisher = {EDP Sciences},
title = {Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals},
url = {http://eudml.org/doc/250741},
volume = {16},
year = {2010},
}

TY - JOUR
AU - Freitas, Pedro
AU - Siudeja, Batłomiej
TI - Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/7//
PB - EDP Sciences
VL - 16
IS - 3
SP - 648
EP - 676
AB - We prove some new upper and lower bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals. In particular, we improve Pólya and Szegö's [Annals of Mathematical Studies 27 (1951)] lower bound for quadrilaterals and extend Hersch's [Z. Angew. Math. Phys. 17 (1966) 457–460] upper bound for parallelograms to general quadrilaterals.
LA - eng
KW - Dirichlet eigenvalues; polygons; variational bounds
UR - http://eudml.org/doc/250741
ER -

References

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  1. P. Antunes and P. Freitas, New bounds for the principal Dirichlet eigenvalue of planar regions. Experiment. Math.15 (2006) 333–342.  
  2. P. Antunes and P. Freitas, A numerical study of the spectral gap. J. Phys. A41 (2008) 055201.  
  3. D. Borisov and P. Freitas, Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions on thin planar domains. Ann. Inst. H. Poincaré Anal. Non Linéaire26 (2009) 547–560.  
  4. P. Freitas, Upper and lower bounds for the first Dirichlet eigenvalue of a triangle. Proc. Amer. Math. Soc.134 (2006) 2083–2089.  
  5. P. Freitas, Precise bounds and asymptotics for the first Dirichlet eigenvalue of triangles and rhombi. J. Funct. Anal.251 (2007) 376–398.  
  6. J. Hersch, Constraintes rectilignes parallèles et valeurs propres de membranes vibrantes. Z. Angew. Math. Phys.17 (1966) 457–460.  
  7. W. Hooker and M.H. Protter, Bounds for the first eigenvalue of a rhombic membrane. J. Math. Phys.39 (1960/1961) 18–34.  
  8. E. Makai, On the principal frequency of a membrane and the torsional rigidity of a beam, in Studies in mathematical analysis and related topics, Essays in honor of George Pólya, Stanford Univ. Press, Stanford (1962) 227–231.  
  9. P.J. Méndez-Hernández, Brascamp-Lieb-Luttinger inequalities for convex domains of finite inradius. Duke Math. J.113 (2002) 93–131.  
  10. G. Pólya and G. Szegö, Isoperimetric inequalities in mathematical physics, Annals of Mathematical Studies27. Princeton University Press, Princeton (1951).  
  11. M.H. Protter, A lower bound for the fundamental frequency of a convex region. Proc. Amer. Math. Soc.81 (1981) 65–70.  
  12. C.K. Qu and R. Wong, “Best possible” upper and lower bounds for the zeros of the Bessel fuction Jv(x). Trans. Amer. Math. Soc.351 (1999) 2833–2859.  
  13. B. Siudeja, Sharp bounds for eigenvalues of triangles. Michigan Math. J.55 (2007) 243–254.  
  14. B. Siudeja, Isoperimetric inequalities for eigenvalues of triangles. Ind. Univ. Math. J. (to appear).  

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