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Bounds for the first Dirichlet eigenvalue of trianglesand quadrilaterals

Published online by Cambridge University Press:  02 July 2009

Pedro Freitas  and
Batłomiej Siudeja
Pedro Freitas
Affiliation:
Department of Mathematics, Faculdade de Motricidade Humana (TU Lisbon) and Group of Mathematical Physics of the University of Lisbon, Complexo Interdisciplinar, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal. freitas@cii.fc.ul.pt
Batłomiej Siudeja
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. siudeja@illinois.edu
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Abstract

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

Keywords

Dirichlet eigenvaluespolygonsvariational bounds
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 3 , July 2010 , pp. 648 - 676
Copyright
© EDP Sciences, SMAI, 2009

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References

Antunes, P. and Freitas, P., New bounds for the principal Dirichlet eigenvalue of planar regions. Experiment. Math. 15 (2006) 333–342. CrossRef
Antunes, P. and Freitas, P., A numerical study of the spectral gap. J. Phys. A 41 (2008) 055201. CrossRef
Borisov, D. and Freitas, P., Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions on thin planar domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009) 547–560. CrossRef
Freitas, P., Upper and lower bounds for the first Dirichlet eigenvalue of a triangle. Proc. Amer. Math. Soc. 134 (2006) 2083–2089. CrossRef
Freitas, P., Precise bounds and asymptotics for the first Dirichlet eigenvalue of triangles and rhombi. J. Funct. Anal. 251 (2007) 376–398. CrossRef
Hersch, J., Constraintes rectilignes parallèles et valeurs propres de membranes vibrantes. Z. Angew. Math. Phys. 17 (1966) 457–460. CrossRef
W. Hooker and M.H. Protter, Bounds for the first eigenvalue of a rhombic membrane. J. Math. Phys. 39 (1960/1961) 18–34.
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.
Méndez-Hernández, P.J., Brascamp-Lieb-Luttinger inequalities for convex domains of finite inradius. Duke Math. J. 113 (2002) 93–131. CrossRef
G. Pólya and G. Szegö, Isoperimetric inequalities in mathematical physics, Annals of Mathematical Studies 27. Princeton University Press, Princeton (1951).
Protter, M.H., A lower bound for the fundamental frequency of a convex region. Proc. Amer. Math. Soc. 81 (1981) 65–70. CrossRef
Qu, C.K. and Wong, R., “Best possible” upper and lower bounds for the zeros of the Bessel fuction J v (x). Trans. Amer. Math. Soc. 351 (1999) 2833–2859. CrossRef
Siudeja, B., Sharp bounds for eigenvalues of triangles. Michigan Math. J. 55 (2007) 243–254. CrossRef
B. Siudeja, Isoperimetric inequalities for eigenvalues of triangles. Ind. Univ. Math. J. (to appear).