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.
Keywords: Dirichlet eigenvalues, polygons, variational bounds
@article{COCV_2010__16_3_648_0,
author = {Freitas, Pedro and Siudeja, Bat{\l}omiej},
title = {Bounds for the first {Dirichlet} eigenvalue of triangles and quadrilaterals},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {648--676},
year = {2010},
publisher = {EDP Sciences},
volume = {16},
number = {3},
doi = {10.1051/cocv/2009018},
mrnumber = {2674631},
zbl = {1205.35174},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2009018/}
}
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 PY - 2010 SP - 648 EP - 676 VL - 16 IS - 3 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2009018/ DO - 10.1051/cocv/2009018 LA - en ID - COCV_2010__16_3_648_0 ER -
%0 Journal Article %A Freitas, Pedro %A Siudeja, Batłomiej %T Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 648-676 %V 16 %N 3 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2009018/ %R 10.1051/cocv/2009018 %G en %F COCV_2010__16_3_648_0
Freitas, Pedro; Siudeja, Batłomiej. Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 3, pp. 648-676. doi: 10.1051/cocv/2009018
[1] and , New bounds for the principal Dirichlet eigenvalue of planar regions. Experiment. Math. 15 (2006) 333-342. | Zbl
[2] and , A numerical study of the spectral gap. J. Phys. A 41 (2008) 055201. | Zbl
[3] and , Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions on thin planar domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009) 547-560. | Zbl | Numdam
[4] , Upper and lower bounds for the first Dirichlet eigenvalue of a triangle. Proc. Amer. Math. Soc. 134 (2006) 2083-2089. | Zbl
[5] , Precise bounds and asymptotics for the first Dirichlet eigenvalue of triangles and rhombi. J. Funct. Anal. 251 (2007) 376-398. | Zbl
[6] , Constraintes rectilignes parallèles et valeurs propres de membranes vibrantes. Z. Angew. Math. Phys. 17 (1966) 457-460. | Zbl
[7] and , Bounds for the first eigenvalue of a rhombic membrane. J. Math. Phys. 39 (1960/1961) 18-34. | Zbl
[8] , 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] , Brascamp-Lieb-Luttinger inequalities for convex domains of finite inradius. Duke Math. J. 113 (2002) 93-131. | Zbl
[10] and , Isoperimetric inequalities in mathematical physics, Annals of Mathematical Studies 27. Princeton University Press, Princeton (1951). | Zbl
[11] , A lower bound for the fundamental frequency of a convex region. Proc. Amer. Math. Soc. 81 (1981) 65-70. | Zbl
[12] and , “Best possible” upper and lower bounds for the zeros of the Bessel fuction Jv(x). Trans. Amer. Math. Soc. 351 (1999) 2833-2859. | Zbl
[13] , Sharp bounds for eigenvalues of triangles. Michigan Math. J. 55 (2007) 243-254. | Zbl
[14] , Isoperimetric inequalities for eigenvalues of triangles. Ind. Univ. Math. J. (to appear).
Cité par Sources :
