Upper and lower bounds for the first Dirichlet eigenvalue of a triangle
Author:
Pedro Freitas
Journal:
Proc. Amer. Math. Soc. 134 (2006), 20832089
MSC (2000):
Primary 35P15; Secondary 35J05
Published electronically:
January 6, 2006
MathSciNet review:
2215778
Fulltext PDF Free Access
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Additional Information
Abstract: We prove some new upper and lower bounds for the first Dirichlet eigenvalue of a triangle in terms of the lengths of its sides.
 [AF]
P. Antunes and P. Freitas, New bounds for the principal Dirichlet eigenvalue of planar regions, preprint.
 [BW]
Robert
Brooks and Peter
Waksman, The first eigenvalue of a scalene
triangle, Proc. Amer. Math. Soc.
100 (1987), no. 1,
175–182. MR
883424 (88f:58147), http://dx.doi.org/10.1090/S00029939198708834241
 [CD]
PeiKun
Chang and Dennis
DeTurck, On hearing the shape of a
triangle, Proc. Amer. Math. Soc.
105 (1989), no. 4,
1033–1038. MR 953738
(89h:58194), http://dx.doi.org/10.1090/S00029939198909537387
 [Ga]
M.
Gaudin, Vers le spectre du triangle?, J. Physique
48 (1987), no. 10, 1633–1650 (French, with
English summary). MR 923667
(89e:58113)
 [Go]
Thomas
Gorin, Generic spectral properties of right triangle
billiards, J. Phys. A 34 (2001), no. 40,
8281–8295. MR 1873184
(2002j:82006), http://dx.doi.org/10.1088/03054470/34/40/306
 [M]
E.
Makai, On the principal frequency of a membrane and the torsional
rigidity of a beam, Studies in mathematical analysis and related
topics, Stanford Univ. Press, Stanford, Calif., 1962,
pp. 227–231. MR 0167004
(29 #4277)
 [O]
Robert
Osserman, A note on Hayman’s theorem on the bass note of a
drum, Comment. Math. Helv. 52 (1977), no. 4,
545–555. MR 0459099
(56 #17297)
 [Po]
G.
Pólya, Two more inequalities between physical and
geometrical quantities, J. Indian Math. Soc. (N.S.)
24 (1960), 413–419 (1961). MR 0133059
(24 #A2895)
 [PoSz]
G.
Pólya and G.
Szegö, Isoperimetric Inequalities in Mathematical
Physics, Annals of Mathematics Studies, no. 27, Princeton University
Press, Princeton, N. J., 1951. MR 0043486
(13,270d)
 [Pr]
M.
H. Protter, A lower bound for the fundamental
frequency of a convex region, Proc. Amer. Math.
Soc. 81 (1981), no. 1, 65–70. MR 589137
(82b:35113), http://dx.doi.org/10.1090/S00029939198105891372
 [AF]
 P. Antunes and P. Freitas, New bounds for the principal Dirichlet eigenvalue of planar regions, preprint.
 [BW]
 R. Brooks and P. Waksman, The first eigenvalue of a scalene triangle, Proc. Amer. Math. Soc. 100 (1987), 175182. MR 0883424 (88f:58147)
 [CD]
 P.K. Chang and D. Deturck, On hearing the shape of a triangle, Proc. Amer. Math. Soc. 105 (1989), 10331038. MR 0953738 (89h:58194)
 [Ga]
 M. Gaudin, Vers le spectre du triangle?, J. Physique 48 (1987), 16331650. MR 0923667 (89e:58113)
 [Go]
 T. Gorin, Generic spectral properties of right triangle billiards, J. Phys. A 34 (2001), 82818295. MR 1873184 (2002j:82006)
 [M]
 E. Makai, On the principal frequency of a membrane and the torsional rigidity of a beam, pp. 227231 in Studies in mathematical analysis and related topics, Essays in honor of George Pólya, Stanford Univ. Press, Stanford 1962. MR 0167004 (29:4277)
 [O]
 R. Osserman, A note on Hayman's theorem on the bass note of a drum, Comment. Math. Helvetici 52 (1977), 545555. MR 0459099 (56:17297)
 [Po]
 G. Pólya, Two more inequalities between physical and geometrical quantities, J. Indian Math. Soc. (N.S.) 24 (1960), 413419. MR 0133059 (24:A2895)
 [PoSz]
 G. Pólya and G. Szegö, Isoperimetric inequalities in mathematical physics, Annals of Mathematical Studies 27, Princeton University Press, Princeton, 1951. MR 0043486 (13:270d)
 [Pr]
 M. H. Protter, A lower bound for the fundamental frequency of a convex region, Proc. Amer. Math. Soc. 81 (1981), 6570. MR 0589137 (82b:35113)
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Additional Information
Pedro Freitas
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049001 Lisboa, Portugal
Address at time of publication:
Faculdade de Motricidade Humana and Mathematical Physics Group of the University of Lisbon, Complexo Interdisciplinar, Av. Prof. Gama Pinto 2, P1649003 Lisboa, Portugal
Email:
pfreitas@math.ist.utl.pt, freitas@cii.fc.ul.pt
DOI:
http://dx.doi.org/10.1090/S0002993906083390
PII:
S 00029939(06)083390
Received by editor(s):
September 9, 2004
Received by editor(s) in revised form:
February 16, 2005
Published electronically:
January 6, 2006
Communicated by:
Carmen C. Chicone
Article copyright:
© Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
