Upper and lower bounds for eigenvalues of the Laplacian on a spherical cap
Author:
Frank E. Baginski
Journal:
Quart. Appl. Math. 48 (1990), 569-573
MSC:
Primary 35P15; Secondary 35J05, 73H05, 73K15
DOI:
https://doi.org/10.1090/qam/1074972
MathSciNet review:
MR1074972
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: In the following, we derive upper and lower bounds for eigenvalues of the Laplacian on a domain that is a spherical cap whose angular width $2{\vartheta _0}$ is less than $\pi$. While previous work of this nature seems to focus on the principle eigenvalue, our results apply to any eigenvalue when $0 < {\vartheta _0} < \pi /2$. In addition, some of our results also apply to spherical caps for which $0 < {\vartheta _0} < \pi$. When our estimates for the principle eigenvalue are compared to the results of [4, 8], we find that our upper and lower bounds are sharper.
- Frank E. Baginski, Ordering the zeroes of Legendre functions $P^m_\nu (z_0)$ when considered as a function of $\nu $, J. Math. Anal. Appl. 147 (1990), no. 1, 296โ308. MR 1044702, DOI https://doi.org/10.1016/0022-247X%2890%2990400-A
- Frank E. Baginski, Comparison theorems for the $\nu $-zeroes of Legendre functions $P^m_\nu (z_0)$ when $-1<z_0<1$, Proc. Amer. Math. Soc. 111 (1991), no. 2, 395โ402. MR 1043402, DOI https://doi.org/10.1090/S0002-9939-1991-1043402-X
- Frank E. Baginski, The buckling of elastic spherical caps, J. Elasticity 25 (1991), no. 2, 159โ192. MR 1111366, DOI https://doi.org/10.1007/BF00042464
- G. Del Grosso, A. Gerardi, and F. Marchetti, A diffusion model for patch formation on cellular surfaces, Appl. Math. Optim. 7 (1981), no. 2, 125โ135. MR 616503, DOI https://doi.org/10.1007/BF01442110
E. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Cambridge University Press, London, 1931
- Andrea Laforgia and Martin E. Muldoon, Some consequences of the Sturm comparison theorem, Amer. Math. Monthly 93 (1986), no. 2, 89โ94. MR 827581, DOI https://doi.org/10.2307/2322698
- Frithiof I. Niordson, Shell theory, North-Holland Series in Applied Mathematics and Mechanics, vol. 29, North-Holland Publishing Co., Amsterdam, 1985. MR 807152
- Mark A. Pinsky, The first eigenvalue of a spherical cap, Appl. Math. Optim. 7 (1981), no. 2, 137โ139. MR 616504, DOI https://doi.org/10.1007/BF01442111
- Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
- Gรกbor Szegล, Orthogonal polynomials, 3rd ed., American Mathematical Society, Providence, R.I., 1967. American Mathematical Society Colloquium Publications, Vol. 23. MR 0310533
G. Szegรถ, Inequalities for the zeroes of Legendre polynomials and related functions, Trans. Amer. Math. Soc. 39, 1โ17 (1936)
- H. F. Weinberger, A first course in partial differential equations with complex variables and transform methods, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1965. MR 0180739
F. Baginski, Ordering the zeroes of the Legendre functions $P_v^m\left ( {{z_0}} \right )$ when considered as a function of v, J. Math. Anal. Appl. 147 (1), 296โ308 (1990)
F. Baginski, Comparison theorems for the $\nu$-zeroes of Legendre function $P_v^m\left ( {{z_0}} \right )$ when $- 1 < {z_0} < \\ 1$, Proc. Amer. Math. Soc., to appear
F. Baginski, The buckling of elastic spherical caps, (preprint)
G. Del Grosso, A. Gerardi, and F. Marchetti, A diffusion model for patch formation on cellular surfaces, Appl. Math. Optim. 7, 125โ135 (1981)
E. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Cambridge University Press, London, 1931
A. Laforgia and M. E. Muldoon, Some consequences of the Sturm comparison theorem, Am. Math. Monthly 93, 89โ94 (1986)
F. I. Niordson, Shell Theory, Elsevier Science Publishers B. V., Amsterdam, 1985
M. Pinsky, The first eigenvalue of a spherical cap, Appl. Math. Optim 7, 137โ139 (1981)
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983
G. Szegรถ, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, New York, 1939
G. Szegรถ, Inequalities for the zeroes of Legendre polynomials and related functions, Trans. Amer. Math. Soc. 39, 1โ17 (1936)
H. F. Weinberger, A First Course in Partial Differential Equations with Complex Variables and Transform Methods, Wiley, New York, 1965
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
35P15,
35J05,
73H05,
73K15
Retrieve articles in all journals
with MSC:
35P15,
35J05,
73H05,
73K15
Additional Information
Article copyright:
© Copyright 1990
American Mathematical Society
