Upper and lower bounds for eigenvalues of the Laplacian on a spherical cap
Frank E. Baginski · Quarterly of Applied Mathematics · 1990
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 ϑ 0 2{\vartheta _0} is less than π \pi . While previous work of this nature seems to focus on the principle eigenvalue, our results apply to any eigenvalue when 0 > ϑ 0 > π / 2 0 > {\vartheta _0} > \pi /2 . In addition, some of our results also apply to spherical caps for which 0 > ϑ 0 > π 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.