International Journal of Mathematics and Mathematical Sciences
Volume 13 (1990), Issue 3, Pages 591-598
doi:10.1155/S0161171290000825
Hearing the shape of membranes: further results
Department of Mathematics, Zagazig University, Faculty of Science, Zagazig, Egypt
Received 18 January 1989; Revised 5 May 1989
Copyright © 1990 E. M. E. Zayed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The spectral function θ(t)=∑m=1∞exp(−tλm), t>0 where {λm}m=1∞ are the eigenvalues of the Laplacian in Rn, n=2 or 3, is studied for a variety of domains. Particular attention is given to circular and spherical domains with the impedance boundary conditions ∂u∂r+γju=0 on Γj (or Sj), j=1,…,J where Γj and Sj, j=1,…,J are parts of the boundaries of these domains respectively, while γj, j=1,…,J are positive constants.