International Journal of Mathematics and Mathematical Sciences
Volume 13 (1990), Issue 3, Pages 591-598

Hearing the shape of membranes: further results

E. M. E. Zayed

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.


The spectral function θ(t)=m=1exp(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 ur+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.