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

E.M.E. Zayed

Department of Mathematics, Zagazig University, Faculty of Science, Zagazig, Egypt

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.