Hearing the shape of membranes: further results - E. M. E. Zayed (International Journal of Mathematics and Mathematical Sciences)
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.