Eigenvalues of the Laplacian for the third boundary value problem: An extension to higher dimensions
Let \(\Omega:=\{(r,\theta,\phi):\) \(a\leq r\leq b\), \(0\leq \theta \leq \pi\), \(0\leq \phi \leq 2\pi \}\) be a spherical shell in \({\mathbb{R}}^ 3\). Suppose the spectrum of eigenvalues \(0<\lambda_ 1\leq \lambda_ 2\leq...\leq \lambda_ j\leq...\to \infty\) is known exactly for the impedance problem \[ (\Delta +\lambda)u=0\quad in\quad \Omega,\quad | (\partial u/\partial r+\gamma_ 1u)_{r=a}=0,\quad | \partial u/\partial r+\gamma_ 2u|_{r=b}=0, \] where \(\gamma_ 1\) and \(\gamma_ 2\) are positive constants. The basic problem is that of determining the geometry of the spherical shell \(\Omega\) and the impedances \(\gamma_ 0\) and \(\gamma_ 1\). This problem is attacked by a careful analysis of the asymptotic behaviour of the trace function \(\theta (f):=\sum^{\infty}_{j=1}e^{-\lambda_ jt}\) for small positive t.
scientific article; zbMATH DE number 4063690
scientific article