Heat equation for an arbitrary doubly-connected region in \(R^ 2\) with mixed boundary conditions (Q916917)

Let \(\Omega\) be an arbitrary doubly-connected region in \(R^ 2\) surrounded internally by a simply connected bounded domain \(\Omega_ 1\) with a smooth boundary \(\partial \Omega_ 1\) and externally by a simply connected bounded domain \(\Omega_ 2\) with a smooth boundary \(\partial \Omega_ 2\). Suppose that the eigenvalues \(0<\lambda_ 1\leq \lambda_ 2\leq...\leq \lambda_ j\leq...\to \infty\) as \(j\to \infty\) are given exactly for the equation \((\Delta +\lambda)u=0\) in \(\Omega\) together with one of the following mixed boundary conditions \(\partial u/\partial n_ 1=0\) on \(\partial \Omega_ 1\), \(u=0\) on \(\partial \Omega_ 2\), \(u=0\) on \(\partial \Omega_ 1\), \(\partial u/\partial n_ 2=0\) on \(\partial \Omega_ 2,\) where \(\partial /\partial n_ 1\) and \(\partial /\partial n_ 2\) denote differentiations along the inward pointing normals to \(\partial \Omega_ 1\) and \(\partial \Omega_ 2\) respectively, while \(\Delta\) is the Laplace operator in \(R^ 2\). Determine the geometry of \(\Omega\) from the expansion of \(\theta (t)=\sum^{\infty}_{j=1}\exp (-t\lambda_ j)\) as \(t\to 0\).