A duality theorem for an overdetermined eigenvalue problem (Q1898029)

The overdetermined eigenvalue problem for the Laplacian \[ \Delta u + \lambda u = 0 \quad \text{in } \Omega, \qquad u = b, \;{\partial u \over \partial n} = c \quad \text{on } \partial \Omega \tag{P} \] is studied in a bounded domain \(\Omega\), for \(\lambda\), \(b > 0\). Call \(w\) a Helmholtz function of wave number \(\lambda\) in \(\Omega\), if \(w\) satisfies the equation \(\Delta w + \lambda w = 0\). The main result of this paper is: There exists a solution \(u \in C^2 (\Omega) \cap C^1 (\overline \Omega)\) of (P) if and only if \(\int_\Omega wdx = d \int_{\partial \Omega} wd \sigma\) for all Helmholtz functions \(w\) of wave number \(\lambda\) \((d\) is an arbitrary fixed constant).