Deformations of balls in Schiffer's conjecture for Riemannian symmetric spaces (Q1817269)

Let us recall that the Schiffer problem is to decide whether in \(\mathbb{R}^n\) the balls are the only simply connected bounded domains for which the overdetermined Dirichlet boundary value problem \(\Delta u+\lambda u=0\) in \(\Omega\), \(u=1\) and \({\partial u\over\partial n}=0\) on \(\partial\Omega\), has a solution. The first author [J. Math. Anal. Appl. 178, No. 1, 269-279 (1993; Zbl 0793.35063)] in \(\mathbb{R}^2\) and \textit{T. Kobayashi} [Commun. Anal. Geom. 1, No. 4, 515-541 (1993; Zbl 0847.35090)] in \(\mathbb{R}^n\) proved (using different methods) that the balls are isolated among classes of domains \(\Omega\) that have such overdetermined eigenvalues. In the present paper, the authors give a beautiful proof, exploiting in a very clever and non-elementary way some elementary ideas from the reviewer [J. Anal. Math. 37, 128-144 (1980; Zbl 0449.35024)] that this result is still true in any non-compact symmetric space of rank 1. (The same proof also works in \(\mathbb{R}^n\), it is just simpler).