An overdetermined problem in nonlinear elliptic equations of second order (Q1309440)

The authors study the equation \(\Delta_ p u= 0\) in \(\Omega= \Omega_ 0\backslash \Omega_ 1\), where \(\Omega_ i\) are convex and \(u= i\) on \(\partial\Omega_ i\). If in addition \(x\cdot \nabla u= d_ i=\text{const.}\) is prescribed on \(\partial \Omega_ i\), then \(\Omega_ i\) must be concentric balls. The proofs are in the spirit of \textit{H. Weinberger} [Arch. Ration. Mech. Anal. 43, 319-320 (1971; Zbl 0222.31008)] or \textit{N. Garofalo} and \textit{J. L. Lewis} [Am. J. Math. 111, No. 1, 9-33 (1989; Zbl 0681.35016)].