Remarks on an overdetermined boundary value problem

We modify and extend proofs of Serrin’s symmetry result for overdetermined boundary value problems from the Laplace-operator to a general quasilinear operator and remove a strong ellipticity assumption in Philippin (Maximum principles and eigenvalue problems in partial differential equations (Knoxville, TN, 1987), Longman Sci. Tech., Pitman Res. Notes Math. Ser., Harlow, 175, pp. 34–48, 1988) and a growth assumption in Garofalo and Lewis (A symmetry result related to some overdetermined boundary value problems, Am. J. Math. 111, 9–33, 1989) on the diffusion coefficient A, as well as a starshapedness assumption on Ω in Fragalà et al. (Overdetermined boundary value problems with possibly degenerate ellipticity: a geometric approach. Math. Zeitschr. 254, 117–132, 2006).