We characterize rotationally symmetric solutions to the Serrin problem on ring-shaped domains in $\mathbb R^n$ (n ≥ 3). Our proof relies on a comparison geometry argument. In particular, by taking advantage of a suitable conformal reformulation of our problem, we derive key gradient estimates, which prove to be crucial to obtain our main rigidity result. The comparison technique can be fruitfully extended to handle quasilinear operators. This will be discussed in the second part of the talk, where we focus on the case of the p-Laplacian torsion problem. This talk is based on joint works with V. Agostiniani, S. Borghini, L. Mazzieri and A. Pinamonti. This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.