We provide a characterization of rotationally symmetric solutions to the Serrin problem on ring-shaped domains in $R^n$ ($n ≥ 3$). Our approach is based on a comparison-geometry argument. By exploiting a suitable conformal reformulation of our problem, we obtain sharp gradient estimates, which play a central role in establishing our main rigidity theorem. The comparison method can be effectively extended to deal with quasi- linear 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.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006