Isometries between Sobolev spaces



Let Ω1 and Ω2 be bounded, connected open sets in equation image with continuous boundary, and let p > 2. We show that every positive linear isometry T from W1, p1) to W1, p2) that satisfies equation image corresponds to a rigid motion of the space, i.e., Tu = u○ξ for an isometry ξ of equation image, and more precisely ξ(Ω2) = Ω1. We also prove similar results for less regular domains, and we obtain partial results also for p = 2.