Bi-Sobolev mappings and Kp-distortions in the plane

In this paper we introduce the class of the inner p-quasiconformal mappings, that are homeomorphisms f:D⟶ontoD, f∈Wloc1,1(D;D), where D⊂R2 is the unit disk, such that there exists a constant Kp≥0 for which the following distortion inequality|Df(x)|p≤Kp|Jf(x)|p−1a.e.x∈Dis satisfied. The study of such mappings is motivated by the fact that their inverses satisfy the distortion inequality introduced in [11]. Here we give a characterization of them so that their components solve a suitable uniformly elliptic p-harmonic system. Moreover, for mappings satisfying the previous distortion inequality with Kp=Kp,f(x) not necessarily constant, we identify the homeomorphism f whose p-distortion function Kp,f(x) is minimal in L1 norm.