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 $f:\mathbb{D}\stackrel{onto}{?}\mathbb{D}$, $f\in W_{\mathrm{loc}}^{1,1}(\mathbb{D};\mathbb{D})$, where $\mathbb{D}?{\mathbb{R}}^2$ is the unit disk, such that there exists a constant ${\mathbb{K}}_p\ge 0$ for which the following distortion inequality

$|Df(x){|}^p\le {\mathbb{K}}_p|J_f(x){|}^{p?1}\text{a.e.}x\in \mathbb{D}$

is 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 ${\mathbb{K}}_p={\mathbb{K}}_{p,f}(x)$ not necessarily constant, we identify the homeomorphism f whose p-distortion function ${\mathbb{K}}_{p,f}(x)$ is minimal in $L^1$ norm.