We study the \(L^p\) -mean distortion functionals, \(\begin{aligned}{{\mathcal {E}}}_p[f] = \int _{\mathbb {Y}} K^p_f(z) \; dz, \end{aligned}\) for Sobolev homeomorphisms where \(\mathbb {X}\) and \({\mathbb {Y}}\) are bounded simply connected domains, and f coincides with a given boundary map \(f_0 :\partial {\mathbb {Y}} \rightarrow \partial {\mathbb {X}}\) . Here, \(K_f(z)\) denotes the pointwise distortion function of f. It is conjectured that for every \(1< p < \infty \) , the functional \(\mathcal {E}_p\) admits a minimizer that is a diffeomorphism. We prove that if such a diffeomorphic minimizer exists, then it is unique within the class of diffeomorphisms \(f:\mathbb {Y}\xrightarrow {\textrm{onto}}\mathbb {X}\) with \(f|_{\partial \mathbb {Y}}=f_0\) and \(L^p\) -mean distortion that minimizes \(\mathcal {E}_p\) .
