Uniqueness of diffeomorphic minimizers of $$L^p$$-mean distortion

Abstract We study the $$L^p$$ L p -mean distortion functionals, $$\begin{aligned}{{\mathcal {E}}}_p[f] = \int _{\mathbb {Y}} K^p_f(z) \; dz, \end{aligned}$$ E p [ f ] = ∫ Y K f p ( z ) d z , for Sobolev homeomorphisms "Equation missing" where $$\mathbb {X}$$ X and $${\mathbb {Y}}$$ Y are bounded simply connected domains, and f coincides with a given boundary map $$f_0 :\partial {\mathbb {Y}} \rightarrow \partial {\mathbb {X}}$$ f 0 : ∂ Y → ∂ X . Here, $$K_f(z)$$ K f ( z ) denotes the pointwise distortion function of f . It is conjectured that for every $$1< p < \infty $$ 1 < p < ∞ , the functional $$\mathcal {E}_p$$ 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 $$f:\mathbb {Y}\xrightarrow {\textrm{onto}}\mathbb {X}$$ f : Y → onto X with $$f|_{\partial \mathbb {Y}}=f_0$$ f | ∂ Y = f 0 and $$L^p$$ L p