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

Abstract

We study the $L^p$-mean distortion functionals, \[{\cal E}_p[f] = \int_\mathbb Y K^p_f(z) \; dz, \] for Sobolev homeomorphisms $f:\overline{\mathbb{Y}}\stackrel{\mathrm{onto}}{\to }\overline{\mathbb{X}}$ where $\mathbb{X}$ and $\mathbb{Y}$ are bounded simply connected Lipschitz domains, and $f$ coincides with a given boundary map $f_0:\partial \mathbb{Y}\to \partial \mathbb{X}$. 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.