Higher regularity and uniqueness for inner variational equations

We study local minima of the $p$-conformal energy functionals, $${\mathsf{E}}_{\mathcal{A}}^{\ast }(h):={\int }_{\text{ID}}\mathcal{A}(\text{IK}(w,h))J(w,h)dw,h{\mathrm{\mid }}_{\text{IS}}=h_0{\mathrm{\mid }}_{\text{IS}},$$ defined for self mappings $h:\text{ID}\to \text{ID}$ with finite distortion of the unit disk with prescribed boundary values $h_0$. Here $\text{IK}(w,h)=\frac{\mathrm{\parallel }Dh(w){\mathrm{\parallel }}^2}{J(w,h)}$ is the pointwise distortion functional, and $\mathcal{A}:[1,\mathrm{\infty })\to [1,\mathrm{\infty })$ is convex and increasing with $\mathcal{A}(t)\approx t^p$ for some $p\ge 1$, with additional minor technical conditions. Note $\mathcal{A}(t)=t$ is the Dirichlet energy functional. Critical points of ${\mathsf{E}}_{\mathcal{A}}^{\ast }$ satisfy the Ahlfors-Hopf inner-variational equation $${\mathcal{A}}^{\mathrm{\prime }}(\text{IK}(w,h))h_w\overline{h_{\text{wbar}}}=\mathrm{\Phi }$$ where $\mathrm{\Phi }$ is a holomorphic function. Iwaniec, Kovalev and Onninen established the Lipschitz regularity of critical points. Here we give a sufficient condition to ensure that a local minimum is a diffeomorphic solution to this equation, and that it is unique. This condition is necessarily satisfied by any locally quasiconformal critical point, and is basically the assumption $\text{IK}(w,h)\in L^1(\text{ID})\cap L_{loc}^r(\text{ID})$ for some $r>1$.