Higher regularity and uniqueness for inner variational equations

Abstract

We study local minima of the p-conformal energy functionals,

$$\begin{aligned} {\mathsf {E}}_{{{\mathcal {A}}}}^*(h){:}{=}\int _{\mathbb D}{{{\mathcal {A}}}}({\mathbb K}(w,h)) \;J(w,h) \; dw,\quad h|_{\mathbb S}=h_0|_{\mathbb S}, \end{aligned}$$

defined for self mappings $h:{\mathbb D}\rightarrow {\mathbb D}$ with finite distortion of the unit disk with prescribed boundary values $h_0$. Here ${\mathbb K}(w,h) = \frac{\Vert Dh(w)\Vert ^2}{J(w,h)}$ is the pointwise distortion functional, and ${{{\mathcal {A}}}}:[1,\infty )\rightarrow [1,\infty )$ 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 ${\mathsf {E}}_{{{\mathcal {A}}}}^*$ satisfy the Ahlfors-Hopf inner-variational equation

$$\begin{aligned} {{{\mathcal {A}}}}'({\mathbb K}(w,h)) h_w \overline{h_{{\overline{w}}}} = \Phi \end{aligned}$$

where $\Phi$ 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 ${\mathbb K}(w,h)\in L^1({\mathbb D})\cap L^r_{loc}({\mathbb D})$ for some $r>1$.