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 \(r>1\).