Higher regularity and uniqueness for inner variational equations (2007.15150v1)

Abstract: We study local minima of the $p$-conformal energy functionals, [ \mathsf{E}{\cal A}^\ast(h):=\int\ID {\cal A}(\IK(w,h)) \;J(w,h) \; dw,\quad h|\IS=h_0|\IS, ] defined for self mappings $h:\ID\to\ID$ with finite distortion of the unit disk with prescribed boundary values $h_0$. Here $\IK(w,h) = \frac{|Dh(w)|^2}{J(w,h)} $ is the pointwise distortion functional, and ${\cal A}:[1,\infty)\to [1,\infty)$ is convex and increasing with ${\cal A}(t)\approx t^p$ for some $p\geq 1$, with additional minor technical conditions. Note ${\cal A}(t)=t$ is the Dirichlet energy functional. Critical points of $\mathsf{E}{\cal A}^\ast$ satisfy the Ahlfors-Hopf inner-variational equation [ {\cal A}'(\IK(w,h)) h_w \overline{h\wbar} = \Phi ] 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 $\IK(w,h)\in L^1(\ID)\cap L^r_{loc}(\ID)$ for some $r>1$.