内变分方程的高阶正则性和唯一性
Higher regularity and uniqueness for inner variational equations
摘要：
本文研究了具有给定边界值$h_0$的单位圆盘有限变形的自映射$h:\ID\to\ID$的$p$ -共形能量泛函\[ \mathsf{E}_{\cal A}^\ast(h):=\int_\ID {\cal A}(\IK(w,h)) \;J(w,h) \; dw,\quad h|_\IS=h_0|_\IS, \]的局部极小。这里$\IK(w,h) = \frac{\|Dh(w)\|^2}{J(w,h)} $是逐点失真泛函，并且对于某些$p\geq 1$，${\cal A}:[1,\infty)\to [1,\infty)$是凸的并且随着${\cal A}(t)\approx t^p$而增加，具有附加的次要技术条件。注意${\cal A}(t)=t$是狄利克雷能量泛函。$\mathsf{E}_{\cal A}^\ast$的临界点满足Ahlfors-Hopf内变分方程\[ {\cal A}'(\IK(w,h)) h_w \overline{h_\wbar} = Φ\]，其中$Φ$是全纯函数。Iwaniec，Kovalev和Onninen建立了临界点的Lipschitz正则性。在这里，我们给出了一个充分条件，以确保一个局部极小值是这个方程的一个非纯解，并且它是唯一的。这个条件是任何局部拟共形临界点都必须满足的，并且基本上是对某些$r>1$的$\IK(w,h)\in L^1(\ID)\cap L^r_{loc}(\ID)$假设。
英文摘要：
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 \[ {\cal A}'(\IK(w,h)) h_w \overline{h_\wbar} = Φ\] where $Φ$ 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 $r>1$.
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