论文标题
内部变异方程的较高规律性和唯一性
Higher regularity and uniqueness for inner variational equations
论文作者
论文摘要
我们研究$ p $ - 符合能量功能的本地最小值，\ [\ mathsf {e} _ {\ cal a}^\ ast（h）：= \ int_ \ id {\ cal a}（\ cal a}（\ ik（w，h）） dw，\ Quad H | _ \ is = h_0 | _ \ is，\]为自映射$ h：\ id \ to \ id $定义，具有有限的单位磁盘的有限失真，带有规定的边界值$ h_0 $。这里$ \ ik（w，h）= \ frac {\ | dh（w）\ |^2} {j（w，w，h）} $是点的变形函数，$ {\ cal a}：[1，\ infty）\ to [1，\ infty）$ convex and convex and convex and convex and p p p $ can}（ $ p \ geq 1 $，具有其他较小的技术条件。注意$ {\ cal a}（t）= t $是Dirichlet Energy功能。 $ \ Mathsf {e} _ {\ cal a}^\ ast $的关键点满足ahlfors-hopf inner-variational方程\ [{\ cal a}'（\ ik（w，w，h））h_w \ edline { Iwaniec，Kovalev和Onninen确立了Lipschitz的临界点。在这里，我们提供足够的条件，以确保局部最小值是该方程式的差异解决方案，并且它是唯一的。任何本地列出的临界点都必须满足此条件，并且基本上是l^1（\ id）\ cap l^r_ {loc}（\ id）的假设$ \ ik（w，h）\对于某些$ r> 1 $。
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 $r>1$.
