论文标题
$ l^p $teichmüller理论：关键点的存在和规律性
The $L^p$ Teichmüller theory: Existence and regularity of critical points
论文作者
论文摘要
我们研究了$ p $ - 符合能量功能的最小化器，\ [\ mathsf {e} _p（f）：= \ int_ \ id \ id \ ik \ ik^p（z，f）\，dz，dz，\ quad f | _ \ is = f _0 | _ \ iS = f _0 | _ \ is，in Selferited $ f ins $ f ins $ f ins $ f：边界值$ f_0 $。 Here \[ \IK(z,f) = \frac{\|Df(z)\|^2}{J(z,f)} = \frac{1+|μ_f(z)|^2}{1-|μ_f(z)|^2}\] is the pointwise distortion functional and $μ_f(z)$ is the Beltrami coefficient of $ f $。我们表明，对于准对称边界数据，限制机制$ p \ to \ infty $恢复了极端准符号映射的经典Teichmüller理论（在AHLFORS的一部分中），对于$ p \ to1 $，恢复了谐波映射理论。 $ \ mathsf {e} _p $的关键点总是满足内部变化分布方程\ [2p \ int_ \ id \ ik \ ik^p \ \; \ frac {\ overline {μ_f}} {1+ dz = \ int_ \ id \ ik^p \; φ_z\; dz，\ quad \forallφ\ in C_0^\ infty（\ id）。 \]我们在{\ em a a a Priori}的规律性类中建立了最小化的存在，$ w^{1，\ frac {2p} {p+1}}}}（\ id）$ $，并显示这些最小值的人具有伪级 - $ w^{1,2}（1,2}（1,2}（\ id）$ sudddipt $ s $（几乎到处都是。然后，我们提供足够的条件，以确保解决方案对分配方程的平滑度$ c^{\ ifty}（\ id）$。例如，对于任何$ r> p+1 $，$ \ ik（z，f）\在l^r_ {loc}（\ id）$中，足以暗示分布方程的解决方案是局部差异性。进一步的$ \ ik（w，h）\在l^1（\ id）$中将暗示$ h $是同构的，并且这些结果共同产生了差异最小的。对于内部变异方程的临界点，我们显示出如此较高的规律性假设。
We study minimisers of the $p$-conformal energy functionals, \[ \mathsf{E}_p(f):=\int_\ID \IK^p(z,f)\,dz,\quad f|_\IS=f_0|_\IS, \] defined for self mappings $f:\ID\to\ID$ with finite distortion and prescribed boundary values $f_0$. Here \[ \IK(z,f) = \frac{\|Df(z)\|^2}{J(z,f)} = \frac{1+|μ_f(z)|^2}{1-|μ_f(z)|^2}\] is the pointwise distortion functional and $μ_f(z)$ is the Beltrami coefficient of $f$. We show that for quasisymmetric boundary data the limiting regimes $p\to\infty$ recover the classical Teichmüller theory of extremal quasiconformal mappings (in part a result of Ahlfors), and for $p\to1$ recovers the harmonic mapping theory. Critical points of $\mathsf{E}_p$ always satisfy the inner-variational distributional equation \[ 2p\int_\ID \IK^p\;\frac{\overline{μ_f}}{1+|μ_f|^2}φ_\zbar \; dz=\int_\ID \IK^p \; φ_z\; dz,\quad\forallφ\in C_0^\infty(\ID ). \] We establish the existence of minimisers in the {\em a priori} regularity class $W^{1,\frac{2p}{p+1}}(\ID)$ and show these minimisers have a pseudo-inverse - a continuous $W^{1,2}(\ID)$ surjection of $\ID$ with $(h\circ f)(z)=z$ almost everywhere. We then give a sufficient condition to ensure $C^{\infty}(\ID)$ smoothness of solutions to the distributional equation. For instance $\IK(z,f)\in L^r_{loc}(\ID)$ for any $r>p+1$ is enough to imply the solutions to the distributional equation are local diffeomorphisms. Further $\IK(w,h)\in L^1(\ID)$ will imply $h$ is a homeomorphism, and together these results yield a diffeomorphic minimiser. We show such higher regularity assumptions to be necessary for critical points of the inner variational equation.
