$L^p$ Teichmüller理论：临界点的存在性和正则性
The $L^p$ Teichmüller theory: Existence and regularity of critical points
摘要：
我们研究极小的$p$ -共形能量泛函，\[ \mathsf{E}_p(f):=\int_\ID \IK^p(z,f)\,dz,\quad f|_\IS=f_0|_\IS, \]定义的自映射$f:\ID\to\ID$有限失真和规定的边界值$f_0$。这里\[ \IK(z,f) = \frac{\|Df(z)\|^2}{J(z,f)} = \frac{1+|μ_f(z)|^2}{1-|μ_f(z)|^2}\]是逐点失真泛函，$μ_f(z)$是$f$的Beltrami系数。我们表明，准对称边界数据的限制制度$p\to\infty$恢复经典Teichmüller理论的极值拟共形映射（部分结果Ahlfors），$p\to1$恢复调和映射理论。$\mathsf{E}_p$的临界点总是满足内变分分布方程\[ 2p\int_\ID \IK^p\;\frac{\overline{μ_f}}{1+|μ_f|^2}φ_\zbar \; dz=\int_\ID \IK^p \; φ_z\; dz,\quad\forallφ\in C_0^\infty(\ID ). \].我们建立了极小元在{ \em a priori}正则类$W^{1,\frac{2p}{p+1}}(\ID)$中的存在性，并证明了这些极小元有一个伪逆-$\ID$的连续$W^{1,2}(\ID)$满射，几乎处处有$(h\circ f)(z)=z$.然后给出了分布方程解的$C^{\infty}(\ID)$光滑性的充分条件。例如，对于任何$r>p+1$，$\IK(z,f)\in L^r_{loc}(\ID)$都足以暗示分布方程的解是局部同态。进一步的$\IK(w,h)\in L^1(\ID)$将意味着$h$是同胚，并且这些结果一起产生同胚极小元。我们表明，这种更高的正则性假设是必要的临界点的内部变分方程。
英文摘要：
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.
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