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arXiv 0712.3580math.APmath.CV
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一类微分不等式解的梯度的先验估计与拟共形映射

A priori estimate of gradient of a solution to certain differential inequality and quasiconformal mappings

David Kalaj

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24 pages
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摘要：

我们将证明{ \it Poisson微分不等式} $|Δu(x)|\le a|\nabla u(x)|^2+b$，$x\in B^{n}$，其中$a,b<\infty$和$u|_{S^{n-1}}\in C^{1,α}(S^{n-1}, \Bbb R^m)$解的梯度的一个全局估计。如果$m=1$和$a\le (n+1)/(|u|_\infty4n\sqrt n)$，则$|\nabla u| $是先验有界的。这推广了E.海因茨（\cite{EH}）和伯恩斯坦（\cite{BS}）的飞机。这些结果的应用产生的定理，这是本文的主要结果：一个拟共形映射的单位球到一个域的$C^2$光滑边界，满足泊松微分不等式，是Lipschitz连续的。这将作者Mateljević和Pavlović的一些结果从复平面推广到空间。

英文摘要：

We will prove a global estimate for the gradient of the solution to the {\it Poisson differential inequality} $|Δu(x)|\le a|\nabla u(x)|^2+b$, $x\in B^{n}$, where $a,b<\infty$ and $u|_{S^{n-1}}\in C^{1,α}(S^{n-1}, \Bbb R^m)$. If $m=1$ and $a\le (n+1)/(|u|_\infty4n\sqrt n)$, then $|\nabla u| $ is a priori bounded. This generalizes some similar results due to E. Heinz (\cite{EH}) and Bernstein (\cite{BS}) for the plane. An application of these results yields the theorem, which is the main result of the paper: A quasiconformal mapping of the unit ball onto a domain with $C^2$ smooth boundary, satisfying the Poisson differential inequality, is Lipschitz continuous. This extends some results of the author, Mateljević and Pavlović from the complex plane to the space.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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