arXiv 1802.017962018-02-08math.AP

具有二次梯度增长的半线性椭圆型方程组的正则性

The regularity of a semilinear elliptic system with quadratic growth of gradient

Weiyong He, Ruiqi Jiang

备注：: 15 pages

摘要：

本文研究了具有临界非线性项的半线性椭圆方程组\begin{equation}\label{sys01} Δu=Q(x, u, \nabla u), \end{equation}，对于$u: \mathbb{R}^n\rightarrow \mathbb{R}^K$，$Q$在$\nabla u$中具有二次增长.我们的工作是由调和映射和双调和映射的椭圆系统激发的。当$n=2$，这样的系统一般不具有光滑的正则性$W^{1, 2}$弱解，由一个著名的例子J. Frehse.调和映射的经典结果，F. Hélein（for $n=2$）和F. Béthuel（对于$n\geq 3$），证明了调和映射的$W^{1, n}$弱解总是光滑的.将Béthuel的结果推广到上述一般系统，证明了上述系统的$W^{1, n}$弱解对于$n\geq 3$是光滑的.对于扩张双调和映射的具有临界非线性项的四阶半线性椭圆方程组，我们证明了一个类似的结果：对于$n\geq 5$，该方程组的$W^{2, n/2}$弱解总是光滑的.我们还构造了各种例子，这些例子表明我们的正则性结果在各种意义下都是最优的。

英文摘要：

In this paper, we study semilinear elliptic systems with critical nonlinearity of the form \begin{equation}\label{sys01} Δu=Q(x, u, \nabla u), \end{equation} for $u: \mathbb{R}^n\rightarrow \mathbb{R}^K$, $Q$ has quadratic growth in $\nabla u$. Our work is motivated by elliptic systems for harmonic map and biharmonic map. When $n=2$, such a system does not have smooth regularity in general for $W^{1, 2}$ weak solutions, by a well-known example of J. Frehse. Classical results of harmonic map, proved by F. Hélein (for $n=2$) and F. Béthuel (for $n\geq 3$), assert that a $W^{1, n}$ weak solution of harmonic map is always smooth. We extend Béthuel's result to above general system, that a $W^{1, n}$ weak solution of above system is smooth for $n\geq 3$. For a fourth order semilinear elliptic system with critical nonlinearity which extends biharmonic map, we prove a similar result, that a $W^{2, n/2}$ weak solution of such system is always smooth, for $n\geq 5$. We also construct various examples, and these examples show that our regularity results are optimal in various sense.

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摘要：

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