二阶椭圆型方程组的$p$ -椭圆性条件及其在Lamé和齐次化问题中的应用
The $p$-ellipticity condition for second order elliptic systems and applications to the Lamé and homogenisation problems
摘要：
$p$ -椭圆性的概念最近在提高我们对标量复值椭圆偏微分方程边值问题可解性问题的理解方面发挥了重要作用。特别地，$p$ -椭圆性的存在确保了此类方程的解的更高的正则性。本文将$p$ -椭圆性的概念推广到二阶椭圆方程组。回想一下，对于系统来说，没有单一的椭圆性概念，而是出现了一个更复杂的图像，其中包含不同强度的椭圆性条件，例如勒让德，勒让德-阿达玛和积分条件。当考虑$p$ -椭圆率时，出现类似的画面。本文定义了三个新的$p$ -椭圆度概念，建立了它们之间的关系，并证明了它们在求解边值问题中的重要作用.这些重要的作用证明了建立外推结果的可解性的$L^p$狄利克雷问题的椭圆系统，然后在两个不同的情况下应用这一结果：一个为Lamé系统的线性弹性和另一个在均匀化理论。
英文摘要：
The notion of $p$-ellipticity has recently played a significant role in improving our understanding of issues of solvability of boundary value problems for scalar complex valued elliptic PDEs. In particular, the presence of $p$-ellipticity ensures higher regularity of solutions of such equations. In this work we extend the notion of $p$-ellipticity to second order elliptic systems. Recall that for systems, there is no single notion of ellipticity, rather a more complicated picture emerges with ellipticity conditions of varying strength such as the Legendre, Legendre-Hadamard and integral conditions. A similar picture emerges when $p$-ellipticity is considered. In this paper, we define three new notions of $p$-ellipticity, establish relationships between them and show that each of them does play an important role in solving boundary value problems. These important roles are demonstrated by establishing extrapolation results for solvability of the $L^p$ Dirichlet problem for elliptic systems, followed by applications of this result in two different scenarios: one for the Lamé system of linear elasticity and another in the theory of homogenization.
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