论文标题
二阶椭圆系统和针对Lamé和同质化问题的$ P $ - 纤毛条件
The $p$-ellipticity condition for second order elliptic systems and applications to the Lamé and homogenisation problems
论文作者
论文摘要
$ p $ - eliptipticity的概念最近在提高我们对标量复合物有价值椭圆形PDE的边界价值问题的理解方面发挥了重要作用。特别是，存在$ p $ eLlipticity可以确保更高的此类方程解决方案的规律性。 在这项工作中，我们将$ p $ - ellipticity的概念扩展到二阶椭圆系统。回想一下，对于系统，没有一个单一的椭圆形概念，而是出现了更复杂的图片，其椭圆度条件有不同的强度，例如Legendre，Legendre-Hadamard和整体条件。当考虑$ p $ ellipticity时，出现了类似的图片。在本文中，我们定义了三个新的概念$ p $ - ellipticity，并在它们之间建立关系，并表明每个概念在解决边界价值问题中确实发挥了重要作用。 通过建立推外结果来证明这些重要的作用，以解决椭圆系统的$ l^p $ dirichlet问题，然后在两种不同的情况下应用这种结果：一种用于线性弹性的唇线系统，另一个用于同质化理论。
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.
